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10^Infinity * 10^-Infinity = 1

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Dan Perez

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Nov 23, 2002, 7:47:59 AM11/23/02
to
I made the mistake of inserting "+" instead of "*" in a previous
equation on another thread.

So;

10^Infinity * 10^-Infinity = 1

or

10^Infinity/10^Infinity = 1

or

Infinity * 0 = 1

Yours truly,


Dan Perez

John Zinni

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Nov 23, 2002, 9:08:14 AM11/23/02
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"Dan Perez" <fasterthanlig...@nospamyahoo.com> wrote in message
news:3ddf7960$0$17649$2c3e...@news.voyager.net...

This is utter nonsense.

Infinity is NOT a number!!! You cannot toss it around as such.

Infinity is a PROCESS and requires some special care or you end up with
gibberish like "Infinity * 0 = 1"

--
Cheers
John Zinni

Dirk Van de moortel

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Nov 23, 2002, 9:34:39 AM11/23/02
to

"John Zinni" <j_z...@sympatico.ca> wrote in message news:OZLD9.36334$hK2.1...@news20.bellglobal.com...

It is not even a process.
That's what we can say about infinities:

- The statement:
For all M>0, there is a delta>0, such that
the following implication holds for all x:
|x-a| < delta ==> f(x) > M
is abbreviated to:
limit{ x --> a; f(x) } = +infinity

- The statement:
For all epilon>0, there is a N>0, such that
the following implication holds for all x:
x > N ==> | f(x) - a | < epsilon
is abbreviated to:
limit{ x --> +infinity; f(x) } = a

- The statement:
For all M>0, there is a N>0, such that
the following implication holds for all x:
x > N ==> f(x) > M
is abbreviated to:
limit{ x --> +infinity; f(x) } = +infinity

And some analogous definitions for "-infinity".

No infinite numbers.
No processes.
Just abbreviations of statements.

Dirk Vdm

John Zinni

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Nov 23, 2002, 9:49:03 AM11/23/02
to
"Dirk Van de moortel" <dirkvand...@ThankS-NO-SperM.hotmail.com> wrote
in message news:3mMD9.15958$Ti2....@afrodite.telenet-ops.be...

Well said.

I guess the process I had in mind was taking a limit.
(its been a couple of decades since 1st year calculus :-)

--
Cheers
John Zinni

Dirk Van de moortel

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Nov 23, 2002, 10:08:59 AM11/23/02
to

"John Zinni" <j_z...@sympatico.ca> wrote in message news:1AMD9.36343$hK2.1...@news20.bellglobal.com...

[snip]

> I guess the process I had in mind was taking a limit.
> (its been a couple of decades since 1st year calculus :-)

One of the best years, don't you think? :-)

I made the remark because the term "process" reminded me of
my sons when they came to me in a slightly bewildered state,
with the question:
"how on earth can you *approach* infinity???"

See also the thread "The Infinitesimal"
http://groups.google.com/groups?&as_umsgid=qi2I8.96666$Ze.1...@afrodite.telenet-ops.be

Schol,
Dirk Vdm


John Zinni

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Nov 23, 2002, 10:35:22 AM11/23/02
to
"Dirk Van de moortel" <dirkvand...@ThankS-NO-SperM.hotmail.com> wrote
in message news:fSMD9.15992$Ti2....@afrodite.telenet-ops.be...

That's great!!! :-))
Were you able to set there minds at ease?

> One of the best years, don't you think? :-)

Certainly one of the most exciting.

I'd have to say my favourite years were near the end of undergrad, though,
when, what we think of in high school and 1st year as separate and distinct
branches of mathematics start to come together in subtle and intricate ways.
It was truly beautiful to experience.

--
Cheers
John Zinni

Titanpoint

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Nov 23, 2002, 11:03:13 AM11/23/02
to

Now that I know this, what do I do?

Dirk Van de moortel

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Nov 23, 2002, 11:08:59 AM11/23/02
to

"John Zinni" <j_z...@sympatico.ca> wrote in message news:IfND9.36360$hK2.1...@news20.bellglobal.com...

> "Dirk Van de moortel" <dirkvand...@ThankS-NO-SperM.hotmail.com> wrote
> in message news:fSMD9.15992$Ti2....@afrodite.telenet-ops.be...
> >
> > "John Zinni" <j_z...@sympatico.ca> wrote in message
> news:1AMD9.36343$hK2.1...@news20.bellglobal.com...
> >
> > [snip]
> >
> > > I guess the process I had in mind was taking a limit.
> > > (its been a couple of decades since 1st year calculus :-)
> >
> > One of the best years, don't you think? :-)
> >
> > I made the remark because the term "process" reminded me of
> > my sons when they came to me in a slightly bewildered state,
> > with the question:
> > "how on earth can you *approach* infinity???"
> >
> > See also the thread "The Infinitesimal"
> >
> http://groups.google.com/groups?&as_umsgid=qi2I8.96666$Ze.1...@afrodite.tel
> enet-ops.be
> >
> > Schol,
> > Dirk Vdm
>
> > I made the remark because the term "process" reminded me of
> > my sons when they came to me in a slightly bewildered state,
> > with the question:
> > "how on earth can you *approach* infinity???"
>
> That's great!!! :-))
> Were you able to set there minds at ease?

Sure, they loved the idea of the calculator challenge.

>
> > One of the best years, don't you think? :-)
>
> Certainly one of the most exciting.
>
> I'd have to say my favourite years were near the end of undergrad, though,
> when, what we think of in high school and 1st year as separate and distinct
> branches of mathematics start to come together in subtle and intricate ways.
> It was truly beautiful to experience.

Absolutely :-)
I wish I could relive them for real in stead of from memory.

Dirk Vdm


John Zinni

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Nov 23, 2002, 11:51:01 AM11/23/02
to
"Titanpoint" <titan...@nospammissthisout.myrealbox.com> wrote in message
news:pan.2002.11.23....@nospammissthisout.myrealbox.com...

From "Infinity * 0 = 1" we can "prove" just about anything we like. For
example:

Start with:

Infinity * 0 = 1

Multipy each side by 2:

2*(Infinity * 0) = 2*1

2*(Infinity * 0) = 2

Divide each side by (Infinity * 0):

2=2/(Infinity * 0)

2=2/Infinity * 2/0

2=2/0 * 2/Infinity

Now 2/0 = Infinity and 2/Infinity = 0, substituting in we have:

2 = Infinity*0

But we know that Infinity*0 = 1

So:

2 = 1

QED

--
Cheers
John Zinni


John Zinni

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Nov 23, 2002, 12:15:07 PM11/23/02
to
From "Infinity * 0 = 1" we can "prove" just about anything we like. For
example:

Start with:

Infinity * 0 = 1

Multipy each side by 2:

2*(Infinity * 0) = 2*1

2*(Infinity * 0) = 2

Divide each side by (Infinity * 0):

2=2/(Infinity * 0)

2=1/Infinity * 2/0

2=2/0 * 1/Infinity

Now 2/0 = Infinity and 1/Infinity = 0, substituting in we have:

2 = Infinity*0

But we know that Infinity*0 = 1

So:

2 = 1

QED

Jeesh, talk about rusty!!!

--
Cheers
John Zinni

Phil

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Nov 23, 2002, 6:16:47 PM11/23/02
to
How do you get from


> 10^Infinity/10^Infinity = 1

to

> Infinity * 0 = 1

If 1/infinity is *defined* as 0 -- and from certain points of view this
makes real sense -- then I agree with, but I still don't see how you made
this particular transition.

Phil


John Zinni

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Nov 23, 2002, 6:57:18 PM11/23/02
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"Phil" <tu...@jump.net> wrote in message news:3DE00C81...@jump.net...

Yep, he wasn't entirely clear, so allow me to restate this nonsense
"correctly".
(I'm not exactly sure why he is using 10^Infinity so I will just use
Infinity)

Start with

x/x = 1

x * 1/x = 1

Let x = Infinity, so we have

Infinity * 1/Infinity = 1

Now 1/Infinity = 0, substituting in we get

Infinity * 0 = 1

How can you argue with "logic" (??!?) like that.

--
Cheers
John Zinni


Phil

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Nov 23, 2002, 7:58:28 PM11/23/02
to
John Zinni wrote:

> "Phil" <tu...@jump.net> wrote in message news:3DE00C81...@jump.net...
> > How do you get from
> >
> >
> > > 10^Infinity/10^Infinity = 1
> >
> > to
> >
> > > Infinity * 0 = 1
> >
> > If 1/infinity is *defined* as 0 -- and from certain points of view this
> > makes real sense -- then I agree with, but I still don't see how you made
> > this particular transition.
> >
> > Phil
>
> Yep, he wasn't entirely clear, so allow me to restate this nonsense
> "correctly".
> (I'm not exactly sure why he is using 10^Infinity so I will just use
> Infinity)
>
> Start with
>
> x/x = 1
>
> x * 1/x = 1
>
> Let x = Infinity, so we have
>
> Infinity * 1/Infinity = 1
>
> Now 1/Infinity = 0, substituting in we get

Thanks, I thought that this might be the "hidden step" in the process, but also
I thought that maybe I was just overlooking something (which often happens to
me even with simple algebra problems).

> Infinity * 0 = 1
>
> How can you argue with "logic" (??!?) like that.

Okay, now I'm wondering what the point is. If 1/infinity is *defined* as 0 then
yes, infinity * 0 = 1. So?

Phil


Robert Kolker

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Nov 23, 2002, 9:10:50 PM11/23/02
to

Phil wrote:
>
>>Infinity * 0 = 1
>
>
> If 1/infinity is *defined* as 0 -- and from certain points of view this
> makes real sense -- then I agree with, but I still don't see how you made
> this particular transition.

That dawg don't hunt. Consider lim (n->00) (2/n) = 0.

Shall we also conclude that 2/infinity = 0?

I doubt it.

Bob Kolker


>
> Phil
>
>

John Zinni

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Nov 23, 2002, 9:35:25 PM11/23/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE02454...@jump.net...

So??? ... So???

So if you buy that "1/infinity = 0" and "infinity * 0 = 1" you are well on
your way to "proving" just about anything you want.

Tell you what. Accepting the above, if I can "prove" to you that the total
value of the .com stock I bought in the late 90's is tending to infinity
even though the price per share is tending to zero, will you buy it from me
for the price I paid?

--
Cheers
John Zinni

Jim

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Nov 24, 2002, 2:22:25 AM11/24/02
to

"Dan Perez" <fasterthanlig...@nospamyahoo.com> wrote in message
news:3ddf7960$0$17649$2c3e...@news.voyager.net...

You assertion that "Infinity * 0 = 1" is a major foopah. Consider the
following example:

let f(x) = 1/(x-1)
let g(x) = x^2 - 1

Now as x approaches 1, f(x) approaches infinity and g(x) approaches zero.
According to your logic f(x) * g(x) should approach 1.

NOT

Work it out and you will see that f(x) * g(x) approaches 2 as x approaches
1.

The fact is that not all infinities are created equal. Some functions
approach infinity faster than others. Evaluation of ratios and products
involving such functions have to be worked out on a case by case basis.


Phil

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Nov 24, 2002, 4:23:49 AM11/24/02
to

John Zinni wrote:

> From "Infinity * 0 = 1" we can "prove" just about anything we like. For
> example:
>
> Start with:
>
> Infinity * 0 = 1
>
> Multipy each side by 2:
>
> 2*(Infinity * 0) = 2*1
>
> 2*(Infinity * 0) = 2
>
> Divide each side by (Infinity * 0):
>
> 2=2/(Infinity * 0)
>
> 2=1/Infinity * 2/0
>
> 2=2/0 * 1/Infinity
>
> Now 2/0 = Infinity and 1/Infinity = 0, substituting in we have:

Wait a minute. Dan's argument includes the (admittedly unstated) assumption
that 1/infinity = 0, and you can't simply change this definition in the
middle of an argument/proof and expect to get valid results. Either
1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
Let's substitute just 1/infinity = 0 and its algebraic alternative, 1/0 =
infinity, and see what happens:

2 = 2/0 * 1/Infinity
2 = 2 * 1/0 * 1/Infinity
2 = 2 * Infinity * 1/Infinity
2 = 2 * Infinity/Infinity
2 = 2 * 1

Not a problem anymore, is it? It's only when you *redefine* infinity in the
middle of an argument that you have problems. Well, infinity is so complex
that you can always find problems, but at least you avoid these kinds of
problems! This is the mistake that Cantor made in his "proofs" that the
number of even numbers is exactly equal to the number of integers; he
redefined the denumerable set in the middle of his proofs. Of course this
resulted in weird answers! He cheated! Unfortunately, being a modern
scientist, the idea of a "shift error," in which you unconsciously shift
from one viewpoint or definition to another in the middle of an argument,
thereby incorrectly concluding that facts which apply to the second
viewpoint actually apply to the first viewpoint, was unknown to Cantor. It
sounds like a ridiculous claim, but the truth is that both Aristotle and
Newton would have spotted these errors in a cold minute.


> 2 = Infinity*0
>
> But we know that Infinity*0 = 1
>
> So:
>
> 2 = 1
>
> QED

> Jeesh, talk about rusty!!!

Makes me feel much better about my many similar errors!

Phil

> --
> Cheers
> John Zinni

Dirk Van de moortel

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Nov 24, 2002, 4:39:34 AM11/24/02
to

"Phil" <tu...@jump.net> wrote in message news:3DE09AC2...@jump.net...

You think so?

> but at least you avoid these kinds of
> problems! This is the mistake that Cantor made in his "proofs" that the
> number of even numbers is exactly equal to the number of integers;

You think so?

> he
> redefined the denumerable set in the middle of his proofs.

You think so?

> Of course this
> resulted in weird answers! He cheated!

You think so?

> Unfortunately, being a modern
> scientist, the idea of a "shift error," in which you unconsciously shift
> from one viewpoint or definition to another in the middle of an argument,
> thereby incorrectly concluding that facts which apply to the second
> viewpoint actually apply to the first viewpoint, was unknown to Cantor.

You think so?

> It
> sounds like a ridiculous claim, but the truth is that both Aristotle and
> Newton would have spotted these errors in a cold minute.

You think so?

So you don't merely *act* like an idiot, do you?

Dirk Vdm


Phil

unread,
Nov 24, 2002, 6:30:16 AM11/24/02
to
Geez Dirk, did I really piss you off that bad with my joke about the Maladjusted
Posters list? I thought you were just having fun with it, and that Stephen, not
realizing that, way overreacted.

Seriously, if you have an intelligent, scientific point to make about my
reasoning -- as opposed to merely making personal attacks on me -- feel free and
I'll see if I can find an answer incorporating a similar level of scientific
intellectual integrity, as opposed to debating tricks, political put-downs,
personal insults. You know, the usual dialectic crap that has nothing to do with
science, truth, accuracy, understanding, etc.

Maybe I caught you on a bad day? This was considerably more harsh and devoid of
rationality than what I usually see from you. We often disagree, and I usually
think (or at least claim) that your comments use forms of reasoning more
appropriate to debates rather than science, but at least I usually have to work
hard to refute them! This was just weird...

Phil

John Zinni

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Nov 24, 2002, 8:21:21 AM11/24/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE09AC2...@jump.net...
>
>
> John Zinni wrote:
>
> > From "Infinity * 0 = 1" we can "prove" just about anything we like. For
> > example:
> >
> > Start with:
> >
> > Infinity * 0 = 1
> >
> > Multipy each side by 2:
> >
> > 2*(Infinity * 0) = 2*1
> >
> > 2*(Infinity * 0) = 2
> >
> > Divide each side by (Infinity * 0):
> >
> > 2=2/(Infinity * 0)
> >
> > 2=1/Infinity * 2/0
> >
> > 2=2/0 * 1/Infinity
> >
> > Now 2/0 = Infinity and 1/Infinity = 0, substituting in we have:
>
> Wait a minute. Dan's argument includes the (admittedly unstated)
assumption
> that 1/infinity = 0, and you can't simply change this definition in the
> middle of an argument/proof and expect to get valid results. Either
> 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
> Let's substitute just 1/infinity = 0 and its algebraic alternative, 1/0 =
> infinity, and see what happens:
>
> 2 = 2/0 * 1/Infinity
> 2 = 2 * 1/0 * 1/Infinity
> 2 = 2 * Infinity * 1/Infinity

OK, how 'bout this. From the last line above:

2 = 2 * Infinity * 1/Infinity

2 = Infinity * (2 * 1/Infinity)
2 = Infinity * (2 * 0)
(Surly you accept that 2 * 0 = 0 ???)
2 = Infinity * 0
2 = 1


> 2 = 2 * Infinity/Infinity
> 2 = 2 * 1
>
> Not a problem anymore, is it? It's only when you *redefine* infinity in
the
> middle of an argument that you have problems. Well, infinity is so complex
> that you can always find problems, but at least you avoid these kinds of
> problems! This is the mistake that Cantor made in his "proofs" that the
> number of even numbers is exactly equal to the number of integers; he

> redefined tche denumerable set in the middle of his proofs.

What the hell you talking 'bout???

Its VERY straight forward to prove that the cardinality(even numbers) =
cardinality(integers)


>Of course this
> resulted in weird answers! He cheated!

He CHEATED???

Prove it?


>Unfortunately, being a modern
> scientist, the idea of a "shift error," in which you unconsciously shift
> from one viewpoint or definition to another in the middle of an argument,
> thereby incorrectly concluding that facts which apply to the second
> viewpoint actually apply to the first viewpoint, was unknown to Cantor.

What the hell does this mean???

David McAnally

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Nov 24, 2002, 9:27:06 AM11/24/02
to
"John Zinni" <j_z...@sympatico.ca> writes:

>"Phil" <tu...@jump.net> wrote in message news:3DE09AC2...@jump.net...
>>

>> Not a problem anymore, is it? It's only when you *redefine* infinity in
>the
>> middle of an argument that you have problems. Well, infinity is so complex
>> that you can always find problems, but at least you avoid these kinds of
>> problems! This is the mistake that Cantor made in his "proofs" that the
>> number of even numbers is exactly equal to the number of integers; he
>> redefined tche denumerable set in the middle of his proofs.

>What the hell you talking 'bout???

>Its VERY straight forward to prove that the cardinality(even numbers) =
>cardinality(integers)

>>Of course this
>> resulted in weird answers! He cheated!

>He CHEATED???

>Prove it?

Perhaps Phil is unaware that two sets are defined to have the cardinality
if there is a bijection between them. Or perhaps he doesn't know what a
bijection between sets is? I'd be interested to see him justify that
Cantor's bijection between the even numbers and the natural numbers, or
the bijection between the natural numbers and the integers, is not a
bijection.

David McAnally

--------------

Ahmed Ouahi, Architect

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Nov 24, 2002, 9:37:43 AM11/24/02
to

....... ...definitely we can eradicate any uncertainity over a convex
bornological algebra, when we do take as a conventional base which is
0.infinity=0, and here we do turn around the geometry of the space, which
it would as it should open an infinity of an

n
alternatives including if and only if as follows could (a x )
would as should

n n
converge bornologicaly to 0 definitely!!!!!!!!!!!!!!!!..................
...


--
Ahmed Ouahi, Architect
Best Regards!


"John Zinni" <j_z...@sympatico.ca> kirjoitti
viestissä:Rn4E9.18$cx4....@news20.bellglobal.com...

Ken S. Tucker

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Nov 24, 2002, 10:51:23 AM11/24/02
to
"Jim" <fowl...@concentric.net> wrote in message news:<arpunh$a...@dispatch.concentric.net>...

Would anyone disagree with
Infinity x Zero = N where Zero < N < Infinity.
Regards Ken S. Tucker

John Zinni

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Nov 24, 2002, 11:06:09 AM11/24/02
to
"Ken S. Tucker" <dyna...@vianet.on.ca> wrote in message
news:2202379a.02112...@posting.google.com...

Yes. Infinity is not a number and cannot be treated as such.

Also x * 0 = 0, no matter what x is.

--
Cheers
John Zinni

David McAnally

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Nov 24, 2002, 11:53:28 AM11/24/02
to
"John Zinni" <j_z...@sympatico.ca> writes:

>"Dan Perez" <fasterthanlig...@nospamyahoo.com> wrote in message
>news:3ddf7960$0$17649$2c3e...@news.voyager.net...

>This is utter nonsense.

>Infinity is NOT a number!!! You cannot toss it around as such.

Cantor's transfinite numbers are infinities. Some models of Formal
Arithmetic in Predicate Calculus include infinite quantities. Surreal
numbers include numbers which are "infinite" (i.e. larger than any finite
quantity - the surreals do not form an Archimedean field). Projective
geometry has genuine points which are "at infinity" (i.e. lines which meet
at those point in the projective geometry become parallel in the affine
geometry), and are therefore non-existent, in the affine
specialization. The extended complex plane has infinity as an element,
and indeed Moebius Transformations require the existence of infinity so
that they can be bijective. So while you are correct and infinity is
certainly not a number, I make the above points to demonstrate in certain
mathematical discussions, infinite quantities have an objective reality,
and can be treated as concrete quantities.

>Infinity is a PROCESS

The process relates to taking limits, as others have pointed out, and in
the context that is true. The above was just a not that in some
mathematical theories (but not the one presently under discussion), there
are concrete infinite quantities.

>and requires some special care or you end up with
>gibberish like "Infinity * 0 = 1"

>--
>Cheers
>John Zinni

David McAnally

--------------

John Zinni

unread,
Nov 24, 2002, 11:58:42 AM11/24/02
to
"David McAnally" <D.McAnally@i'm_a_gnu.uq.net.au> wrote in message
news:arqnjq$qft$1...@bunyip.cc.uq.edu.au...

> "John Zinni" <j_z...@sympatico.ca> writes:
>
> >"Phil" <tu...@jump.net> wrote in message
news:3DE09AC2...@jump.net...
> >>
> >> Not a problem anymore, is it? It's only when you *redefine* infinity in
> >the
> >> middle of an argument that you have problems. Well, infinity is so
complex
> >> that you can always find problems, but at least you avoid these kinds
of
> >> problems! This is the mistake that Cantor made in his "proofs" that the
> >> number of even numbers is exactly equal to the number of integers; he
> >> redefined tche denumerable set in the middle of his proofs.
>
> >What the hell you talking 'bout???
>
> >Its VERY straight forward to prove that the cardinality(even numbers) =
> >cardinality(integers)
>
> >>Of course this
> >> resulted in weird answers! He cheated!
>
> >He CHEATED???
>
> >Prove it?
>
> Perhaps Phil is unaware that two sets are defined to have the cardinality
> if there is a bijection between them. Or perhaps he doesn't know what a
> bijection between sets is?

Perhaps, but he does use terms like "denumerable set" implying that he might
have some knowledge of set theory.


> I'd be interested to see him justify that
> Cantor's bijection between the even numbers and the natural numbers, or
> the bijection between the natural numbers and the integers, is not a
> bijection.

I'd be interested to see this also.


>
> David McAnally
>
> --------------

--
Cheers
John Zinni

John Zinni

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Nov 24, 2002, 12:20:01 PM11/24/02
to
"David McAnally" <D.McAnally@i'm_a_gnu.uq.net.au> wrote in message
news:arr068$ma3$1...@bunyip.cc.uq.edu.au...

Hi David

Thanks for your input. I guess I can, justifiably, be accused of not being
precise in my terminology.

I have just recently set myself the task of relearning some of the
mathematics that I used to know and to be honest Set and Ring Theory were
not my strongest subjects.

It's good to see that there are more than a few people in this news group
that "know of what they speak" (I was beginning to have my doubts). I will
keep an eye out for your posts.

--
Cheers
John Zinni


Stephen Speicher

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Nov 24, 2002, 2:37:18 PM11/24/02
to

I left your entire post intact so that there can be no question
as to what you wrote, and what you mean. You shed the criticism
you receive over your misunderstandings and your misformulations
of relativity, faster than a duck sheds water. I am sure you will
receive a few posts from mathematicians here who might bother to
point out your numerous errors. (I won't bother.) It will be
interesting to see if you fend off the mathematical criticism as
quickly, and wrongly, as you do the criticism you receive
regarding relativity.

--
Stephen
s...@speicher.com

Ignorance is just a placeholder for knowledge.

Printed using 100% recycled electrons.
-----------------------------------------------------------

Dirk Van de moortel

unread,
Nov 24, 2002, 1:41:04 PM11/24/02
to

"Phil" <tu...@jump.net> wrote in message news:3DE0B868...@jump.net...

> Geez Dirk, did I really piss you off that bad with my joke about the Maladjusted
> Posters list?

There's 3 people on this planet that can piss me off:
my wife and my two sons. Even my mother-in-law
couldn't do it if she wanted.

Dirk Vdm

Dirk Van de moortel

unread,
Nov 24, 2002, 1:47:17 PM11/24/02
to

"Stephen Speicher" <s...@speicher.com> wrote in message news:Pine.LNX.4.33.021124...@localhost.localdomain...

> On Sun, 24 Nov 2002, Phil wrote:
> >

[snip]

> > Makes me feel much better about my many similar errors!
> >
>
> I left your entire post intact so that there can be no question
> as to what you wrote, and what you mean. You shed the criticism
> you receive over your misunderstandings and your misformulations
> of relativity, faster than a duck sheds water. I am sure you will
> receive a few posts from mathematicians here who might bother to
> point out your numerous errors. (I won't bother.) It will be
> interesting to see if you fend off the mathematical criticism as
> quickly, and wrongly, as you do the criticism you receive
> regarding relativity.

I won't bother either.
Ross Finlayson on sci.math a few years ago was more than enough.

Dirk Vdm


David McAnally

unread,
Nov 24, 2002, 5:16:27 PM11/24/02
to
dyna...@vianet.on.ca (Ken S. Tucker) writes:

>Would anyone disagree with
>Infinity x Zero = N where Zero < N < Infinity.

0*infinity = 0 in measure theory. In set theory, if x is a transfinite
carinal, then 0*x = x*0 = 0 (you can see this immediately from the fact
that the product of the cardinality of A and the cardinality of B is
defined to be the cardinality of A x B (their Cartesian product)).
If x is an infinite quantity in a nonstandard model of Formal Arithmetic,
then x*0 = 0 by definition, and 0*x = 0 by the Formal Arithmetic version
of the Principle of Induction. If x is an infinite quantity in a field of
surreal numbers (i.e. larger in magnitude than any natural number), then
0*x = x*0 = 0 since the surreals form a field (in fact, if x is an
infinite surreal number, then 1/x is a nonzero infinitesimal surreal
number).

In the way that you are thinking about infinity, the answer is anywhere
between - infinity and infinity, incluse=ive (and this includes zero).
The answer is dependent on how you approach infinity and how you appraoch
zero in the product.

David McAnally

--------------

Phil

unread,
Nov 25, 2002, 4:54:16 AM11/25/02
to
John Zinni wrote:
"Phil" <tu...@jump.net> wrote in message news:3DE09AC2...@jump.net...
[snip]
> > Now 2/0 = Infinity and 1/Infinity = 0, substituting in we have:
>
> Wait a minute. Dan's argument includes the (admittedly unstated) assumption
> that 1/infinity = 0, and you can't simply change this definition in the
> middle of an argument/proof and expect to get valid results. Either
> 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
> Let's substitute just 1/infinity = 0 and its algebraic alternative, 1/0 =
> infinity, and see what happens:
>
> 2 = 2/0 * 1/Infinity
> 2 = 2 * 1/0 * 1/Infinity
> 2 = 2 * Infinity * 1/Infinity

OK, how 'bout this. From the last line above:

2 = 2 * Infinity * 1/Infinity
2 = Infinity * (2 * 1/Infinity)
2 = Infinity * (2 * 0)
(Surly you accept that 2 * 0 = 0 ???)

No, remember that we are *defining* zero as 1/Infinity, so 2 * 0 = 2/Infinity, not 1/Infinity. It then follows that:

2 = Infinity * (2 * 1/Infinity)

2 = 2 * Infinity/Infinity
2 = 2

I repeat, you *cannot* change the definition of something in the middle of a line of reasoning and expect to get valid results. Unfortunately, the fact that we may not be aware of such a change in no way protects us from these errors.

2 = Infinity * 0
2 = 1

> 2 = 2 * Infinity/Infinity
> 2 = 2 * 1
>
> Not a problem anymore, is it? It's only when you *redefine* infinity in the
> middle of an argument that you have problems. Well, infinity is so complex
> that you can always find problems, but at least you avoid these kinds of
> problems! This is the mistake that Cantor made in his "proofs" that the
> number of even numbers is exactly equal to the number of integers; he

> redefined the denumerable set in the middle of his proofs.

What the hell you talking 'bout???

It's VERY straight forward to prove that the cardinality(even numbers) =

cardinality(integers)

>Of course this
> resulted in weird answers! He cheated!

He CHEATED???

Prove it?

Okay, I thought maybe I would get either zero or maybe one minor request for more info on this. Boy was I wrong! Allright John (and others), this response is going to be fairly long, as it is from my (unpublished and unfinished) book on Zeno's paradoxes and the Infinite, which is written at a high school or at most sophomore college level, but here goes. I will mainly just quote straight from my book, unless I need to add comments in [ ] brackets. The symbol for infinity will be replaced with a capital "I".

"Georg Cantor’s researches into the infinite produced more strange, nonsensical, and downright paradoxical results and proofs than perhaps any other researcher into the infinite, including Zeno. We will begin our examination of his work with perhaps the most basic of all infinite sets, namely the set of integers, {1, 2, 3, ..., I}. Cantor himself began with this set, calling it the “denumerable set,” not so much because it consisted of numbers, but because we can use the elements of this set to “count off” or enumerate basically any series of objects. Armed with the denumerable set, Cantor then began to explore the relationships between this set and various other infinite sets of numbers or objects.

Perhaps the simplest comparison is between the integers (the denumerable set) and the even numbers. In any large but finite sequence of numbers, no matter how large that sequence might be, there are almost twice as many integers as even numbers (and exactly twice as many if the sequence contains an even number of integers). This leads to the common sense conclusion that in the infinite set of integers there are also twice as many integers as even numbers. Cantor, however, found a proof that this is not the case.

The standard method for determining whether two sets have the same number of elements is to match or link each element in both sets with a single element in the opposite set, so that no element has more than one link to the other set. If every element then has a single link to an element in the opposite set, with no elements “left over” after the matching process, then the two sets must have the same number of elements. In the case of the integers and the even numbers, Cantor provided the simple matching scheme shown in Figure 7.

                    {2, 4, 6, 8,10,12, ..., I}
                     |  |  |  |  |  |       |
                    {1, 2, 3, 4, 5, 6, ..., I}

                    Figure 7 Cantor’s scheme for matching the integers to the even numbers

Clearly, each element in both sets is linked to a single element in the opposite set, proving that both of these sets have exactly the same number of elements. Of course, this merely proves that these two particular sets of integers and even numbers are equal in size; it does not necessarily follow that the even numbers are as numerous as the integers in the denumerable set itself. Nevertheless, Cantor was able to use this equality in the following proof.

First, Cantor took the set of all positive integers, namely the denumerable set, and matched each of them to a unique even number, exactly as shown in Figure 7. He then noted that the set of even numbers, since it consists of nothing but positive integers, must itself belong to the set of all positive integers, the denumerable set. This is the really important point, because if the set of integers on the bottom of Figure 7 is the denumerable set, and the set of even numbers on the top of Figure 7 belongs to the denumerable set, then the fact that these two particular sets are exactly the same size does prove that the even numbers are as numerous as the integers.

Talk about paradoxical results! In any finite sequence of numbers there are only half as many even numbers as integers, and yet somehow, in the infinite set of integers, the even numbers suddenly manage to be as numerous as the integers, with no “unmatched” integers left over. Of course, as we have seen, truths which apply to the finite realm do not always apply to either the infinite or infinitesimal realms, but this case is a little different from the other cases we have seen. In the Achilles, for example, there may have always been one more halfway point to cross, but at least we kept getting closer to the door! In the case of the integers and even numbers, however, an examination of even the largest finite sequence of numbers shows that the even numbers never become more than half as numerous as the integers. To complete the analogy with the Achilles, it is as if we asked a man to march in place in the middle of the room, without ever moving so much as a single point towards the door, and then expected him to somehow walk out of the room after an infinite amount of time. Even if we take the limitations of our finite minds into account, and acknowledge the greater possibilities within the infinite and infinitesimal realms, there are some things that simply cannot be done. Expecting the even numbers to somehow catch up with the integers, simply because we extend the comparison to infinity, certainly appears to be one of those things that cannot be done in any realm, under
any circumstances.

Since Cantor’s proof relies so much on the ability to match the integers with the even numbers, it obviously makes sense to examine this process in more detail. In particular, we will check to see if other matching schemes exist between the integers and even numbers, or between sets which are similar in some way to Cantor’s matching scheme. Figure 8 shows one example of a different matching scheme between the integers and the even numbers.

                    {   2,    4,    6, ...,      I}
                        |     |     |            |
                    {1, 2, 3, 4, 5, 6, ..., I-1, I}

                Figure 8 An alternative scheme for matching the integers to the even numbers

In this matching scheme, the first even number, namely 2, is matched to the integer 2 instead of 1, the even number 4 is matched to the integer 4, etc. As before, every even number is matched or linked to a single integer, with no even numbers left unaccounted for. But while each integer is also linked to at most a single even number–meaning that a single integer does not account for several even numbers–fully half of the integers are suddenly not accounted for. This matching scheme obviously suggests that the even numbers are only half as numerous as the integers.

An even stronger case for believing that the integers are twice as numerous as the even numbers can be made by using the matching scheme shown in Figure 9. In this figure the top two sets are identical to the two sets shown in Figure 8. The bottom set is an exact copy of the top set, the set of even numbers, except that each element in this set is matched to one of the odd integers, in such a manner that any even number (n) is matched to the odd number (n-1). As a result, every element in all three sets is matched to one and only one element in one of the other sets, with no elements left over. Since the set of integers can fully accommodate two complete sets of the even numbers, it again appears to show that the integers are twice as numerous as
the even numbers.

                    {   2,    4,    6, ...,      I}
                        |     |     |            |
                    {1, 2, 3, 4, 5, 6, ..., I-1, I}
                     |     |     |           |
                    {2,    4,    6, ...,     I    }

                    Figure 9 A second alternative scheme for matching the integers to the even numbers

What makes all of this so strange is that we are using Cantor’s own method of establishing a one-to-one matching scheme between the elements of two sets in order to determine whether they are the same size, but with completely different results. This leads to an obvious question, namely which, if any, of these matching schemes is correct? Unfortunately, the answer is not nearly so obvious, but there is one thing that is certain; the fact that several different matching schemes with different conclusions exist proves that we cannot just take the first matching scheme that comes along and assume that its apparent conclusions are correct! We must have more information, more insight into what is going on here, before we can even begin to think about drawing reliable conclusions from any of these matching schemes.

We can begin by noting that the one-to-one matching method used by Cantor to prove that various sets are equal in size is in fact completely valid and correct. If each and every element in one set is linked to a single element in a second set, and vice-versa, then it does necessarily follow that the two sets have the same number of elements. Therefore, we can correctly conclude that the particular sets of integers and even numbers used in Cantor’s matching scheme and shown in Figure 7 are indeed the same size. Similarly, we can correctly conclude that the particular set of integers used in Figure 9 is indeed twice as large as each of the two particular sets of even numbers used in the same figure, since a state of one-to-one matching exists
only when both sets of even numbers are used. All of these statements and conclusions are true and valid. Therefore, it must be the statements made after this point, such as “there are as many even numbers as integers,” that contain errors.

We can get a better idea of what is occurring by continuing to examine the basic pattern of Cantor’s matching scheme from a variety of different viewpoints.  Figure 10a shows Cantor’s matching scheme for reference. Looking at Figure 10a shows that each of the even numbers is twice as large as the integer to which it is linked. Therefore, we can restate Cantor’s matching scheme from another viewpoint by linking each integer (n) to the number which is twice as large, (2 × n). Figure 10b shows Cantor’s matching scheme from this alternative viewpoint. The only thing that is different is the final value of infinity. Figure 10a uses the same symbol, I, as the last element in both sets, while Figure 10b implies that the last element in the set of even numbers is twice as large as the last element in the set of integers. Furthermore, given that every other integer is matched to a number twice as large, it is difficult to imagine how this can suddenly fail to be the case for the largest elements as well. [10c should show lines linking 1 to 1, 2 to 4, 3 to 6, and infinity to 2 * infinity]

         I – I                    I – 2I (2*I)                        2I
                                                                      /
           .                        .                                /
           .                        .                               /
           .                        .                              /  6
                                                                  I   5
         4 – 8                    4 – 8  (2*4)                    :   4
         3 – 6                    3 – 6  (2*3)                    3 / 3
         2 – 4                    2 – 4  (2*2)                    2   2
         1 – 2                    1 – 2  (2*1)                    1 - 1

                     (a)                                              (b)                                                           (c)

                                    Figure 10 Cantor’s matching scheme and two variations

Figure 10c is the first real change from Cantor’s basic scheme. It is the same as Figure 10b, except that we have added the “missing”odd numbers. Obviously this set no longer matches one-to-one with the set of integers on the left, but it is otherwise identical with the other figures. This actually is not that surprising, since we merely added elements to the set of even numbers, with no changes to the set of integers on the left, to the even numbers in the set on the right, or to the links from the integers to the even numbers. But while Figures 10b and 10c leave Cantor’s matching scheme basically intact, the changes leading to Figure 10c lead to the realization that the reason the even numbers in the right-hand sets are as numerous as the integers in the left-hand sets is because those even numbers initially come from a set of integers that is twice as large as the set of integers on the left.

This line of reasoning also explains the very different results found in Figure 9, where the integers were found to be twice as numerous as the even numbers. In Figure 9, the value of I found in the set of even numbers is not twice as large as the value of I found in the set of integers. It is, in fact, exactly the same. In other words, if we add the “missing” odd numbers to the set of even numbers in Figure 9, we wind up with a set that is not only the same size, but actually identical in every way to the set of integers. Not surprisingly, the fact that we begin with two sets of integers that are the same size eventually leads to the conclusion that the integers are twice as numerous as the even numbers. To summarize, the only way that we can conclude that the even numbers are as numerous as the integers is to initially take those even numbers from a set of integers that is twice as large as the set of integers to which the even numbers will be compared. In other words, we have to cheat.

This fact makes it extremely likely that Cantor’s proof is somehow flawed. To restate, Cantor began with a set of integers which he defined as the denumerable set, the set of all positive integers. He then linked each integer in this denumerable set to an even number, proving that those two particular sets are indeed the same size. After noting that this set of even numbers, since it consists of nothing but positive integers, must also belong to the denumerable set, he concluded that within the denumerable set there are as many even numbers as integers.

Once again, an examination of Figure 10c will help to clarify things. In Cantor’s proof, the set of integers on the left side of Figure 10c is initially defined as the denumerable set, but suppose that we initially define the set of integers on the right side of Figure 10c as being the denumerable set. We then match or link each of the even numbers within this set to an integer from the set of integers on the left side of Figure 10c. This is very similar to what Cantor did, except in reverse, but a little thought reveals that the act of defining the relationship between these two sets in this manner also has the effect of defining the set on the left as being exactly half as large as the denumerable set on the right. In other words, by declaring that each integer in the left set is linked to an even number in the right set, we force the left set to be half the size of the denumerable set on the right. This leads to the question, what makes us think that this form of automatic definition will not occur simply because we initially define the set on the left as being the denumerable set?

This was Cantor’s mistake. He simply assumed that having defined the set on the left as being the denumerable set, it would stay that way, immune from all subsequent redefinitions or other actions. In reality, however, the moment Cantor defined the relationship between his set of integers and the set of even numbers as being one-to-one, he forced that set of integers to be exactly half the size of the denumerable set, quite regardless of the set’s previous definitions. It literally could have been defined as the denumerable set, the set of letters in the alphabet, or even as the man in moon, and it would have made no difference. Whatever this set was before Cantor defined its relationship with the even numbers, afterwards it became an infinite set of integers which was exactly half the size of the denumerable set. Interestingly, Cantor’s observation that the even numbers, since they are all positive integers, must belong to the denumerable set was quite correct. As shown in Figure 10C, once we add the “missing” odd numbers, we do indeed have the denumerable set, a set which is exactly twice as large as the set of integers on the left.

Like most errors, Cantor’s error can be viewed from more than one point of view. To summarize, Cantor began with the denumerable set, but then accidently redefined it as being a set of integers half as large as the denumerable set. Since he never noticed the shift from the denumerable set to this new set of integers, Cantor naturally concluded that a characteristic of this new set, namely that it had the same number of elements as the set of all even numbers, was actually a characteristic of the denumerable set: a classic shift-error. The fact that this error occurred as a result of an accidental redefinition means that we can also view it as a definition error. Since this definition error was caused by shift from one definition to another–as opposed to the more usual case where two people unknowingly use two different definitions for a particular concept–it really is just a specific type of shift error. However, anything that can make it easier to spot a shift error is worthwhile, and by consciously looking for an accidental redefinition, we greatly increase our chances of recognizing this particular type of shift error."

Okay, that's all from my book. The main point is that in the process of taking "the denumerable set" and *defining* it as having a one-to-one link with "the infinite set of even numbers," Cantor unknowingly *redefined* this set of integers as being an infinite set exactly 1/2 the size of the "true" denumerable set.

>Unfortunately, being a modern
> scientist, the idea of a "shift error," in which you unconsciously shift
> from one viewpoint or definition to another in the middle of an argument,
> thereby incorrectly concluding that facts which apply to the second
> viewpoint actually apply to the first viewpoint, was unknown to Cantor.

What the hell does this mean???

Hopefully you now have at least an inkling not only of what a shift error is, but also of how extraordinarily powerful and difficult to detect they can be. Another example I often use is the flight path of an artillery shell. To test how air temperature affects the flight path, we lock a gun's ellevation and fire it at regular intervals from dawn (cold) to the afternoon (hot), being careful to take normal factors such as wind resistance and exit velocity into account. Such a test will show that the shells travel farther in the morning. However, another factor that affects the result is "barrel droop," where the sun heats the top of the gun, causing it to warp slightly downward, even though the elevation controls at the base show no change. If the tester is unaware of this factor, he will unknowingly *shift* from an examination of temperature, which actually decreases with temperature, to an examination of barrel droop, which has a larger effect than temperature. Aristotle, Galileo, Newton, and Gauss were all, I am fairly certain, quite aware of such errors, but as far as I have been able to determine modern scientists literally do not even know that such errors exist, leaving them completely vulnerable to the effects of these errors.

Let me know, on list or off, if you have any questions about one or more particular points on the issue of the equality of various infinite sets.

And to answer David's question, I have no idea what a bijection is, although I will guess that it refers to the ability to establish a one-to-one matching between the elements of two sets.

Phil
 

Phil

unread,
Nov 25, 2002, 5:09:26 AM11/25/02
to
Jim wrote:

I agree that not all infinities are equal, but your example does not
necessarily lead to the problem you state. If zero is *defined* as being
1/Infinity, then f(x) = 1/(1 -1) = 1/0 = Infinity, as you state. However,

g(x) = (x - 1)(x + 1)
g(x) = (1 - 1)(1 + 1) = 0 * 2 = 2 * Infinity, so

g(x) * f(x) = (2 * Infinity) * (1/Infinity) = 2 * (Infinity/Infinity) = 2

The statement that g(x) approaches infinity as x approaches 1, although it is
certainly true, does not clearly state whether and how this statement defines
zero. Is it 1/Infinity, 2/Infinity? Do you see what I am saying? The line of
reasoning you give does the same thing that all other lines of reasoning which
prove that 2 = 1 do that I have seen; it *redefines* zero or infinity in the
middle of the line of reasoning without actually making the existence of that
redefinition clear. Since the individual following this line of reasoning does
not notice this shift from one definition of zero to another, he naturally
concludes that there is a problem with defining zero as being 1/Infinity, when
the actual problem is that a term, namely zero, has been *redefined* in the
middle of the line of reasoning, something which is virtually guaranteed to
lead to faulty conclusion.

Phil


David McAnally

unread,
Nov 25, 2002, 8:09:36 AM11/25/02
to
Phil <tu...@jump.net> writes:

>John Zinni wrote:

>> "Phil" <tu...@jump.net> wrote in message news:3DE09AC2...@jump.net...

>[snip]

>> > > Now 2/0 = Infinity and 1/Infinity = 0, substituting in we have:
>> >
>> > Wait a minute. Dan's argument includes the (admittedly unstated) assumption
>> > that 1/infinity = 0, and you can't simply change this definition in the
>> > middle of an argument/proof and expect to get valid results. Either
>> > 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
>> > Let's substitute just 1/infinity = 0 and its algebraic alternative, 1/0 =
>> > infinity, and see what happens:
>> >
>> > 2 = 2/0 * 1/Infinity
>> > 2 = 2 * 1/0 * 1/Infinity
>> > 2 = 2 * Infinity * 1/Infinity
>>
>> OK, how 'bout this. From the last line above:
>>
>> 2 = 2 * Infinity * 1/Infinity
>> 2 = Infinity * (2 * 1/Infinity)
>> 2 = Infinity * (2 * 0)
>> (Surly you accept that 2 * 0 = 0 ???)

>No, remember that we are *defining* zero as 1/Infinity, so 2 * 0 = 2/Infinity,
>not 1/Infinity. It then follows that:

That is not how zero is defined. Zero is the additive identity.

>I repeat, you *cannot* change the definition of something in the middle of a
>line of reasoning and expect to get valid results. Unfortunately, the fact that
>we may not be aware of such a change in no way protects us from these errors.

But you have the wrong definition. Zero is the additive identity.

This is wrong. The set of positive integers is {1, 2, 3, ...}. I is not
an element of the set.

The rest of Phil's comments are cut out, firstly because he included his
entire posting TWICE (which was quite inconsiderate of him), and secondly
because the rest of what he had to say only amounted to his refusing to
accept that two sets which have a bijection must have the cardinality, or
that a set can have a proper subset of the same cardinality. Of course,
no finite set has a proper subset of the same cardinality, but that does
not stop the effect happening when you get to infinite sets. I wonder
if Phil can give us a good set-theoretic condition for two sets to have
the same cardinality so that we can see why the even positive numbers do
not have the same cardinality as the positive integers as per Phil's claim.

David McAnally

--------------

John Zinni

unread,
Nov 25, 2002, 8:38:27 AM11/25/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE1F6F9...@jump.net...

How did you manage this line???
I don't think I saw you define 0=Infinity at the beginning of your "proof".
(this, of course, would have incredible implications throughout the hard
sciences!!!).


>
> g(x) * f(x) = (2 * Infinity) * (1/Infinity) = 2 * (Infinity/Infinity) = 2
>
> The statement that g(x) approaches infinity as x approaches 1, although it
is
> certainly true, does not clearly state whether and how this statement
defines
> zero. Is it 1/Infinity, 2/Infinity? Do you see what I am saying? The line
of
> reasoning you give does the same thing that all other lines of reasoning
which
> prove that 2 = 1 do that I have seen; it *redefines* zero or infinity in
the
> middle of the line of reasoning without actually making the existence of
that
> redefinition clear. Since the individual following this line of reasoning
does
> not notice this shift from one definition of zero to another, he naturally
> concludes that there is a problem with defining zero as being 1/Infinity,
when
> the actual problem is that a term, namely zero, has been *redefined* in
the
> middle of the line of reasoning, something which is virtually guaranteed
to
> lead to faulty conclusion.
>
> Phil

--
Cheers
John Zinni

John Zinni

unread,
Nov 25, 2002, 9:18:23 AM11/25/02
to
Hi All

Sorry for the top-post but this is a tech question as opposed to anything to
do with this thread.

Does anyone know why Phil's response to my post (parts of which are quoted
here by David) might not show up in my newsreader?
(This may be a blessing in disguise)

I have tried all "Views" and have even reloaded the entire newsgroup, but it
still does not show up?

Any thoughts?

--
Cheers
John Zinni

"David McAnally" <D.McAnally@i'm_a_gnu.uq.net.au> wrote in message
news:art7eg$hea$1...@bunyip.cc.uq.edu.au...

ste...@nomail.com

unread,
Nov 25, 2002, 9:27:08 AM11/25/02
to
Phil <tu...@jump.net> wrote:


: John Zinni wrote:

:> From "Infinity * 0 = 1" we can "prove" just about anything we like. For
:> example:
:>
:> Start with:
:>
:> Infinity * 0 = 1
:>
:> Multipy each side by 2:
:>
:> 2*(Infinity * 0) = 2*1
:>
:> 2*(Infinity * 0) = 2
:>
:> Divide each side by (Infinity * 0):
:>
:> 2=2/(Infinity * 0)
:>
:> 2=1/Infinity * 2/0
:>
:> 2=2/0 * 1/Infinity
:>
:> Now 2/0 = Infinity and 1/Infinity = 0, substituting in we have:

: Wait a minute. Dan's argument includes the (admittedly unstated) assumption
: that 1/infinity = 0, and you can't simply change this definition in the
: middle of an argument/proof and expect to get valid results. Either
: 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.

What definition did he change? Do you agree that 1*0=0?
Am I changing a definition if I say that 2*0=0? Do you think
that 1*0=0 and 2*0=0 are mutually exclusive?

Stephen

David McAnally

unread,
Nov 25, 2002, 9:45:29 AM11/25/02
to
ste...@nomail.com writes:

>Phil <tu...@jump.net> wrote:

>: John Zinni wrote:

Of course he does!

David McAnally

--------------

Phil

unread,
Nov 25, 2002, 12:42:19 PM11/25/02
to
John,

I can only see one of them (I messed up and selected the "send both text and
HTML" option), and there have been other times when I could not see someone's
post. I have no idea why.

So, blessing or curse, I'm going to send you another copy! Just to you. ;-)

Phil

Phil

unread,
Nov 25, 2002, 1:04:19 PM11/25/02
to
ste...@nomail.com wrote:

For normal, practical use they can be viewed as being equivalent, but when you
are dealing with infinity the answer is yes, if 0 = 1/Infinity, then 1 * 0 is
*not equal* to 2 * 0! If you believe otherwise then you really do believe that
either (1) 2 = 1, or (2) that the normal laws of algebra do not apply to
situations involving infinity. The latter is in fact the most widely accepted
conclusion, but since these laws only appear to be inconsistent when we deal with
infinity and *assume* that 1 * 0 = 2 * 0, you have a situation where you do not
know, a priori, which of the following two beliefs is flawed (if not both). (1)
the laws of algebra do not apply to the realm of infinity, or (2) even after we
*define* zero as being 1/Infinity, that we can then *assume* that zero also =
2/Infinity. The fact that 0 is being freely redefined during any sequence of
reasoning that "proves" that 1 = 2 looks extremely suspicious to me, to say the
least. My own choice, given the option of throwing out the laws of algebra, or
throwing out the belief that once 0 is defined as 1/Infinity that we can also
define it as being 2/Infinity without any problem, is to junk the latter.

I'm not saying that you (in particular) cannot find good arguments against this,
or even that I am *necessarily* correct, but I think I have a good point, one
which should not be simply dismissed out of hand without further examination
simply because it is not already widely known.

Defensive Phil


Phil

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Nov 25, 2002, 1:09:37 PM11/25/02
to
John Zinni wrote:

Oops! Now it's my turn to get out the rust.

g(x) = (x - 1)(x + 1)

g(x) = (1 - 1)(1 + 1) = 0 * 2 = 2 * (1/Infinity), so
g(x) * f(x) = (2/Infinity) * (Infinity) = 2 * (Infinity/Infinity) = 2

Got my "f 'n g's mixed up" :-(

Dirk Van de moortel

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Nov 25, 2002, 1:13:23 PM11/25/02
to

"Phil" <tu...@jump.net> wrote in message news:3DE1F6F9...@jump.net...

[snip]

> I agree that not all infinities are equal, but your example does not
> necessarily lead to the problem you state. If zero is *defined* as being
> 1/Infinity, then f(x) = 1/(1 -1) = 1/0 = Infinity, as you state. However,
>
> g(x) = (x - 1)(x + 1)
> g(x) = (1 - 1)(1 + 1) = 0 * 2 = 2 * Infinity, so
>
> g(x) * f(x) = (2 * Infinity) * (1/Infinity) = 2 * (Infinity/Infinity) = 2
>
> The statement that g(x) approaches infinity as x approaches 1, although it is
> certainly true, does not clearly state whether and how this statement defines
> zero. Is it 1/Infinity, 2/Infinity? Do you see what I am saying? The line of
> reasoning you give does the same thing that all other lines of reasoning which
> prove that 2 = 1 do that I have seen; it *redefines* zero or infinity in the
> middle of the line of reasoning without actually making the existence of that
> redefinition clear. Since the individual following this line of reasoning does
> not notice this shift from one definition of zero to another, he naturally
> concludes that there is a problem with defining zero as being 1/Infinity, when
> the actual problem is that a term, namely zero, has been *redefined* in the
> middle of the line of reasoning, something which is virtually guaranteed to
> lead to faulty conclusion.
>
> Phil

Brilliant:
http://users.pandora.be/vdmoortel/dirk/Physics/ImmortalFumbles.html#Infinity
Title: "If zero is defined as being 1/infinity"

Dirk Vdm


ste...@nomail.com

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Nov 25, 2002, 1:24:50 PM11/25/02
to
Phil <tu...@jump.net> wrote:
: ste...@nomail.com wrote:

:> Phil <tu...@jump.net> wrote:
:>
:> : Wait a minute. Dan's argument includes the (admittedly unstated) assumption


:> : that 1/infinity = 0, and you can't simply change this definition in the
:> : middle of an argument/proof and expect to get valid results. Either
:> : 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
:>
:> What definition did he change? Do you agree that 1*0=0?
:> Am I changing a definition if I say that 2*0=0? Do you think
:> that 1*0=0 and 2*0=0 are mutually exclusive?

: For normal, practical use they can be viewed as being equivalent, but when you
: are dealing with infinity the answer is yes, if 0 = 1/Infinity, then 1 * 0 is
: *not equal* to 2 * 0!

No one ever defined 0 as 1/Infinity. They defined 1/Infinity
as 0. Defining 1/Infinity = 2/Infinity = 3/Infinity = 0
is no more problematic than defining 1*0 = 2*0 = 3*0 =0.
Do you think there is a problem with the fact that 1*0 = 2*0 = 3*0 = 0?

: If you believe otherwise then you really do believe that


: either (1) 2 = 1, or (2) that the normal laws of algebra do not apply to
: situations involving infinity.

Yes, the normal laws do not apply. In particular infinity-infinity
and infinity/infinity are undefined. If you define these to be
0 and 1, then you can generate all sorts of contradictions,
as I and other posters have shown.

: The latter is in fact the most widely accepted


: conclusion, but since these laws only appear to be inconsistent when we deal with
: infinity and *assume* that 1 * 0 = 2 * 0, you have a situation where you do not
: know, a priori, which of the following two beliefs is flawed (if not both). (1)
: the laws of algebra do not apply to the realm of infinity, or (2) even after we
: *define* zero as being 1/Infinity, that we can then *assume* that zero also =
: 2/Infinity. The fact that 0 is being freely redefined during any sequence of
: reasoning that "proves" that 1 = 2 looks extremely suspicious to me, to say the
: least. My own choice, given the option of throwing out the laws of algebra, or
: throwing out the belief that once 0 is defined as 1/Infinity that we can also
: define it as being 2/Infinity without any problem, is to junk the latter.

: I'm not saying that you (in particular) cannot find good arguments against this,
: or even that I am *necessarily* correct, but I think I have a good point, one
: which should not be simply dismissed out of hand without further examination
: simply because it is not already widely known.

: Defensive Phil

Honestly, I do not see what your point is. No one is defining 0 as
1/infinity, and noone is redefining what 0 is in the middle
of a proof. We all know what 0 is. The question is, what is 1/infinity?
And there is no problem defining things such that 1/infinity = 2/infinity
= 3/infinity = 0.

Stephen

Phil

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Nov 25, 2002, 2:12:08 PM11/25/02
to
David McAnally wrote:

> Phil <tu...@jump.net> writes:
>
> >John Zinni wrote:
>
> >> "Phil" <tu...@jump.net> wrote in message news:3DE09AC2...@jump.net...
>
> >[snip]
>
> >> > > Now 2/0 = Infinity and 1/Infinity = 0, substituting in we have:
> >> >
> >> > Wait a minute. Dan's argument includes the (admittedly unstated) assumption
> >> > that 1/infinity = 0, and you can't simply change this definition in the
> >> > middle of an argument/proof and expect to get valid results. Either
> >> > 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
> >> > Let's substitute just 1/infinity = 0 and its algebraic alternative, 1/0 =
> >> > infinity, and see what happens:
> >> >
> >> > 2 = 2/0 * 1/Infinity
> >> > 2 = 2 * 1/0 * 1/Infinity
> >> > 2 = 2 * Infinity * 1/Infinity
> >>
> >> OK, how 'bout this. From the last line above:
> >>
> >> 2 = 2 * Infinity * 1/Infinity
> >> 2 = Infinity * (2 * 1/Infinity)
> >> 2 = Infinity * (2 * 0)
> >> (Surly you accept that 2 * 0 = 0 ???)
>
> >No, remember that we are *defining* zero as 1/Infinity, so 2 * 0 = 2/Infinity,
> >not 1/Infinity. It then follows that:
>
> That is not how zero is defined. Zero is the additive identity.

Actually yours is *a* definition of zero, not the one and only, godlike and complete
definition that provides total understanding in all situations. Come one David, you
know that most concepts can be neither completely described nor understood from a
single point of view. The definition of zero as the additive identity is made from
within a specific context, and to assume that no other contexts for zero exist, or
that this definition is complete and sufficient for all other contexts, is unfounded.

Oh really? So when I walk out of a room I do not actually cross *all* of the halfway
points between myself and the door, but only "a large number of them?" One of the
reasons that infinity is such a difficult subject is that it is literally
unimaginable, rather like the 4th spatial dimension. We can and do work with such
concepts all the time, but they are difficult and our thoughts are prone to error,
because they remain beyond our ability to fully visualize or understand. In the case
of Infinity, we love to "water it down" and bring it within the limits of our finite
minds by thinking of it as just "a large number, as large as we please." One of the
first things the Achilles taught me, however, is that Infinity is more than that.
Infinity is not just "a large number, as large as we please," infinity is *all* of
something.

After you jump up and down an infinite number of times; how many more times can you
jump up and down? None, your jumping days are over. After you think an infinite
number of thoughts, how many more thoughts can you have? None, your brain just shut
down, permanently. After you have crossed the infinite set of halfway points between
yourself and the door, how many more halfway points can you cross? None, you just
walked out the door. And once you have a set containing the infinite set of integers,
how many more integers exist? None, because your set already contains *all* of them.

In the finite realm to which our minds are limited, this is nonsense. How can there
be a "last point" between us and the door? How can infinity literally be "the last
integer?" In our own finite realm these things are impossible, but infinity is not
limited to the finite realm, and things that are true in our realm are not
necessarily true in the infinite or infinitesimal realms. This simplest example i
know of is walking out the door. Do you believe that you cross *all* of the halfway
points between you and the door every time you do this or not? If the answer is yes,
then the set of halfway points between you and the door does not merely contain "a
large number, as large as we please," it contains *all* of the halfway points. The
fact that this is literally incomprehensible (but not inconceivable!) to your finite
mind is something that you will just have to get used to if you want to have an even
halfway accurate understanding of infinity. I don't like it any more than you, but
that's too bad, because any attempt to water infinity down to something more
comprehensible and acceptable to our finite minds invariably leads to serious errors,
such as the idea that we can never leave a room, or that we must "mystically
dematerialize" from the room and then "rematerialize" outside the door, leaving an
infinite number of halfway points uncrossed.


> The rest of Phil's comments are cut out, firstly because he included his
> entire posting TWICE (which was quite inconsiderate of him),

Sorry about that.

> and secondly
> because the rest of what he had to say only amounted to his refusing to
> accept that two sets which have a bijection must have the cardinality, or
> that a set can have a proper subset of the same cardinality. Of course,
> no finite set has a proper subset of the same cardinality, but that does
> not stop the effect happening when you get to infinite sets. I wonder
> if Phil can give us a good set-theoretic condition for two sets to have
> the same cardinality so that we can see why the even positive numbers do
> not have the same cardinality as the positive integers as per Phil's claim.

David, Cantor came up with a proof, based on the premises of mathematics, which led
him to conclude that in the denumerable set, the number of even integers is exactly
equal to the number of integers. Now, we can simply *assume* that this proof is
correct, in which case you should indeed ignore the rest of my post, or we can test
this proof, to see if it contains one or more flaws. In such a test, however, we
cannot continue to *assume* that it is correct!

The core of Cantor's argument was the act of establishing a one-to-one matching
(bijection?) between two sets which Cantor claimed both belonged to the (one and
only) denumerable set. I easily produced another matching scheme that fully accounted
for all of the even numbers, while leaving fully half of the integers "unmatched."
Given the ability to produce more than one matching scheme with different results, I
think that any honest person would have to at least question whether the conclusions
which *appear* to follow from just one of those matching schemes, either Cantor's or
mine, are necessarily correct. I then provided a line of reasoning which showed that
Cantor's reasoning included an act, specifically the manner in which he matched the
elements of the two sets, that inherently *defines* his "denumerable set of integers"
as being exactly half the size of his "denumerable set of even numbers."

Now, you are more than welcome to criticize and possibly correct my premises and/or
the line of reasoning which I used to come to this conclusion, but to sit here and
say, a priori, that *since* Cantor's conclusions are correct, that the flaw which I
found in his reasoning must be incorrect, is a "circular proof," and therefore
meaningless.

As for your request for me to provide "a good set-theoretic condition for two sets to
have the same cardinality," it's a good request (assuming I correctly understand your
terms), but I have a better idea. Let's take one thing at a time, stick to the
question of whether Cantor's reasoning did indeed contain a hidden redefinition of
one of his sets, and save the bigger fish (the entire question of cardinality) to fry
for later. In the first place, you are asking a massive question which, if I get
involved with it at all, will require equally massive responses. In the second place,
shifting to this second (admittedly related) subject does not address the issue of
whether Cantor redefined one of his terms in the middle of his proof, and I want to
see if you are willing to address this point in a ruthless, analytical way that does
not involve assuming that all of Cantor's conclusions must be accepted as givens.

Phil


Dirk Van de moortel

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Nov 25, 2002, 2:21:07 PM11/25/02
to

"Phil" <tu...@jump.net> wrote in message news:3DE2762C...@jump.net...

O help.

Dirk Vdm


Phil

unread,
Nov 25, 2002, 2:28:07 PM11/25/02
to
ste...@nomail.com wrote:

> Phil <tu...@jump.net> wrote:
> : ste...@nomail.com wrote:
>
> :> Phil <tu...@jump.net> wrote:
> :>
> :> : Wait a minute. Dan's argument includes the (admittedly unstated) assumption
> :> : that 1/infinity = 0, and you can't simply change this definition in the
> :> : middle of an argument/proof and expect to get valid results. Either
> :> : 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
> :>
> :> What definition did he change? Do you agree that 1*0=0?
> :> Am I changing a definition if I say that 2*0=0? Do you think
> :> that 1*0=0 and 2*0=0 are mutually exclusive?
>
> : For normal, practical use they can be viewed as being equivalent, but when you
> : are dealing with infinity the answer is yes, if 0 = 1/Infinity, then 1 * 0 is
> : *not equal* to 2 * 0!
>
> No one ever defined 0 as 1/Infinity. They defined 1/Infinity
> as 0. Defining 1/Infinity = 2/Infinity = 3/Infinity = 0
> is no more problematic than defining 1*0 = 2*0 = 3*0 =0.
> Do you think there is a problem with the fact that 1*0 = 2*0 = 3*0 = 0?

It appears to me that the answer is yes, but see below for why.

My point is that the normal laws of algebra *do* apply *if* you do not assume that
1/infinity = 2/infinity = 3/infinity = 0. To put it another way, I defy you (or
challenge you) to come up with an example that violates the laws of algebra, when that
example assumes that the relationship between infinity and zero is 0 = 1/infinity, and
which does *not* assume that 1/infinity = 2/infinity = 3/infinity = 0.

From still another point of view, my point is that we do not have to give up the laws
of algebra when dealing with the infinite realm *if* we are willing to give up the idea
that 1/infinity = 2/infinity = 3/infinity = 0. Given that option (asuming that it
really is an option, of course), I would much rather give up the latter than the
former. In any case, if it really is impossible for any of us to come up with something
that violates the laws of algebra *without* assuming that 1/infinity = 2/infinity =
3/infinity = 0, then the very fact that this assumption does lead to violations of the
laws of algebra is itself an indication that this assumption may well be false, and
that we should investigate further.

Phil


ste...@nomail.com

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Nov 25, 2002, 2:39:59 PM11/25/02
to
Phil <tu...@jump.net> wrote:
: ste...@nomail.com wrote:

:> Phil <tu...@jump.net> wrote:


:> : ste...@nomail.com wrote:
:>
:> :> Phil <tu...@jump.net> wrote:
:> :>
:> :> : Wait a minute. Dan's argument includes the (admittedly unstated) assumption
:> :> : that 1/infinity = 0, and you can't simply change this definition in the
:> :> : middle of an argument/proof and expect to get valid results. Either
:> :> : 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
:> :>
:> :> What definition did he change? Do you agree that 1*0=0?
:> :> Am I changing a definition if I say that 2*0=0? Do you think
:> :> that 1*0=0 and 2*0=0 are mutually exclusive?
:>
:> : For normal, practical use they can be viewed as being equivalent, but when you
:> : are dealing with infinity the answer is yes, if 0 = 1/Infinity, then 1 * 0 is
:> : *not equal* to 2 * 0!
:>
:> No one ever defined 0 as 1/Infinity. They defined 1/Infinity
:> as 0. Defining 1/Infinity = 2/Infinity = 3/Infinity = 0
:> is no more problematic than defining 1*0 = 2*0 = 3*0 =0.
:> Do you think there is a problem with the fact that 1*0 = 2*0 = 3*0 = 0?

: It appears to me that the answer is yes, but see below for why.

You do think there is a problem with saying that 1*0 = 2*0?
Is there a problem saying that 3*4=2*6?

<snip>

:>
:> Honestly, I do not see what your point is. No one is defining 0 as


:> 1/infinity, and noone is redefining what 0 is in the middle
:> of a proof. We all know what 0 is. The question is, what is 1/infinity?
:> And there is no problem defining things such that 1/infinity = 2/infinity
:> = 3/infinity = 0.

: My point is that the normal laws of algebra *do* apply *if* you do not assume that
: 1/infinity = 2/infinity = 3/infinity = 0. To put it another way, I defy you (or
: challenge you) to come up with an example that violates the laws of algebra, when that
: example assumes that the relationship between infinity and zero is 0 = 1/infinity, and
: which does *not* assume that 1/infinity = 2/infinity = 3/infinity = 0.

Using just the definition that 0=1/infinity and the laws of algebra
we get

2/infinity = (1+1)/infinity = 1/infinity + 1/infinity = 0 + 0 = 0

This violates your assumption that 2/infinity != 1/infinity.
Or do you not believe that 0 + 0 = 0? You are defining 0
in a strange way, so maybe 0 + 0 equals something else.


: From still another point of view, my point is that we do not have to give up the laws


: of algebra when dealing with the infinite realm *if* we are willing to give up the idea
: that 1/infinity = 2/infinity = 3/infinity = 0.


No. We do not have to give up the laws of algebra if
1/infinity = 2/infinity = 3/infinity = 0. You cannot find
a contradiction if you just define x/infinity = 0 for all finite x.
It is a lot like saying that x*0 = 0 for all finite x.
The problems start when you define things like infinity-infinity
and 0 * infinity.

Stephen

Phil

unread,
Nov 25, 2002, 2:44:09 PM11/25/02
to
ste...@nomail.com wrote:

You may be right, but I've got to run some errands (either that, or I'm looking for an excuse
to think hard about what you are saying for a bit), so I'll have to get back a few hours from
now.

Phil


John Zinni

unread,
Nov 25, 2002, 2:58:58 PM11/25/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE279EC...@jump.net...

[bunch of crap sniped]

>To put it another way, I defy you (or
> challenge you) to come up with an example that violates the laws of
algebra, when that
> example assumes that the relationship between infinity and zero is 0 =
1/infinity, and
> which does *not* assume that 1/infinity = 2/infinity = 3/infinity = 0.

OK. This will be my last foray into this thread (I grow weary of this
game!!!).

From above:

0=1/infinity

From another post:

> Oops! Now it's my turn to get out the rust.
>

> g(x) = (x - 1)(x + 1)

> g(x) = (1 - 1)(1 + 1) = 0 * 2 = 2 * (1/Infinity), so
> g(x) * f(x) = (2/Infinity) * (Infinity) = 2 * (Infinity/Infinity) = 2

this seems to imply that you believe that:

Infinity/Infinity=1

and that:

1-1=0

Since you talk about "the laws of algebra" above, I will take a leap of
faith and assume that you will let me add 1 to each side of this equation,
so we have:

1=0+1

substituting in 1/infinity for 0 and infinity/infinity for 1 on the right
side, gives us:

1 = 1/infinity + infinity/infinity

1 = (1+infinity)/infinity

Multiplying each side by infinity yields:

infinity = 1 + infinity

subtracting infinity from each side gives us

0 = 1


OK. Tell me what's wrong with this one???


[more crap sniped]

--
Cheers
John Zinni

Randy Poe

unread,
Nov 25, 2002, 5:52:57 PM11/25/02
to
Phil wrote:

>
> John Zinni wrote:
> Wait a minute. Dan's argument includes the (admittedly unstated) assumption
> that 1/infinity = 0, and you can't simply change this definition in the
> middle of an argument/proof and expect to get valid results. Either
> 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.

If 1/infinity = 0, then it follows that a/infinity = 0 for all a.

a*(1/infinity) = a/infinity = a*0 = 0.

- Randy

Randy Poe

unread,
Nov 25, 2002, 5:57:52 PM11/25/02
to

OK. So you are saying 2*0 is not 0? Or what are you claiming
is the value of 2/infinity?

How about (2/infinity)-(1/infinity)? By the rules of
finite algebra, I say this is 1/infinity which we have
said is 0.

If 0 is the additive identity, that makes 2/infinity
equal to 1/infinity.

I'm a little confused about the rules of the system you
are proposing.

- Randy

John Zinni

unread,
Nov 25, 2002, 6:35:05 PM11/25/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE26120...@jump.net...

> John,
>
> I can only see one of them (I messed up and selected the "send both text
and
> HTML" option), and there have been other times when I could not see
someone's
> post. I have no idea why.
>
> So, blessing or curse, I'm going to send you another copy! Just to you.
;-)
>
> Phil

Hi Phil

OK, I've taken a look at what you sent me and WOW, that's quite a bit of
gobbledygook you've written.

One question?

YOU define the set of {+ive integers} to be:

{1, 2, 3, ..., I}, were I is infinity and is the largest element of the set.

In figure 10c you map the set of {+ive integers} to the set of {even +ive
integers} such that:

1->2, 2->4, 3->6, ..., x->2x, ..., and finally I->2I

Please tell me how 2I is an element of the {even +ive integers} but NOT an
element of the {+ive integers}???

In the set of {rational numbers} is 1/I twice as big as 1/(2I)???

--
Cheers
John Zinni

John Zinni

unread,
Nov 25, 2002, 6:55:17 PM11/25/02
to
"Randy Poe" <rp...@nospam.com> wrote in message
news:aru9k...@enews3.newsguy.com...

> Phil wrote:
> >
> > John Zinni wrote:

*** Please note: I (John Zinni) did not write this ***

Cheers
John Zinni

Phil

unread,
Nov 25, 2002, 8:34:37 PM11/25/02
to
ste...@nomail.com wrote:

> Phil <tu...@jump.net> wrote:
> : ste...@nomail.com wrote:
>
> :> Phil <tu...@jump.net> wrote:
> :> : ste...@nomail.com wrote:
> :>
> :> :> Phil <tu...@jump.net> wrote:
> :> :>
> :> :> : Wait a minute. Dan's argument includes the (admittedly unstated) assumption
> :> :> : that 1/infinity = 0, and you can't simply change this definition in the
> :> :> : middle of an argument/proof and expect to get valid results. Either
> :> :> : 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.
> :> :>
> :> :> What definition did he change? Do you agree that 1*0=0?
> :> :> Am I changing a definition if I say that 2*0=0? Do you think
> :> :> that 1*0=0 and 2*0=0 are mutually exclusive?
> :>
> :> : For normal, practical use they can be viewed as being equivalent, but when you
> :> : are dealing with infinity the answer is yes, if 0 = 1/Infinity, then 1 * 0 is
> :> : *not equal* to 2 * 0!
> :>
> :> No one ever defined 0 as 1/Infinity. They defined 1/Infinity
> :> as 0. Defining 1/Infinity = 2/Infinity = 3/Infinity = 0
> :> is no more problematic than defining 1*0 = 2*0 = 3*0 =0.
> :> Do you think there is a problem with the fact that 1*0 = 2*0 = 3*0 = 0?
>
> : It appears to me that the answer is yes, but see below for why.
>
> You do think there is a problem with saying that 1*0 = 2*0?

When 0 is defined as being 1/infinity, yes, there is a problem. Most of the examples posted
here go away when we stop assuming that 1/infinity = 2/infinity.

> Is there a problem saying that 3*4=2*6?

Not that I am aware of.

> <snip>
>
> :>
> :> Honestly, I do not see what your point is. No one is defining 0 as
> :> 1/infinity, and noone is redefining what 0 is in the middle
> :> of a proof. We all know what 0 is. The question is, what is 1/infinity?
> :> And there is no problem defining things such that 1/infinity = 2/infinity
> :> = 3/infinity = 0.
>
> : My point is that the normal laws of algebra *do* apply *if* you do not assume that
> : 1/infinity = 2/infinity = 3/infinity = 0. To put it another way, I defy you (or
> : challenge you) to come up with an example that violates the laws of algebra, when that
> : example assumes that the relationship between infinity and zero is 0 = 1/infinity, and
> : which does *not* assume that 1/infinity = 2/infinity = 3/infinity = 0.
>
> Using just the definition that 0=1/infinity and the laws of algebra
> we get
>
> 2/infinity = (1+1)/infinity = 1/infinity + 1/infinity = 0 + 0 = 0
>
> This violates your assumption that 2/infinity != 1/infinity.

Well, I "defied" you to come up with an example, and I'll be damned if you and John didn't
both promptly do so. Glad I didn't bet any money ...

Okay, here's the situation as I see it:

(1) If we assume that 1/infinity = 2/infinity = 0, we get into trouble, because we can prove
that 1 = 2 and such. Stating that 1/infinity = 0, but that it is not equal to 2/infinity
stops one class of problems from coming up with nonsensical answers, but not all problems.

(2) If 0 is defined (as David said) such that x + 0 = x, then there is *no way* that
1/infinity = 0, as you and John have just conclusively demonstrated.


> Or do you not believe that 0 + 0 = 0? You are defining 0
> in a strange way, so maybe 0 + 0 equals something else.

The "zero" I have been using is 1/infinity, and in this case 0 + 0 does not equal 0. However,
it clearly makes no sense to define 0 in such a way that x + 0 does not equal x, so I am
going to have to agree with all of you that there is no way that 1/infinity, whatever it is,
equals 0, where 0 is defined (as it should be) as having the property that x + 0 = x.

Well, if the shoe fits, put it on! Thanks Stephen, you and John have corrected and improved
my understanding in this matter. I usually find it most convenient to define infinity such
that there are exactly an infinite number of points on the line segment from 0 to 1 (and yes,
under this definition there are 2 * infinity points from 0 to 2), which then makes 1/infinity
the "distance" from one point to the next (this is meaningless in the finite realm, but not
in the infinitesimal realm). I should have realized, however, that even here, 1/infinity
cannot possibly have the property such that 1 + 1/infinity = 1. (This gets really fun if you
further declare that the 1st point to the right of 0 corresponds to 1, the 2nd point to 2,
and the 10000...(infinitly many zeros)...0000th point -- defined as the last of the
infinitely many points -- corresponds to 1. You get a sort of "decimal continuum" that
behaves very well algebraically, even when dealing with finite quantities of points.)

Okay, I am officially joining those who say that anyone who claims that 1/infinity = 0 is
full of crap.

> : From still another point of view, my point is that we do not have to give up the laws
> : of algebra when dealing with the infinite realm *if* we are willing to give up the idea
> : that 1/infinity = 2/infinity = 3/infinity = 0.
>
> No. We do not have to give up the laws of algebra if
> 1/infinity = 2/infinity = 3/infinity = 0. You cannot find
> a contradiction if you just define x/infinity = 0 for all finite x.
> It is a lot like saying that x*0 = 0 for all finite x.
> The problems start when you define things like infinity-infinity
> and 0 * infinity.

This I will have to think about some more, but at first glance, it looks like you are correct
here as well.

Phil


Phil

unread,
Nov 25, 2002, 8:41:29 PM11/25/02
to
David McAnally wrote:

> Phil <tu...@jump.net> writes:
>
> >No, remember that we are *defining* zero as 1/Infinity, so 2 * 0 = 2/Infinity,
> >not 1/Infinity. It then follows that:
>
> That is not how zero is defined. Zero is the additive identity.

Well, Stephen and John both came up with examples that prove to me that the claim
that 0 = 1/infinity is unavoidably inconsistent with the claim that x + 0 = x, so you
are right, and I was wrong (I am now a firm believer that anyone who says that
1/infinity = 0 is full of crap!).

However, I still don't see a logical flaw in my criticism of Cantor's proof
concerning the even numbers and the integers. Well, we will see. Time may or may not
prove me wrong on that one as well.

Phil

Phil

unread,
Nov 25, 2002, 8:48:55 PM11/25/02
to

John Zinni wrote:

Not a damn thing.

Well I challenged Stephen to give an example, and you and Stephen both
delivered. I gave a more complete response to Stephen's post, but what it
amounts to is that I now believe that the claim that 1/infinity = 0 is
completely incompatible with the claim that x + 0 = x, which makes it
(1/infinity = 0) flat out wrong in my book.

Still waiting to see if anyone comes up with a logical flaw in my criticism of
Cantor's proof concerning the number of even numbers versus the number of
integers. :-)

Beaten but Wiser Phil


ste...@nomail.com

unread,
Nov 25, 2002, 9:09:22 PM11/25/02
to
Phil <tu...@jump.net> wrote:
: David McAnally wrote:

:> Phil <tu...@jump.net> writes:
:>
:> >No, remember that we are *defining* zero as 1/Infinity, so 2 * 0 = 2/Infinity,
:> >not 1/Infinity. It then follows that:
:>
:> That is not how zero is defined. Zero is the additive identity.

: Well, Stephen and John both came up with examples that prove to me that the claim
: that 0 = 1/infinity is unavoidably inconsistent with the claim that x + 0 = x, so you
: are right, and I was wrong (I am now a firm believer that anyone who says that
: 1/infinity = 0 is full of crap!).

: However, I still don't see a logical flaw in my criticism of Cantor's proof
: concerning the even numbers and the integers. Well, we will see. Time may or may not
: prove me wrong on that one as well.

: Phil

You are wrong about Cantor as well. I do not understand why you
think that either Cantor's proofs or the proofs that about
0 * infinity = 1 leading to contradictions involve a "shift".
You still do not seem to understand that defining 1/infinity as 0
does not cause any problems.

I will now try to convince you that you are wrong about Cantor.

Two sets are said to have the same cardinality if there exists
a bijection between the sets. This is a definition. |A|=|B|
if there exists a bijection f: A -> B. This definition never
changes during any proof of the countability or uncountability
of sets. You may not like that definition, but Cantor used
it consistently.

Anyway, consider the sets
A= b{a,b}* (the set of all strings beginning with a b followed
by any number of a's and b's.)
B= b{a,b}*a (the set of all strings beginning with a b, followed
by any number of a's and b's, and ending with an a.)

There is a bijection between these sets

A = ( b, ba, bb, baa, bab, bba, bbb, baaa, baab, baba, ...

B = ( ba, baa, bba, baaa, baba, bbaa, bbba, baaaa, baaba, babaa, ...

so |A|=|B|, which is pretty intuitive. If we simply added an 'a' to
the end of each string in A, it should not change the number
of elements in the set.

What if we replace every 'a' with '0' and every 'b' with '1'.
We then get two new sets

C = ( 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, ...

D = ( 10, 100, 110, 1000, 1010, 1100, 1110, 10000, 10010, 10100, ...

Simply changing the symbols in the string should not change
the size of the sets. If you have a roomful of people and you
change all of their names, you would not expect the number of
people in the room to change. So it follows that |C|=|A|=|B|=|D|.

But what if we interpret C and D as sets of binary numbers?

C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

D'= ( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

Clearly |C'|=|C|=|D|=|D'|. Why would how we interpret the elements of
a set affect the number of elements in a set?

If you do not agree, what exactly do you not agree with?
Do you think that interpreting a set of strings as binary
numbers suddenly causes half of the elements to vanish?
Do you think that renaming the elements of a set affects
the number of elements in the set? Do you think that
adding a symbol to each element in a set affects the
number of elements in the set? None of the above are true
for finite sets. Why should they not be true for infinite
sets?

Stephen

Stephen Speicher

unread,
Nov 25, 2002, 10:30:08 PM11/25/02
to
On Sun, 24 Nov 2002, John Zinni wrote:

> "David McAnally" <D.McAnally@i'm_a_gnu.uq.net.au> wrote in message

> news:arr068$ma3$1...@bunyip.cc.uq.edu.au...
> >
> > Cantor's transfinite numbers are infinities. ...
>
> It's good to see that there are more than a few people in this news group
> that "know of what they speak" (I was beginning to have my doubts). I will
> keep an eye out for your posts.
>

David is, indeed, a knowledgeable poster, and I wish we heard
more from him on detailed technical areas.

--
Stephen
s...@speicher.com

Ignorance is just a placeholder for knowledge.

Printed using 100% recycled electrons.
-----------------------------------------------------------

John Zinni

unread,
Nov 25, 2002, 9:23:54 PM11/25/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE2D32D...@jump.net...

HALLELUJAH!!!


> Well I challenged Stephen to give an example, and you and Stephen both
> delivered. I gave a more complete response to Stephen's post, but what it
> amounts to is that I now believe that the claim that 1/infinity = 0 is
> completely incompatible with the claim that x + 0 = x, which makes it
> (1/infinity = 0) flat out wrong in my book.
>
> Still waiting to see if anyone comes up with a logical flaw in my
criticism of
> Cantor's proof concerning the number of even numbers versus the number of
> integers. :-)

I'll have to take another look, but you are already in trouble with your
definition of the set of {+ive integers}.

>
> Beaten but Wiser Phil

Cheers
John Zinni

John Zinni

unread,
Nov 25, 2002, 9:52:28 PM11/25/02
to
"Stephen Speicher" <s...@speicher.com> wrote in message
news:Pine.LNX.4.33.02112...@localhost.localdomain...

> On Sun, 24 Nov 2002, John Zinni wrote:
>
> > "David McAnally" <D.McAnally@i'm_a_gnu.uq.net.au> wrote in message
> > news:arr068$ma3$1...@bunyip.cc.uq.edu.au...
> > >
> > > Cantor's transfinite numbers are infinities. ...
> >
> > It's good to see that there are more than a few people in this news
group
> > that "know of what they speak" (I was beginning to have my doubts). I
will
> > keep an eye out for your posts.
> >
>
> David is, indeed, a knowledgeable poster, and I wish we heard
> more from him on detailed technical areas.

Yes, I took the liberty of looking him up on the web and this is indeed
true.

I also did a search on the "Surreal Numbers" that he mentioned in his post
because I had never heard of them before. I found this readable (???) 50
page introduction to them.
http://www.tondering.dk/claus/sur12.pdf
They are indeed "surreal"!!!

I am kicking around a question about them that may have cosmological
implications (if I am understanding things correctly and this is by no means
certain) and will therefore be appropriate to this group. I need to work on
it a little to ensure that it is, hopefully, well posed.

Cheers
John Zinni


Stephen Speicher

unread,
Nov 26, 2002, 12:42:39 AM11/26/02
to
On Mon, 25 Nov 2002, John Zinni wrote:
>
> I also did a search on the "Surreal Numbers" that he mentioned in his post
> because I had never heard of them before. I found this readable (???) 50
> page introduction to them.
> http://www.tondering.dk/claus/sur12.pdf
> They are indeed "surreal"!!!
>
> I am kicking around a question about them that may have cosmological
> implications (if I am understanding things correctly and this is by no means
> certain) and will therefore be appropriate to this group. I need to work on
> it a little to ensure that it is, hopefully, well posed.
>

You might enjoy Tony Smith's pages, which are chock-full of some
of the more interesting mathematics related to quantum theory and
relativity. See his discussion on surreal numbers at

http://www.innerx.net/personal/tsmith/surreal.html

Afterwards search starting at

http://www.innerx.net/personal/tsmith/

I'm willing to bet you will learn about some groups which you
never thought existed. There is also a lot of material on
Clifford algebras, and alas, we recently lost Pertti Lounesto, a
true expert in the field, whose book "Clifford Algebras and
Spinors", Cambridge University Press, 1997/99 remains, in my
humble opinion, the finest introduction to the subject.

Phil

unread,
Nov 26, 2002, 4:34:40 AM11/26/02
to

ste...@nomail.com wrote:

> Phil <tu...@jump.net> wrote:
> : David McAnally wrote:
>
> :> Phil <tu...@jump.net> writes:
> :>
> :> >No, remember that we are *defining* zero as 1/Infinity, so 2 * 0 = 2/Infinity,
> :> >not 1/Infinity. It then follows that:
> :>
> :> That is not how zero is defined. Zero is the additive identity.
>
> : Well, Stephen and John both came up with examples that prove to me that the claim
> : that 0 = 1/infinity is unavoidably inconsistent with the claim that x + 0 = x, so you
> : are right, and I was wrong (I am now a firm believer that anyone who says that
> : 1/infinity = 0 is full of crap!).
>
> : However, I still don't see a logical flaw in my criticism of Cantor's proof
> : concerning the even numbers and the integers. Well, we will see. Time may or may not
> : prove me wrong on that one as well.
>
> : Phil
>
> You are wrong about Cantor as well. I do not understand why you
> think that either Cantor's proofs or the proofs that about
> 0 * infinity = 1 leading to contradictions involve a "shift".
> You still do not seem to understand that defining 1/infinity as 0
> does not cause any problems.

If 1/infinity = 0, and also 0 + x = x, then

1/infinity + 1/infinity = 1/infinity

1/infinity * (1 + 1) = 1/infinity * (1)

2 = 1

Now at this point, having been soundly thrashed concerning my belief that consistently
defining 0 as 1/infinity never leads to problems, I am not willing to declare absolutely
that the above sequence cannot be legitimately rewritten so that it is not a problem, but
it certainly looks like a problem to me.

> I will now try to convince you that you are wrong about Cantor.
>
> Two sets are said to have the same cardinality if there exists
> a bijection between the sets. This is a definition. |A|=|B|
> if there exists a bijection f: A -> B. This definition never
> changes during any proof of the countability or uncountability
> of sets. You may not like that definition, but Cantor used
> it consistently.

Okay, but am I not correct in saying that two sets have the same cardinality if and only if
it is possible to match all of the elements of one set with all of the elements of the
other set, with no elements in either set "left over" (left unmatched or unpaired)? A set
has the same cardinality as the denumerable set (aleph 0?) *only* if it is possible to pair
off each element of the set in question with an element from the denumerable set. I think
we are in complete agreement on these points, but I want to make sure.

> Anyway, consider the sets
> A= b{a,b}* (the set of all strings beginning with a b followed
> by any number of a's and b's.)
> B= b{a,b}*a (the set of all strings beginning with a b, followed
> by any number of a's and b's, and ending with an a.)
>
> There is a bijection between these sets
>
> A = ( b, ba, bb, baa, bab, bba, bbb, baaa, baab, baba, ...
>
> B = ( ba, baa, bba, baaa, baba, bbaa, bbba, baaaa, baaba, babaa, ...
>
> so |A|=|B|, which is pretty intuitive. If we simply added an 'a' to
> the end of each string in A, it should not change the number
> of elements in the set.
>
> What if we replace every 'a' with '0' and every 'b' with '1'.
> We then get two new sets
>
> C = ( 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, ...
>
> D = ( 10, 100, 110, 1000, 1010, 1100, 1110, 10000, 10010, 10100, ...
>
> Simply changing the symbols in the string should not change
> the size of the sets. If you have a roomful of people and you
> change all of their names, you would not expect the number of
> people in the room to change. So it follows that |C|=|A|=|B|=|D|.
>
> But what if we interpret C and D as sets of binary numbers?
>
> C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
>
> D'= ( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
>
> Clearly |C'|=|C|=|D|=|D'|. Why would how we interpret the elements of
> a set affect the number of elements in a set?

Beautifully written, this is as good a way as I have seen for stopping the natural tendency
to think that "1" must be paired with the "first" element of the other set, along with the
idea that the fact that an element happens to look like a symbol representing a particular
number (or position on the real number line) should tell us something about which element
of another set it should be paired with (be honest, someone else wrote this and you just
copied it, right? ;-) ).

> If you do not agree, what exactly do you not agree with?

I agree with everything you have said so far.

> Do you think that interpreting a set of strings as binary
> numbers suddenly causes half of the elements to vanish?

Nope.

> Do you think that renaming the elements of a set affects
> the number of elements in the set? Do you think that
> adding a symbol to each element in a set affects the
> number of elements in the set?

Nope to all of the above.

> None of the above are true
> for finite sets. Why should they not be true for infinite
> sets?

They obviously are true for infinite sets. My objection refers to another possible type of
error altogether. I will use your diagrams to demonstrate:

C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

D'= ( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

This is the normal pairing used to demonstrate that the set of even numbers has the same
cardinality as the set of integers. Now, I will pair these same two sets, but in a
different way (adding vertical lines to emphasize the pairing).

C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
| | | | |

D'= ( 2, 4, 6, 8, 10, ...

As before, every element of the set of even numbers has been paired with an element of the
set of integers. There are no elements from the set of even numbers left unpaired. However,
fully half of the elements from the set of integers are not paired with an element from the
set of integers, suggesting that these two sets do not have the same cardinality. A similar
but slightly stronger example follows:

E'= ( 2, 4, 6, 8, 10, ...


| | | | |
C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
| | | | |

D'= ( 2, 4, 6, 8, 10, ...

Here we have two sets of the even numbers, and every element from both sets is paired with
a single element from the set of integers. In this case, no elements from any set are left
unpaired. However, the fact that the set of integers has the same cardinality as the two
sets of even numbers put together proves that *this particular* set of integers has twice
as many elements as *these two particular* sets of even numbers (and the emphasis on
"particular" is both important and necessary).

Does this by itself falsify Cantor's conclusion that in the denumerable set, the number of
even numbers is exactly equal to the number of integers? No it does not. However, it should
damn sure give you pause, and make you wonder why the two different methods for pairing the
elements of these sets (and both methods are completely valid) lead to such different
conclusions.

Let me start by stating that the *particular sets* used in Cantor's example do indeed have
the same cardinality, and the *particular sets* used in my examples do not. The method of
establishing a one-to-one correspondence between the elements of two sets is too basic and
reliable to allow for any other conclusion. Therefore, any flaws or errors *must* have
occurred in the conclusions which followed after a bijection or lack thereof was
established for the sets in question (the two conclusions were "in the denumerable set, the
even numbers are as numerous as the integers," and "in the denumerable set, the even
numbers are half as numerous as the integers"). I hope I am using the term "bijection"
correctly; I assume it means the ability to establish a one-to-one correspondence between
the elements of two sets with no elements from either set left over, thereby proving that
the two sets have the same cardinality.

One possible error is an accidental redefinition. For example, let X = the denumerable set,
the set of all positive integers. Now let Y = a set with just one element, which we
*define* as being a positive integer which is *not* contained in X (for convenience, we
will call it "larger" than any element in X). Obviously these two definitions are
incompatible, but suppose for a moment that someone accepted that the element in Y was
indeed a positive integer which was greater than any element in X. In the process of
accepting this definition, he will, whether he realizes it or not, effectively *redefine* X
as being the set of all positive integers except one, namely the element contained in Y.
This is an unavoidable consequence of accepting the definition for Y. The act of accepting
this definition for Y *automatically* redefines X, because X's original definition is
incompatible with the definition for Y.

The question we have to ask now, of course, is whether a similar error exists in Cantor's
proof that *in the denumerable set itself*, the even numbers are as numerous as the
integers. We will begin with the diagram for Cantor's proof, adding some information.

C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
| | | | | | | | | |
D'= ( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

Cantor began by *defining* C' as being the denumerable set, the set of all positive
integers, after which he then *defined* D' as the set of all even numbers. He then provided
a method for matching every element in C' with a unique element in D'. The question we have
to ask is whether any of these steps contain "incompatible definitions," which have the
effect of *redefining* C' into being something other than the denumerable set. The diagram
below provides the answer.

C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
| | | | | | | | | |
D'= ( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...

At first, pretend that the line of numbers just below the line for D' is not there. We
start out with the normal diagram for cantor's proof, in which the infinite set of integers
C' does indeed have exactly the same number of elements as the infinite set of even numbers
D', just as Cantor said. We also acknowledge that C' was originally defined as being the
denumerable set. Now let's add another infinite set of numbers, the set of odd numbers, to
D'. Nothing has changed regarding the matching between the elements of C' and the even
numbers of D'. These elements were matched before, and they remain matched. Only the newly
added odd numbers in D' are not matched with any of the elements in C'.

Now let's look at D'. It contains the complete infinite set of even numbers, and it
contains the complete infinite set of odd numbers ... it's the denumerable set! It is the
denumerable set, and it contains exactly twice as many elements as C', the set which was
*intially defined* as being the denumerable set. In the process of declaring that each
element of C' was matched with a unique element from D', Cantor *accidently redefined* C'
as being an infinite set of integers containing exactly half as many elements as the
denumerable set. This was his mistake, and it is the reason why it was so easy to find
alternate matching schemes that led to the conclusion that there are half as many even
numbers as integers. There really are half as many even numbers as integers! In my
diagrams, the process of first defining C' as the denumerable set and then showing that
only half of its elements are needed to account for the entire set of even numbers in D'
(or E') does *not* involve an accidental redefinition -- at least, not as far as I can see!
-- which then really does prove that the even numbers are half as numerous.

And if that isn't good enough for you, let me share with you a jewel that was given to me
by the Dichotomy. Consider a room with 1 door, 8 m long from the door to the opposite wall.
Starting from the wall, the point halfway to the door, 4 m from the door, is defined as the
1/2 point. From there, the next halfway point to the door, 2 m from the door, is the 1/4
point, with infinitely many more halfway points 1/8 at 1 m, 1/16 at 0.5 m, 1/32 at 0.25 m,
etc. remaining to the door (so far this is actually the Achilles). Call this set of halfway
points from the center of the room (inclusive) to the door, set Z. A person walking into
the room (switching to the Dichotomy) will initially cross an infinite number of these
halfway points in an infinitesimal distance and in an infinitesimal amount of time, after
which he will begin to cross the remaining halfway points at a finite rate until he reaches
the last halfway point, at the center of the room. An interesting point is that at the 1/8
point, 1 m into the room, he will have crossed all but two of the halfway points contained
in the infinite set Z. So, to those who claim that there are just as many halfway points
from the 1/8 point to the door as there are from the center point to the door (i.e., (Z -
2) = Z) -- and of course, if infinity/2 = infinity, equivalent to the number of evens and
integers, then without question (infinity - 2) = infinity -- my response is to walk 1 m
into the room and say, "If these two sets are equal, then why aren't I standing in the
middle of the room?" ;-)

Phil


Ahmed Ouahi, Architect

unread,
Nov 26, 2002, 7:38:56 AM11/26/02
to

.......... ... I have had never private myself in culturing
sciences.......... ...As around seventy two years of a thought, made by
days and nights, I have had only constate my entire
ignorance!!!!!!!!!!!!!!.......... ...

--OMAR KHAYYAM, 11th Century
Poet, Mathematician and Astronom

--
Ahmed Ouahi, Architect
Best Regards!


"Phil" <tu...@jump.net> kirjoitti viestissä:3DE3404E...@jump.net...

John Zinni

unread,
Nov 26, 2002, 8:51:35 AM11/26/02
to
"Stephen Speicher" <s...@speicher.com> wrote in message
news:Pine.LNX.4.33.02112...@localhost.localdomain...
> You might enjoy Tony Smith's pages, which are chock-full of some
> of the more interesting mathematics related to quantum theory and
> relativity. See his discussion on surreal numbers at
>
> http://www.innerx.net/personal/tsmith/surreal.html
>
> Afterwards search starting at
>
> http://www.innerx.net/personal/tsmith/

Hi Stephen

Thanks for the links. I have taken a quick look at the first one and it
appears to answer (or at least touch on) the questions I had in mind but was
not able to properly pose yet. Exciting stuff. I've got some reading to
do!!!

>
> I'm willing to bet you will learn about some groups which you
> never thought existed. There is also a lot of material on
> Clifford algebras, and alas, we recently lost Pertti Lounesto, a
> true expert in the field, whose book "Clifford Algebras and
> Spinors", Cambridge University Press, 1997/99 remains, in my
> humble opinion, the finest introduction to the subject.
>
> --
> Stephen
> s...@speicher.com
>
> Ignorance is just a placeholder for knowledge.
>
> Printed using 100% recycled electrons.
> -----------------------------------------------------------
>

--
Cheers
John Zinni

Randy Poe

unread,
Nov 26, 2002, 10:20:56 AM11/26/02
to
Phil wrote:

>
> ste...@nomail.com wrote:
>>Two sets are said to have the same cardinality if there exists
>>a bijection between the sets. This is a definition. |A|=|B|
>>if there exists a bijection f: A -> B. This definition never
>>changes during any proof of the countability or uncountability
>>of sets. You may not like that definition, but Cantor used
>>it consistently.
>
>
> Okay, but am I not correct in saying that two sets have the same cardinality if and only if
> it is possible to match all of the elements of one set with all of the elements of the
> other set, with no elements in either set "left over" (left unmatched or unpaired)?

Yes. The short version of this statement is "there exists a bijection
between the two sets."

> A set
> has the same cardinality as the denumerable set (aleph 0?) *only* if it is possible to pair
> off each element of the set in question with an element from the denumerable set. I think
> we are in complete agreement on these points, but I want to make sure.
>

[snip]

> C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
>
> D'= ( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
>
> This is the normal pairing used to demonstrate that the set of even numbers has the same
> cardinality as the set of integers. Now, I will pair these same two sets, but in a
> different way (adding vertical lines to emphasize the pairing).
>
> C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
> | | | | |
> D'= ( 2, 4, 6, 8, 10, ...

I think I see where you're going.

>
> As before, every element of the set of even numbers has been paired with an element of the
> set of integers. There are no elements from the set of even numbers left unpaired. However,
> fully half of the elements from the set of integers are not paired with an element from the
> set of integers, suggesting that these two sets do not have the same cardinality.

Nope. That's not the definition you accepted.

The definition is "there exists a mapping which is a bijection",
not "ALL mappings between the sets are bijections."

It still allows the possibility that "there also exists other mappings
which are not bijections." You've just illustrated above that
there is a mapping between Z and 2Z which is a bijection,
so that meets the definition you accepted above.

What would you do with the following example?
C'= { 1, 2, 3, 4, 5, 6,...}
| | | | | |
C'= {1, 2, 3, 4, 5, 6, 7, ...}

This mapping takes every element of C' to an element of C',
but it leaves one element (1) unmapped. Thus, by your
intuition, it proves that C' is smaller than itself, by
one element.

One property of infinite sets, in fact one way to define
"infinite set", is that there exists a bijection between
S and a proper subset of itself.

- Randy

John Zinni

unread,
Nov 26, 2002, 11:19:29 AM11/26/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE3404E...@jump.net...

[snip]

> Okay, but am I not correct in saying that two sets have the same
cardinality if and only if
> it is possible to match all of the elements of one set with all of the
elements of the
> other set, with no elements in either set "left over" (left unmatched or
unpaired)? A set
> has the same cardinality as the denumerable set (aleph 0?) *only* if it is
possible to pair
> off each element of the set in question with an element from the
denumerable set. I think
> we are in complete agreement on these points, but I want to make sure.

No, this is not true.

Given two sets A and B, |A| = |B| if (NOT only if) there is a (at least one)
bijection between A and B.

A bijection means that there is a 1 to 1 mapping of A onto B and that the
inverse mapping of B onto A is also 1 to 1 (hence the "bi" in bijection, I
think)

So, if we can find at least one bijection between A and B then |A| = |B|.

If there is a bijection between A and B, the fact that we can find other
mappings between A and B that are not bijections does not constitute a
counterexample to |A| = |B| since we know that there is still "at least
one" bijection between A and B.


Now, if we let A = {+ive integers} and B = {+ive integers} and construct a
mapping such that:

1->2, 2->4, 3->6, ..., n->2n, ...

(This is essentially what you do somewhere in your post)

So, every element in A is mapped to one and only one element of B but we
still have half of the elements of B left over.

You would like to conclude from this that |A| is one half of |B|, but this
is not the case because we know that there is a bijection between A and B,
namely:

1->1, 2->2, 3->3, ..., n->n, ...

[snip]

--
Cheers
John Zinni

John Zinni

unread,
Nov 26, 2002, 11:49:41 AM11/26/02
to
Now, lets see if I can convince you that Infinity (I) is NOT an element of
the {+ive integers).
(I don't think that this is going to be rigorous but we'll give it a shot
anyway)

You would like to define the set of {+ive integers} as:

Z+ = {1, 2, 3, ..., n, ..., I}

Lets consider the set of {+ive rationals}

Each element of the {+ive rationals} has the form a/b were a, b are elements
of Z+

Lets set a=1 (1 is certainly an element of Z+)

I think it is clear that:

1/b=q, were b is an element of Z+ and 0<q<=1 (q is finite and rational)

Now since b can be any element of Z+ lets select I

1/I=q, were 0<q<=1 (q is finite an rational)

This fits in well with the fact that we have convinced you that 1/I!=0

Given x = y, were x, y are elements of {+ive rationals}

1/x = 1/y and 1/x, 1/y are also elements of {+ive rationals}

So we have

I = 1/q

q is finite and rational so 1/q is finite and rational but I = 1/q which
contradicts the definition of I

So I CANNOT be an element of Z+.

--
Cheers
John Zinni

John Zinni

unread,
Nov 26, 2002, 12:09:26 PM11/26/02
to
"John Zinni" <j_z...@sympatico.ca> wrote in message
news:2bNE9.285$Nm.1...@news20.bellglobal.com...

> "Phil" <tu...@jump.net> wrote in message news:3DE3404E...@jump.net...
>
> [snip]
>
> > Okay, but am I not correct in saying that two sets have the same
> cardinality if and only if
> > it is possible to match all of the elements of one set with all of the
> elements of the
> > other set, with no elements in either set "left over" (left unmatched or
> unpaired)? A set
> > has the same cardinality as the denumerable set (aleph 0?) *only* if it
is
> possible to pair
> > off each element of the set in question with an element from the
> denumerable set. I think
> > we are in complete agreement on these points, but I want to make sure.
>
> No, this is not true.
>
> Given two sets A and B, |A| = |B| if (NOT only if) there is a (at least
one)
> bijection between A and B.

Opps! Turns out it should be iff.

The rest below stands though (I think).

ste...@nomail.com

unread,
Nov 26, 2002, 9:16:20 AM11/26/02
to
Phil <tu...@jump.net> wrote:


: ste...@nomail.com wrote:

:> Phil <tu...@jump.net> wrote:
:> : David McAnally wrote:
:>
:> :> Phil <tu...@jump.net> writes:
:> :>
:> :> >No, remember that we are *defining* zero as 1/Infinity, so 2 * 0 = 2/Infinity,
:> :> >not 1/Infinity. It then follows that:
:> :>
:> :> That is not how zero is defined. Zero is the additive identity.
:>
:> : Well, Stephen and John both came up with examples that prove to me that the claim
:> : that 0 = 1/infinity is unavoidably inconsistent with the claim that x + 0 = x, so you
:> : are right, and I was wrong (I am now a firm believer that anyone who says that
:> : 1/infinity = 0 is full of crap!).
:>
:> : However, I still don't see a logical flaw in my criticism of Cantor's proof
:> : concerning the even numbers and the integers. Well, we will see. Time may or may not
:> : prove me wrong on that one as well.
:>
:> : Phil
:>
:> You are wrong about Cantor as well. I do not understand why you
:> think that either Cantor's proofs or the proofs that about
:> 0 * infinity = 1 leading to contradictions involve a "shift".
:> You still do not seem to understand that defining 1/infinity as 0
:> does not cause any problems.

: If 1/infinity = 0, and also 0 + x = x, then

: 1/infinity + 1/infinity = 1/infinity

: 1/infinity * (1 + 1) = 1/infinity * (1)

Where did this step come from? What rule did you use?

: 2 = 1

: Now at this point, having been soundly thrashed concerning my belief that consistently
: defining 0 as 1/infinity never leads to problems, I am not willing to declare absolutely
: that the above sequence cannot be legitimately rewritten so that it is not a problem, but
: it certainly looks like a problem to me.

No, because you used something other than the fact that 1/infinity=0.
I will let you figure out what you did wrong.


:> I will now try to convince you that you are wrong about Cantor.


:>
:> Two sets are said to have the same cardinality if there exists
:> a bijection between the sets. This is a definition. |A|=|B|
:> if there exists a bijection f: A -> B. This definition never
:> changes during any proof of the countability or uncountability
:> of sets. You may not like that definition, but Cantor used
:> it consistently.

: Okay, but am I not correct in saying that two sets have the same cardinality if and only if
: it is possible to match all of the elements of one set with all of the elements of the
: other set, with no elements in either set "left over" (left unmatched or unpaired)? A set
: has the same cardinality as the denumerable set (aleph 0?) *only* if it is possible to pair
: off each element of the set in question with an element from the denumerable set. I think
: we are in complete agreement on these points, but I want to make sure.


That is the definition. Note it does not say that no other mappings
exist between the two sets. Obviously all sorts of mappings exist.

<snip>

: Beautifully written, this is as good a way as I have seen for stopping the natural tendency


: to think that "1" must be paired with the "first" element of the other set, along with the
: idea that the fact that an element happens to look like a symbol representing a particular
: number (or position on the real number line) should tell us something about which element
: of another set it should be paired with (be honest, someone else wrote this and you just
: copied it, right? ;-) ).

No. That is an example I cooked up, although I did get some inspiration
from the folks over at sci.math. Refutations of Cantor are relatively
common over there.

:> If you do not agree, what exactly do you not agree with?

: I agree with everything you have said so far.

:> Do you think that interpreting a set of strings as binary
:> numbers suddenly causes half of the elements to vanish?

: Nope.

:> Do you think that renaming the elements of a set affects
:> the number of elements in the set? Do you think that
:> adding a symbol to each element in a set affects the
:> number of elements in the set?

: Nope to all of the above.

:> None of the above are true
:> for finite sets. Why should they not be true for infinite
:> sets?

: They obviously are true for infinite sets. My objection refers to another possible type of
: error altogether. I will use your diagrams to demonstrate:

But if you agree with the above, then you agree that there
are the set of even integers and the set of integers have
the same cardinality.


: C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

: D'= ( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

: This is the normal pairing used to demonstrate that the set of even numbers has the same
: cardinality as the set of integers. Now, I will pair these same two sets, but in a
: different way (adding vertical lines to emphasize the pairing).

: C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
: | | | | |
: D'= ( 2, 4, 6, 8, 10, ...


Yes, but that has nothing to do with the definition of cardinality.
Consider


C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
| | | | |

C'= ( 1, 2, 3, 4, 5, ...

I can map C' to itself as above. Does this mean that |C'|<|C'|.

A mapping cannot change the number of elements in a set.
It is obviously not true for finite sets, why would it be
true for infinite sets?

<snip>

: Here we have two sets of the even numbers, and every element from both sets is paired with


: a single element from the set of integers. In this case, no elements from any set are left
: unpaired. However, the fact that the set of integers has the same cardinality as the two
: sets of even numbers put together proves that *this particular* set of integers has twice
: as many elements as *these two particular* sets of even numbers (and the emphasis on
: "particular" is both important and necessary).

There are no "particular" sets. If two sets have the same elements,
then they are the same, and if two sets are the same,
then they have the same cardinality. And all your example shows
is that the cardinality of the union of two countably infinite
sets equals the cardinality of another countably infinite set.
Does it surprise you that infinity + infinity = infinity ?

None of the rest of your stuff is worth responding to.
There is no way that defining a mapping between two sets
redefines either set. Likewise, interpreting strings as
binary numbers does not make half of them vanish. There is no
"shift" in Cantor's proof, other than your unwillingness to
accept a seemingly unintuitive result. Last, Zeno's paradox and
Cantor really have nothing to do with each other.

Stephen

Phil

unread,
Nov 27, 2002, 2:08:15 AM11/27/02
to
John Zinni wrote:

> Now, lets see if I can convince you that Infinity (I) is NOT an element of
> the {+ive integers).
> (I don't think that this is going to be rigorous but we'll give it a shot
> anyway)
>
> You would like to define the set of {+ive integers} as:
>
> Z+ = {1, 2, 3, ..., n, ..., I}

I will digress for a minute and point out that "I" is not meant to be a
variable, but the symbol for infinity, and as such, should not be treated like
some arbitrary large number, even a number as large as we please. One of the
first things the Achilles taught me is that infinity is *not* "a large number,
as large as we please." The infinite set of halfway points between us and the
door includes *all* of the halfway points between us and the door. This
necessarily follows from the fact that if we (or our mathematical equivalents)
merely crossed "a large number of halfway points, as large as we please," then
we really couldn't ever leave the room, because there would always be infinitely
more halfway points between us and the door. To repeat the examples I gave to
David, however,

"After you jump up and down an infinite number of times, how many more times can


you jump up and down? None, your jumping days are over. After you think an

infinite number of thoughts, how many more thoughts can you think? None, your
mind just shut down, permanently. And after you have crossed the infinite set of
halfway points between you and the door, how many more halfway points can you


cross? None, you just walked out the door."

To our minds this seems impossible, and for good reason; in the finite realm
where our minds operate, this *isn't true*. There are always more halfway
points, more thoughts, more numbers, but things that are true in the finite
realm are not always true in either the infinite or infinitesimal realms. The
unavoidable fact is that we can indeed walk out of a room, and in the process we
cross not "as large a number of halfway points as we please," but rather *all*
of the halfway points. To refer to infinity as a number as large as we please is
an understandable attempt to bring something that is literally incomprehensible
into the finite realm, where our minds operate. To do so is a mistake, however,
because it makes an already extremely difficult subject even more prone to
errors and misunderstanding.

> Lets consider the set of {+ive rationals}
>
> Each element of the {+ive rationals} has the form a/b were a, b are elements
> of Z+
>
> Lets set a=1 (1 is certainly an element of Z+)
>
> I think it is clear that:
>
> 1/b=q, were b is an element of Z+ and 0<q<=1 (q is finite and rational)
>
> Now since b can be any element of Z+ lets select I
>
> 1/I=q, were 0<q<=1 (q is finite an rational)

Whoa, where did you get the idea that q is finite? If I is a finite number, even
a number "as large as we please," then yes, q is also finite, but I is not
finite, forcing 1/q to be infinitesimal. The term "infinitesimal" is somewhat
crowded, because it can refer to the "realm of points," in which we mentally
picure points on a number line strung out like beads on a necklace (from this
perspective, any finite length in the "normal" realm appears to be infinitely
long), or it can refer to objects which are infinitely larger than points, but
infinitely smaller than any finite length, a concept that has been usually been
rejected over the last 2,400 years or so (Cantor also rejected them). In this
case, however, it simply refers to something that is smaller than any finite
length whatsoever, something that always *appears* to have a length of zero when
viewed from the finite realm (even if we use a "mathematical microscope" that
can use any *finite* magnification whatsoever). One possible result for
1/infinity is that it represents a movement of a single point in the
infinitesimal realm, although I am still fuzzy on this issue.

> This fits in well with the fact that we have convinced you that 1/I!=0

Except for the finite part, very well indeed! By the way, when did != replace <>
for "not equal?" I haven't keep track for 10 or 20 years, but ...

Phil

Phil

unread,
Nov 27, 2002, 3:22:33 AM11/27/02
to
Randy Poe wrote:

Okay, but understand that since I am using the rules for science, as opposed to the rules for
debate, I will feel free to correct any definitions I may have accepted once it becomes clear
that those definitions have problems. Of course, this does *not* mean that I can change them
merely to win an argument; there have to be real flaws before that is permitted.

In this case, I think I see two flaws. First, if I accept that *any* mapping between two sets
that *appears* to be a bijection proves that the two sets have the same cardinality, I am all but
accepting Cantor's conclusion as a premise. Here, I want to see if those mappings are actually
valid, and not merely state that since a supposedly valid mapping exist, that he must be correct.
What you are asking me to do is equivalent to trying to test the parallel postulate while
accepting that the sum of the angles of a triangle is 180 degrees. If I accept the latter, then
the former is automatically true, and I have merely provided a premise, not a proof.

Second, in situations where there appear to be two valid mappings, one which is a bijection and
one which is not, what evidence can you or Cantor provide that shows that the bijection mapping
*must* be correct, and the non-bijection mapping *must* be flawed? I cannot accept that simply
because Cantor provided a mapping that *appears* to show a bijection, that it must be so, even
when other mappings exist which show just the opposite, because I am specifically asking whether
Cantor's mapping *really does* prove the existence of a bijection.

> It still allows the possibility that "there also exists other mappings
> which are not bijections." You've just illustrated above that
> there is a mapping between Z and 2Z which is a bijection,
> so that meets the definition you accepted above.
>
> What would you do with the following example?
> C'= { 1, 2, 3, 4, 5, 6,...}
> | | | | | |
> C'= {1, 2, 3, 4, 5, 6, 7, ...}
>
> This mapping takes every element of C' to an element of C',
> but it leaves one element (1) unmapped. Thus, by your
> intuition, it proves that C' is smaller than itself, by
> one element.

I would use Stephen's point that we can relabel the elements of a set to "redraw" it in one or
two ways that allow us to notice certain facts that would normally be overlooked, and call the
second set D', although I realize that you are *initially* defining the two sets as being the
same.

C'= { 1, 2, 3, 4, 5, 6,...}
| | | | | |

D'= {0, 1, 2, 3, 4, 5, 6,...}

Even here, it should be obvious that you have done the same thing Cantor did: you picked a
mapping scheme that inherently changes the definition of D' from being the infinite set of
integers to being something else, in this case the infinite set of whole numbers. Since you never
noticed this *shift* in the definition of the second set, you attributed a property of this newly
defined set, namely that it contains one more element than C', to C' itself: a shift error.

I will add that I have never seen this type of error discussed in modern science texts, so don't
waste too much of your time looking for carefully explained examples. If they were commonly
understood, Cantor's errors would probably have been found long ago, including the defects
(assuming that they exist) in his proof of the existence of transfinite numbers (hint hint).

Let me restate your example one more time, since it is ideally suited for examining the seriously
cool notion that the infinite set Z, and the infinite set Z + 1, are not the same.

C'= { 1/4, 1/8, 1/16, 1/32, 1/64, 1/128,...}
| | | | | |
D'= {1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128,...}

This is, of course, the Dichotomy, with the point 1/2 corresponding to the middle of the room,
1/4 being the first halfway point to the door, etc. A person walks into the room and enters, in
effect, the "right end' of an infinite set, crossing infinitely many halfway points during the
first instant he is in the room. After any finite time whatsoever, the rate at which he crosses
the halfway points drops to finite values, and he soon comes to the 1/8 point, leaving one more
point to complete the set of halfway points in set C', and two more points in set D'. He takes a
piece of paper, tears it in two, wads both halves up and tosses the first one to the point 1/4,
the second to the point 1/2.

Now, the original paper initially crossed the same set of halfway points for both C' and D' -- in
fact, we can even state that C' and D' contain the same single set of halfway points, with the
exception of 1/2 -- so the two pieces of paper *must* have crossed the same number of halfway
points prior to being tossed. However, the second wad crossed one more halfway point than the
first, and wound up in a physically different position. If these two sets are "really" the same
size, then there is no way that the two wads can be at different locations after crossing *all*
of the points in the sets, in the same direction, and crossing them only once. And yet, Cantor's
"proofs," if we overlook the shift errors, claim not just that (Z - 1) = Z, but that Z/2 = Z, and
even log(Z) = Z, even though the difference between Z and Z/2 is infinitely greater than the
difference between Z and (Z - 1). The difference of a single point out of infinitely many points
is enough to cause a real, noticeable, and physical difference in the location of the two paper
wads in a room. This by itself is a proof that something is seriously wrong with Cantor's proofs,
and you can bet that if those errors haven't been found after 100 years, that they are shift
errors, because I know of no other error that can simultaneously be so powerful in terms of its
effect on the result, and yet so damned difficult to detect.

> One property of infinite sets, in fact one way to define
> "infinite set", is that there exists a bijection between
> S and a proper subset of itself.

This is another restatement of Cantor's conclusions, making it an unacceptable premise if I want
to *test* those conclusions.

Phil


David McAnally

unread,
Nov 27, 2002, 5:17:14 AM11/27/02
to
Phil <tu...@jump.net> writes:

>David McAnally wrote:

>> Phil <tu...@jump.net> writes:
>>
>> >John Zinni wrote:
>>
>> >> "Phil" <tu...@jump.net> wrote in message news:3DE09AC2...@jump.net...
>>
>> >[snip]
>>
>> >> > > Now 2/0 = Infinity and 1/Infinity = 0, substituting in we have:


>> >> >
>> >> > Wait a minute. Dan's argument includes the (admittedly unstated) assumption
>> >> > that 1/infinity = 0, and you can't simply change this definition in the
>> >> > middle of an argument/proof and expect to get valid results. Either
>> >> > 1/infinity = 0 or 2/infinity = 0, take one or the other, but not both.

>> >> > Let's substitute just 1/infinity = 0 and its algebraic alternative, 1/0 =
>> >> > infinity, and see what happens:
>> >> >
>> >> > 2 = 2/0 * 1/Infinity
>> >> > 2 = 2 * 1/0 * 1/Infinity
>> >> > 2 = 2 * Infinity * 1/Infinity
>> >>
>> >> OK, how 'bout this. From the last line above:
>> >>
>> >> 2 = 2 * Infinity * 1/Infinity
>> >> 2 = Infinity * (2 * 1/Infinity)
>> >> 2 = Infinity * (2 * 0)
>> >> (Surly you accept that 2 * 0 = 0 ???)


>>
>> >No, remember that we are *defining* zero as 1/Infinity, so 2 * 0 = 2/Infinity,
>> >not 1/Infinity. It then follows that:
>>
>> That is not how zero is defined. Zero is the additive identity.

>Actually yours is *a* definition of zero, not the one and only,

In Formal Arithmetic, and in Peano's Axioms, zero is an undefined object
which has certain unique properties. In additive groups, rings and
fields, zero is the additive identity. In lattices, zero is the smallest
element (if there is one). In set theory, the ordinal zero (if defined as
a von Neumann ordinal) is just the empty set. The cardinal zero is the
cardinality of the empty set. Of course there are plenty of definitions
of zero (and it may even be undefined, but have definitive properties),
but the fact is that zero is at no point ever defined as 1/Infinity.

>godlike and complete
>definition that provides total understanding in all situations. Come one David, you
>know that most concepts can be neither completely described nor understood from a
>single point of view. The definition of zero as the additive identity is made from
>within a specific context, and to assume that no other contexts for zero exist, or
>that this definition is complete and sufficient for all other contexts, is unfounded.

I believe that I have answered this comment.

>> >I repeat, you *cannot* change the definition of something in the middle of a
>> >line of reasoning and expect to get valid results. Unfortunately, the fact that
>> >we may not be aware of such a change in no way protects us from these errors.
>>
>> But you have the wrong definition. Zero is the additive identity.

Okay, you have the wrong definition. Nobody has ever defined zero as
1/infinity.

>> >> > Not a problem anymore, is it? It's only when you *redefine* infinity in the
>> >> > middle of an argument that you have problems. Well, infinity is so complex
>> >> > that you can always find problems, but at least you avoid these kinds of
>> >> > problems! This is the mistake that Cantor made in his "proofs" that the
>> >> > number of even numbers is exactly equal to the number of integers; he
>> >> > redefined the denumerable set in the middle of his proofs.
>> >>
>> >> What the hell you talking 'bout???
>> >>
>> >> It's VERY straight forward to prove that the cardinality(even numbers) =
>> >> cardinality(integers)
>> >>
>> >> >Of course this
>> >> > resulted in weird answers! He cheated!
>> >>
>> >> He CHEATED???
>> >>
>> >> Prove it?
>>
>> >Okay, I thought maybe I would get either zero or maybe one minor request for
>> >more info on this. Boy was I wrong! Allright John (and others), this response is
>> >going to be fairly long, as it is from my (unpublished and unfinished) book on
>> >Zeno's paradoxes and the Infinite, which is written at a high school or at most
>> >sophomore college level, but here goes. I will mainly just quote straight from
>> >my book, unless I need to add comments in [ ] brackets. The symbol for infinity
>> >will be replaced with a capital "I".
>>
>> >"Georg Cantor's researches into the infinite produced more strange, nonsensical,
>> >and downright paradoxical results and proofs than perhaps any other researcher
>> >into the infinite, including Zeno. We will begin our examination of his work
>> >with perhaps the most basic of all infinite sets, namely the set of integers,
>> >{1, 2, 3, ..., I}.
>>
>> This is wrong. The set of positive integers is {1, 2, 3, ...}. I is not
>> an element of the set.

>Oh really? So when I walk out of a room I do not actually cross *all* of the halfway
>points between myself and the door, but only "a large number of them?"

What has that to do with what I wrote? I stated that infinity is not an
integer. The above argument has nothing to do with that claim.

>One of the
>reasons that infinity is such a difficult subject is that it is literally
>unimaginable, rather like the 4th spatial dimension. We can and do work with such
>concepts all the time, but they are difficult and our thoughts are prone to error,
>because they remain beyond our ability to fully visualize or understand.

What has this to do with whether or not infinity is an integer? Integers
are defined as differences between natural numbers. Infinity is not such
a difference.

>In the case
>of Infinity, we love to "water it down" and bring it within the limits of our finite
>minds by thinking of it as just "a large number, as large as we please."

You might think of it like that, but I don't, and I doubt that you'd find
a working mathematician who does.

>One of the
>first things the Achilles taught me, however, is that Infinity is more than that.

Is this a reference to Zeno? Limits take care of that problem.

>Infinity is not just "a large number, as large as we please,"

We never said it was, but it might do you some good to look up what
mathematics texts have to say about infinity.

>infinity is *all* of
>something.

>After you jump up and down an infinite number of times;

Nobody can jump up and down an infinite number of times.

>how many more times can you
>jump up and down? None, your jumping days are over. After you think an infinite

>number of thoughts, how many more thoughts can you have?

Nobody has ever had an infinite number of thoughts. Now, who's thinking
of infinity as "a very large number"? Certainly not me.

>None, your brain just shut
>down, permanently. After you have crossed the infinite set of halfway points between
>yourself and the door, how many more halfway points can you cross? None, you just
>walked out the door. And once you have a set containing the infinite set of integers,
>how many more integers exist? None, because your set already contains *all* of them.

There is no largest integer. Whatever made you think there was?

>In the finite realm to which our minds are limited, this is nonsense. How can there
>be a "last point" between us and the door?

Try looking up Dedekind cut sometime. The last point is AT the door.

>How can infinity literally be "the last
>integer?"

I never said that infinity was the last integer. That was YOUR claim.
Are you now questioning your own claims?

>In our own finite realm these things are impossible, but infinity is not
>limited to the finite realm, and things that are true in our realm are not
>necessarily true in the infinite or infinitesimal realms.

The integers are each finite, but the set of integers is infinite. Please
recognize the difference between the two.

>This simplest example i
>know of is walking out the door. Do you believe that you cross *all* of the halfway
>points between you and the door every time you do this or not?

Try learning about the real number line some time (Dedekind cuts etc).

>If the answer is yes,
>then the set of halfway points between you and the door does not merely contain "a
>large number, as large as we please," it contains *all* of the halfway points. The
>fact that this is literally incomprehensible (but not inconceivable!) to your finite
>mind is something that you will just have to get used to if you want to have an even
>halfway accurate understanding of infinity. I don't like it any more than you, but
>that's too bad, because any attempt to water infinity down to something more
>comprehensible and acceptable to our finite minds invariably leads to serious errors,
>such as the idea that we can never leave a room, or that we must "mystically
>dematerialize" from the room and then "rematerialize" outside the door, leaving an
>infinite number of halfway points uncrossed.

All this diatribe was prompted by my statement that infinity is not an
integer. At no point in the diatribe above did you actually address the
claim that you were objecting to (i.e. that infinity is not an integer).
Do you believe that if 1, 2, 3, .... are all elements of a set, then
infinity must also an element of the same set, because if that is your
belief then it is wrong (proof: suppose that A is a set with 1, 2, 3, ...
as elements, then A-{infinity} is a set with 1, 2, 3, ... as elements, but
not infinity).

>> The rest of Phil's comments are cut out, firstly because he included his
>> entire posting TWICE (which was quite inconsiderate of him),

>Sorry about that.

>> and secondly
>> because the rest of what he had to say only amounted to his refusing to
>> accept that two sets which have a bijection must have the cardinality, or
>> that a set can have a proper subset of the same cardinality. Of course,
>> no finite set has a proper subset of the same cardinality, but that does
>> not stop the effect happening when you get to infinite sets. I wonder
>> if Phil can give us a good set-theoretic condition for two sets to have
>> the same cardinality so that we can see why the even positive numbers do
>> not have the same cardinality as the positive integers as per Phil's claim.

>David, Cantor came up with a proof, based on the premises of mathematics, which led
>him to conclude that in the denumerable set, the number of even integers is exactly
>equal to the number of integers.

This was proved by the existence of a one-to-one correspondence. Which do
you have trouble with? That there exists a one-to-one correspondence? Or
that the existence of a one-to-one correspondence means that the sets have
the same cardinality?

>Now, we can simply *assume* that this proof is
>correct, in which case you should indeed ignore the rest of my post, or we can test
>this proof, to see if it contains one or more flaws. In such a test, however, we
>cannot continue to *assume* that it is correct!

This means that one has to commit to stating explicitly when one accepts
that sets have the same cardinality, and when one does not accept it.

>The core of Cantor's argument was the act of establishing a one-to-one matching
>(bijection?)

A bijection is a one-to-one correspondence. A mapping is called injective
if it a one-to-one mapping (i.e. f : A -> B is injective if no element of
B is expressible as f(a) for more than one element a of A). A mapping is
called surjective if it is onto (i.e. f : A -> B is surjective if all
elements of B are expressible as f(a) for some a in A). A mapping is
called bijective if it is both injective and surjective. A mapping
f : A -> B is bijective iff every element of B is f(a) for exactly one
element of A, i.e. f is bijective iff it is a one-to-one correspondence.
A mapping is called a bijection if it is bijective.

>between two sets which Cantor claimed both belonged to the (one and
>only) denumerable set.

Where is Cantor's claim that there is only one denumerable set? Give an
explicit reference, with exact quotes.

>I easily produced another matching scheme that fully accounted
>for all of the even numbers, while leaving fully half of the integers "unmatched."
>Given the ability to produce more than one matching scheme with different results, I
>think that any honest person would have to at least question whether the conclusions
>which *appear* to follow from just one of those matching schemes, either Cantor's or
>mine, are necessarily correct.

I think that you'll find that Cantor is correct. The statement is that if
there exists a one-to-one correspondence between two sets, then they have
the same cardinality. Cantor exhibited such a one-to-one correspondence.
Just because you have exhibited a different mapping which isn't a
one-to-one correspondence doesn't invalidate the claim.

>I then provided a line of reasoning which showed that
>Cantor's reasoning included an act, specifically the manner in which he matched the
>elements of the two sets, that inherently *defines* his "denumerable set of integers"
>as being exactly half the size of his "denumerable set of even numbers."

Just what bit of a one-to-one correspondence between sets implying that
they have the same cardinality don't you like?

>Now, you are more than welcome to criticize and possibly correct my premises and/or
>the line of reasoning which I used to come to this conclusion,

Since you won't commit your premises to print, then how do you expect me
to address them?

>but to sit here and
>say, a priori, that *since* Cantor's conclusions are correct, that the flaw which I
>found in his reasoning must be incorrect, is a "circular proof," and therefore
>meaningless.

What circular proof? Cantor defined two sets to have the same cardinality
if there was a one-to-one correspondence between them. Just what don't
you like about that definition?

>As for your request for me to provide "a good set-theoretic condition for two sets to
>have the same cardinality," it's a good request (assuming I correctly understand your
>terms), but I have a better idea.

This is being evasive. You are objecting to some part of the reasoning
which leads to the conclusion that the set of even natural numbers has the
same cardinality as the set of natural numbers. I am trying to see what
your problem with the argument is.

>Let's take one thing at a time, stick to the
>question of whether Cantor's reasoning did indeed contain a hidden redefinition of
>one of his sets,

He defined the set of even natural numbers as the set of even natural
numbers. What is wrong with that? He explicitly exhibited a bijection
between the two sets. I don't understand why you have such a problem with
the concept when you have actually seen the bijection.

>and save the bigger fish (the entire question of cardinality) to fry
>for later. In the first place, you are asking a massive question which, if I get
>involved with it at all, will require equally massive responses. In the second place,
>shifting to this second (admittedly related) subject

No. It is exactly the same subject. You are objecting to a conclusion
about cardinalities. This makes your definition of sets having the same
cardinality very cogent to the discussion.

>does not address the issue of
>whether Cantor redefined one of his terms in the middle of his proof,

What term are you thinking of?

>and I want to
>see if you are willing to address this point in a ruthless, analytical way that does
>not involve assuming that all of Cantor's conclusions must be accepted as givens.

There are two parts to the proof: exhibiting the bijection, and concluding
from that that the sets have the same cardinality. Exactly where is your
problem with the proof. It has to be in one of those two steps. Which?

David McAnally

--------------

David McAnally

unread,
Nov 27, 2002, 5:21:11 AM11/27/02
to
Phil <tu...@jump.net> writes:

>My point is that the normal laws of algebra *do* apply *if* you do not assume that
>1/infinity = 2/infinity = 3/infinity = 0. To put it another way, I defy you (or


>challenge you) to come up with an example that violates the laws of algebra, when that
>example assumes that the relationship between infinity and zero is 0 = 1/infinity, and
>which does *not* assume that 1/infinity = 2/infinity = 3/infinity = 0.

>From still another point of view, my point is that we do not have to give up the laws


>of algebra when dealing with the infinite realm *if* we are willing to give up the idea

>that 1/infinity = 2/infinity = 3/infinity = 0. Given that option (asuming that it
>really is an option, of course), I would much rather give up the latter than the
>former. In any case, if it really is impossible for any of us to come up with something
>that violates the laws of algebra *without* assuming that 1/infinity = 2/infinity =
>3/infinity = 0, then the very fact that this assumption does lead to violations of the
>laws of algebra is itself an indication that this assumption may well be false, and
>that we should investigate further.

Infinity is not a real number. You are treating it as a real number, and
getting mistakes as a result.

David McAnally

--------------

David McAnally

unread,
Nov 27, 2002, 5:35:00 AM11/27/02
to
Phil <tu...@jump.net> writes:

>ste...@nomail.com wrote:

>> Phil <tu...@jump.net> wrote:
>> : ste...@nomail.com wrote:
>>
>> :> Phil <tu...@jump.net> wrote:
>> :> : ste...@nomail.com wrote:
>> :>

>Well, I "defied" you to come up with an example, and I'll be damned if you and John didn't


>both promptly do so. Glad I didn't bet any money ...

>Okay, here's the situation as I see it:

>(1) If we assume that 1/infinity = 2/infinity = 0, we get into trouble, because we can prove
>that 1 = 2 and such. Stating that 1/infinity = 0, but that it is not equal to 2/infinity
>stops one class of problems from coming up with nonsensical answers, but not all problems.

Actually, there is one aspect of mathematics when you can state that
1/infinity = 0, when you take an extended field (e.g. in projective
geometry), when dealing with Moebius Transformations.

>(2) If 0 is defined (as David said) such that x + 0 = x, then there is *no way* that
>1/infinity = 0, as you and John have just conclusively demonstrated.

The limit of 1/x as x approaces zero is infinity. The limit of 1/x as x
approaches zero from above is +infinity. The limit of 1/x as x approaches
zero from below is -infinity.

>> Or do you not believe that 0 + 0 = 0? You are defining 0
>> in a strange way, so maybe 0 + 0 equals something else.

<snip>

>Well, if the shoe fits, put it on! Thanks Stephen, you and John have corrected and improved
>my understanding in this matter. I usually find it most convenient to define infinity such
>that there are exactly an infinite number of points on the line segment from 0 to 1

This is a bad conclusion. There are numerous infinite cardinalities, and
the infinity that describes the number of points is but one of them.
Further, it is a nondenumerable infinity, and is definitely NOT the same
cardinality as the integers or natural numbers (Cantor proved this fact as
well, using what was probably the first of the "diagonal proofs").

>(and yes,
>under this definition there are 2 * infinity points from 0 to 2), which then makes 1/infinity
>the "distance" from one point to the next (this is meaningless in the finite realm, but not
>in the infinitesimal realm). I should have realized, however, that even here, 1/infinity
>cannot possibly have the property such that 1 + 1/infinity = 1. (This gets really fun if you

You seem to believe that two real numbers can be neighbours without any
real number between them. That is wrong. There is always a real number
between distinct real numbers.

>further declare that the 1st point to the right of 0 corresponds to 1, the 2nd point to 2,

There is no such first point or second point.

David McAnally

--------------

David McAnally

unread,
Nov 27, 2002, 5:43:59 AM11/27/02
to
ste...@nomail.com writes:

>Anyway, consider the sets
> A= b{a,b}* (the set of all strings beginning with a b followed
> by any number of a's and b's.)
> B= b{a,b}*a (the set of all strings beginning with a b, followed
> by any number of a's and b's, and ending with an a.)

>There is a bijection between these sets

> A = ( b, ba, bb, baa, bab, bba, bbb, baaa, baab, baba, ...

> B = ( ba, baa, bba, baaa, baba, bbaa, bbba, baaaa, baaba, babaa, ...

>so |A|=|B|, which is pretty intuitive. If we simply added an 'a' to
>the end of each string in A, it should not change the number
>of elements in the set.

Just a quick note to mention there is a similarity between this and part
of the proof of the counterintuitive Banach-Tarski Theorem from the Axiom
of Choice.

David McAnally

--------------

David McAnally

unread,
Nov 27, 2002, 5:48:58 AM11/27/02
to
Phil <tu...@jump.net> writes:

>They obviously are true for infinite sets. My objection refers to another possible type of
>error altogether. I will use your diagrams to demonstrate:

> C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

> D'= ( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

>This is the normal pairing used to demonstrate that the set of even numbers has the same
>cardinality as the set of integers. Now, I will pair these same two sets, but in a
>different way (adding vertical lines to emphasize the pairing).

> C'= ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
> | | | | |
> D'= ( 2, 4, 6, 8, 10, ...

This does not invalidate the other bijection.

David McAnally

--------------

David McAnally

unread,
Nov 27, 2002, 5:52:28 AM11/27/02
to
"John Zinni" <j_z...@sympatico.ca> writes:

>"Phil" <tu...@jump.net> wrote in message news:3DE3404E...@jump.net...

>[snip]

>> Okay, but am I not correct in saying that two sets have the same
>cardinality if and only if
>> it is possible to match all of the elements of one set with all of the
>elements of the
>> other set, with no elements in either set "left over" (left unmatched or
>unpaired)? A set
>> has the same cardinality as the denumerable set (aleph 0?) *only* if it is
>possible to pair
>> off each element of the set in question with an element from the
>denumerable set. I think
>> we are in complete agreement on these points, but I want to make sure.

>No, this is not true.

>Given two sets A and B, |A| = |B| if (NOT only if) there is a (at least one)
>bijection between A and B.

Actually, strictly speaking, it is "if and only if".

>A bijection means that there is a 1 to 1 mapping of A onto B and that the
>inverse mapping of B onto A is also 1 to 1 (hence the "bi" in bijection, I
>think)

Correct.

>So, if we can find at least one bijection between A and B then |A| = |B|.

>If there is a bijection between A and B, the fact that we can find other
>mappings between A and B that are not bijections does not constitute a
>counterexample to |A| = |B| since we know that there is still "at least
>one" bijection between A and B.

Correct.

David McAnally

--------------

David McAnally

unread,
Nov 27, 2002, 5:55:22 AM11/27/02
to
"John Zinni" <j_z...@sympatico.ca> writes:

>"John Zinni" <j_z...@sympatico.ca> wrote in message
>news:2bNE9.285$Nm.1...@news20.bellglobal.com...

>> "Phil" <tu...@jump.net> wrote in message news:3DE3404E...@jump.net...
>>

>> Given two sets A and B, |A| = |B| if (NOT only if) there is a (at least
>one)
>> bijection between A and B.

>Opps! Turns out it should be iff.

Sorry, I hadn't read your retraction before I posted my correction.

David McAnally

--------------

David McAnally

unread,
Nov 27, 2002, 5:57:36 AM11/27/02
to
Phil <tu...@jump.net> writes:

>John Zinni wrote:

>> Now, lets see if I can convince you that Infinity (I) is NOT an element of
>> the {+ive integers).
>> (I don't think that this is going to be rigorous but we'll give it a shot
>> anyway)
>>
>> You would like to define the set of {+ive integers} as:
>>
>> Z+ = {1, 2, 3, ..., n, ..., I}

>I will digress for a minute and point out that "I" is not meant to be a


>variable, but the symbol for infinity, and as such, should not be treated like
>some arbitrary large number, even a number as large as we please. One of the

>first things the Achilles taught me is that infinity is *not* "a large number,


>as large as we please."

What part of "infinity is not an integer" do you have trouble with?

David McAnally

--------------

David McAnally

unread,
Nov 27, 2002, 6:06:29 AM11/27/02
to
Phil <tu...@jump.net> writes:

>Second, in situations where there appear to be two valid mappings, one which is a bijection and
>one which is not, what evidence can you or Cantor provide that shows that the bijection mapping
>*must* be correct, and the non-bijection mapping *must* be flawed?

The non-bijection is not flawed. Why must either be flawed? What is
wrong with the conclusion that both are legitimate mappings?

>I cannot accept that simply
>because Cantor provided a mapping that *appears* to show a bijection,

*Is* a bijection.

>that it must be so, even
>when other mappings exist which show just the opposite,

Your mapping is not a counterexample. It does not show the opposite.
That's only your faulty interpretation of a legitimate mapping.

>because I am specifically asking whether
>Cantor's mapping *really does* prove the existence of a bijection.

Yes, it does. And your mapping, which is perfectly legal, does not
disprove it.

David McAnally

---------------

John Zinni

unread,
Nov 27, 2002, 8:54:08 AM11/27/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE46F83...@jump.net...

> John Zinni wrote:
>
> > Now, lets see if I can convince you that Infinity (I) is NOT an element
of
> > the {+ive integers).
> > (I don't think that this is going to be rigorous but we'll give it a
shot
> > anyway)
> >
> > You would like to define the set of {+ive integers} as:
> >
> > Z+ = {1, 2, 3, ..., n, ..., I}
>
> I will digress for a minute and point out that "I" is not meant to be a
> variable, but the symbol for infinity, and as such, should not be treated
like
> some arbitrary large number, even a number as large as we please.

I thought I was treating it as an integer???

So, is it an integer or not???

If it is, than it should be included in the set.
If it is not, than it should not be in the set.
This should be straight forward enough.

If it is an integer, then why is I+1 not an integer???

You yourself use it as an actual integer in the paper you sent me (which
contradicts your stament above "and as such, should not be treated like some
arbitrary large number, even a number as large as we please". At one point
you map I->I and at another point you map I ->I-1.
Can we map I to an element I+1 in the integers? Is I+2 an integer. blah-blah
blah-blah blah-blah ...


At this point I would accuse you of (either consciously or unconsciously)
redefining the set Z+ wherever necessary.

At some points (although not explicitly stated) you treat it as a finite
set. This is facilitated by your notation for it, namely:
Z+ = {1, 2, 3, ..., n, ..., I} (looks finite to me)

At other points you treat it as an infinite set. You justify this by stating
that the last element (I) is infinity.

Unclear definitions lead to unclear conclusions.

Are you telling me that in the set of {+ive rationals} there are elements
that are infinitesimal???

Do you agree or disagree that 1/b, were b is an element of Z+, is finite?

Do you agree or disagree that I is an element (an actual concrete element)
of Z+?


> The term "infinitesimal" is somewhat
> crowded, because it can refer to the "realm of points," in which we
mentally
> picure points on a number line strung out like beads on a necklace (from
this
> perspective, any finite length in the "normal" realm appears to be
infinitely
> long), or it can refer to objects which are infinitely larger than points,
but
> infinitely smaller than any finite length, a concept that has been usually
been
> rejected over the last 2,400 years or so (Cantor also rejected them).

I have not tried to wrap my head around this yet but perhaps the rejection
is justified.

--
Cheers
John Zinni

John Zinni

unread,
Nov 27, 2002, 10:51:16 AM11/27/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE46F83...@jump.net...

[snip]

> > This fits in well with the fact that we have convinced you that 1/I!=0
>
> Except for the finite part, very well indeed! By the way, when did !=
replace <>
> for "not equal?" I haven't keep track for 10 or 20 years, but ...

"!=" comes from a need to do some programming in my job. In many programming
languages "!=" is "not equal to".

I would also like to revise what I said above.

1/I!=0 still implies that I is a number.

I should have said:

1/I=0 is nonsensical

John Zinni

unread,
Nov 27, 2002, 2:49:48 PM11/27/02
to
"David McAnally" <D.McAnally@i'm_a_gnu.uq.net.au> wrote in message
news:arr068$ma3$1...@bunyip.cc.uq.edu.au...
> "John Zinni" <j_z...@sympatico.ca> writes:
>
> >"Dan Perez" <fasterthanlig...@nospamyahoo.com> wrote in message
> >news:3ddf7960$0$17649$2c3e...@news.voyager.net...
>
> >This is utter nonsense.
>
> >Infinity is NOT a number!!! You cannot toss it around as such.
>
> Cantor's transfinite numbers are infinities. Some models of Formal
> Arithmetic in Predicate Calculus include infinite quantities. Surreal
> numbers include numbers which are "infinite" (i.e. larger than any finite
> quantity - the surreals do not form an Archimedean field).

Hi David

I have been doing a little reading on the Surreal Numbers and trying to get
some grasp of them and a question popped up.

I have read in a couple of places that it is difficult to get a clear
understanding of what is meant by "Integration" and "Differentiation" on
{Surreal}. Would it be accurate to say, that on the {Real}, integration is
an infinite sum of infinitesimal quantities that can yield a finite result.

If so, it seems to me that, with the definition of addition on {Surreal} and
the infinitesimal elements of the {Surreal}, that this is already possible.

So, my question is:
Is it really necessary to figure out what a meaningful definition of
"Integration" is on the {Surreal}?


> Projective
> geometry has genuine points which are "at infinity" (i.e. lines which meet
> at those point in the projective geometry become parallel in the affine
> geometry), and are therefore non-existent, in the affine
> specialization. The extended complex plane has infinity as an element,
> and indeed Moebius Transformations require the existence of infinity so
> that they can be bijective. So while you are correct and infinity is
> certainly not a number, I make the above points to demonstrate in certain
> mathematical discussions, infinite quantities have an objective reality,
> and can be treated as concrete quantities.
>
> >Infinity is a PROCESS
>
> The process relates to taking limits, as others have pointed out, and in
> the context that is true. The above was just a not that in some
> mathematical theories (but not the one presently under discussion), there
> are concrete infinite quantities.
>
> >and requires some special care or you end up with
> >gibberish like "Infinity * 0 = 1"
>
> >--
> >Cheers
> >John Zinni
>
> David McAnally
>
> --------------

--
Cheers
John Zinni

Ken S. Tucker

unread,
Nov 28, 2002, 12:02:21 PM11/28/02
to
> > Would anyone disagree with
> > Infinity x Zero = N where Zero < N < Infinity.
> > Regards Ken S. Tucker
>
> Yes. Infinity is not a number and cannot be treated as such.
>
> Also x * 0 = 0, no matter what x is.

Your comments are invited..
Consider the integral of zero, I'll write this
INTEGRAL(0)*dX = N, N and zero (0) are constants,
it follows 0*INTEGRAL dX = N = INTEGRAL(0)*dX
and since X= INTEGRAL dX, 0*X = N.
Since infinity is the largest variable and can not be a
constant, N must be less (in magnitude) than infinity.
"X" is the only variable here, and may be infinite.
Therefore 0*X = N, where -X<N<X .

Would anyone disagree with this?
Ken S. Tucker

John Zinni

unread,
Nov 28, 2002, 11:32:54 PM11/28/02
to
"Ken S. Tucker" <dyna...@vianet.on.ca> wrote in message
news:2202379a.02112...@posting.google.com...

> > > Would anyone disagree with
> > > Infinity x Zero = N where Zero < N < Infinity.
> > > Regards Ken S. Tucker
> >
> > Yes. Infinity is not a number and cannot be treated as such.
> >
> > Also x * 0 = 0, no matter what x is.
>
> Your comments are invited..
> Consider the integral of zero, I'll write this
> INTEGRAL(0)*dX = N, N and zero (0) are constants,

Hi Ken
I don't think you can really start out this way. I'm assuming here that you
are using the fact that you already know that INT(0)dx equals an arbitrary
constant but the reason we know this is, that at some point we would have
had to have evaluated the following:

f(x) = INT(a)dx were a is some constant
f(x) = a*x + C were C is some arbitrary constant
set a=0, yeilds
f(x) = 0*x + C
f(x) = C

So:

INT(0)dx = C were C is some arbitrary constant

This means that at best, your opening line just says that the arbitrary
constant C = N. We used the fact that 0*x = 0 in order to obtain INT(0)dx =
C so there is no reason to keep 0*x around when you go through the exercise
of doing the same integration again.


> it follows 0*INTEGRAL dX = N = INTEGRAL(0)*dX
> and since X= INTEGRAL dX, 0*X = N.

Note: INT dx = x + C not x

> Since infinity is the largest variable and can not be a
> constant, N must be less (in magnitude) than infinity.
> "X" is the only variable here, and may be infinite.
> Therefore 0*X = N, where -X<N<X .

Could not really follow this part. Could you elaborate.

>
> Would anyone disagree with this?
> Ken S. Tucker

--
Cheers
John Zinni

David McAnally

unread,
Nov 29, 2002, 6:44:17 PM11/29/02
to
dyna...@vianet.on.ca (Ken S. Tucker) writes:

>> > Would anyone disagree with
>> > Infinity x Zero = N where Zero < N < Infinity.
>> > Regards Ken S. Tucker
>>
>> Yes. Infinity is not a number and cannot be treated as such.
>>
>> Also x * 0 = 0, no matter what x is.

>Your comments are invited..
>Consider the integral of zero, I'll write this
>INTEGRAL(0)*dX = N, N and zero (0) are constants,
>it follows 0*INTEGRAL dX = N = INTEGRAL(0)*dX
>and since X= INTEGRAL dX, 0*X = N.

No. Actually integral(0)*dX = 0*integral dX + constant.
This leads to 0*integral dX + constant = N, and the
problem is now solvable by noting that we can take
constant = N and 0*integral dX = 0.

>Since infinity is the largest variable and can not be a
>constant, N must be less (in magnitude) than infinity.
>"X" is the only variable here, and may be infinite.
>Therefore 0*X = N, where -X<N<X .

>Would anyone disagree with this?

Certainly, for the reason that you forgot the arbitrary
constant.

>Ken S. Tucker

David McAnally

--------------

Phil

unread,
Nov 30, 2002, 3:59:38 AM11/30/02
to
Okay, this is getting too spread out, with 10 different "sub-threads"
addressing various related concepts or problems all over the place, so I
will attempt to consolidate a bit. On the other hand, some issues are so
minor that they are best left where they are, so I will also make a few
minor posts "at the point of origin."

We can begin by assuming the existence of three realms, the finite or
"normal" realm, the infinite realm, and the infinitesimal realm. The
finite realm corresponds, to a surprisingly large degree, to what we can
perceive, either in reality or in our imaginations. Any *perceivable*
length has a *finite* length, and, if we make use of "mathematical
microscopes and telescopes" capable of any finite level of magnification,
any finite length is a perceivable length. However, any finite number of
points always appears, even under our mathematical microscope, to have a
length of zero when seen from the finite realm. This must be the case, or
else points would have a finite width, with a finite quantity making up
any finite length line segment. However, we can, in our imagination, make
a "transition of infinity" and *shift* to the infinitesimal realm, where
we can (in our minds) see the individual points strung out like beads on
an endless necklace. Here, we can work with finite numbers of points, but
any finite length from the "normal" realm appears to be infinitely long,
with no way for our mathematical telescope to see the end of it.

Overall, therefore, we can work with finite lengths in the finite realm,
infinitesimal lengths in the infinitesimal realm, and infinite lengths in
the infinite realm (where, for example, the entire set of positive
integers can be seen as a line segment 1 unit long). However, we cannot
work with more than one realm at a time. As seen from the finite realm
*all* finite sequences of points are indistinguishable, with an apparent
length of exactly zero, and all infinite lengths are equally
indistinguishable, with an apparent length of exactly infinity. We can
shift in our imagination from one realm to another, but in each realm we
view things exactly as we do in the finite realm. This introduces biases,
such as a belief that two points right next to each other must have some
"space" between them, since that would normally be the case in our own
finite realm. Of course, even in our realm, if two bowling balls are
right next to each other, and "space" is defined as the ability to place
another bowling ball at some position, then two bowling balls next to
each other do *not* have any space between them. However, we usually use
our intuitive or "normal" definitions of space, which often leads to
errors and/or confusion.

The "spacing" of points on a line segment is an important subject, which
can be partially clarified by using a "mathematical knife," with an
infinitely sharp blade of literally zero thickness, which can move to any
position whatsoever over the line segment and descend. The position where
it contacts the line is a point, also of zero width. Leaving the knife in
contact with the line segment and moving it from one end to the other
causes it to pass over *every possible position* on the segment. If we
define a point as existing at every position passed over by the knife,
then it is clear that the segment is *completely filled* with zero-width
points. Otherwise, we would have to accept that the knife "mystically
dematerialzed" while passing over certain positions, thereby making it
possible to add additional points onto the line segment.

If even we agree with the "bowling ball" analogy above, however, and
accept that there is no room to fit in even a single point onto the
segment after the mathematical knife has passed over the segment, we are
left with the question of how even an infinite number of zero-width
points can completely fill a line-segment. This actually forms another of
Zeno's paradoxes, the Plurality. I will not attempt to completely answer
this question, except to say that Zeno is quite correct in that points
have no ability by themselves to fill anything. The ability of the line
segment to be completely filled with zero-width points is actually a
characteristic of the first dimension, rather than of points. In other
words, the ability of points to completely fill a line is basically a
premise, one which is largely equivalent to the existence of the first
dimension, and of geometric lines. It is not a premise which fits well
with our "finite minds," but the premises of mathematics do not have to
make sense; they just have to be consistent with each other, so that they
do not lead to truly contradictory results. I will at least claim that
the idea of zero-width points completely filling a line is such a
premise.

I will combine John's notation with my own to define, or at least
represent, the set of positive integers, Z+, as:

Z+ = {1, 2, 3, ..., n, ..., (Inf - 1), Inf}

Here "Inf" stands for "infinity," to avoid some of the confusion caused
when I used "I" to represent infinity (since "I" normally refers to an
integer, not infinity). Normally this would be crazy, because "there is
no last integer," and as seen from the finite realm, or when the
existence of a last integer is assumed to *define* a set as being finite
this is quite correct. However, when dealing with infinity we need
*something* similar to the above definition for Z+, as the following
examples will show.

In Zeno's Achilles, we start at one wall, and cross the halfway points to
the door, beginning with the point in the center of the room 1/2, then
the 1/4 point, then 1/8, etc. (this is actually one of several
mathematically equivalent ways in which the Achilles can be stated; I use
this one because it is easiest version to visualize, and very similar to
Zeno's Dichotomy). The point of the Achilles, obviously, is that we can
never leave the room because (1) there are infinitely many halfway points
to cross, and (2) given any "last" halfway point, we could divide the
remaining distance in two and produce still another halfway point to
cross.

The first point can be partly dealt with by observing that as we approach
the door, the *rate* at which we cross the halfway points becomes itself
infinite. Obviously, there is at least the possibility that crossing
points at an infinite rate will allow us to cross an infinite number of
them in a finite period of time. The real first lesson of the Achilles,
however, is that in crossing the infinite set of halfway points between
us and the door, we do not "approach the limit" of the door, or cross "a
large number of halfway points, a number as large as we please," we cross
*all* of the halfway points. In other words, the set of halfway points to
the door cannot be *completely* defined as something that extends
forever, without an end -- although this is in fact the case as seen from
the finite realm -- because the *last* 2, 10, or any other finite number
of halfway points that exist between us and the door do exist. We can
examine a finite set of points at the beginning of the set, or we can
make a different kind of "transition of infinity" and examine a finite
set of points at the end of the set, but in between is something that is
literally beyond the limits of human comprehension, namely infinity
itself.

The proof of a "last point" comes from Zeno's Dichotomy, which is
basically just the Achilles in reverse. In the Dichotomy, the same set of
halfway points from the center of the room to the door is used, but this
time we enter the door and walk toward the center. Zeno's point was that
since an infinite number of halfway points must be crossed before we move
any finite distance whatsoever, we cannot even begin to move, proving
that all motion is impossible. In reality, of course, we can enter a
room, because although an infinite number of halfway points must be
crossed, we initially cross them at an infinite rate. Once we are into
the room, however, there are a *finite* number of halfway points
remaining to the center of the room. The Dichotomy shows that the idea of
"the last two points" out of an infinite set of points has not just a
vague, mathematical meaning, but a physical, real-world meaning as well.
Furthermore, the fact that a person who moves from the door to the 1/4
point is not standing at the 1/2 point proves that the difference of a
single point is enough to distinguish between two infinite sets of
points, and that this difference again has a physical, as well as a
mathematical, validity.

The first lesson of the Dichotomy, therefore, is that when given an
infinite set of elements, adding a single additional element to this set
can produce a *different* infinite set, which has observably different
properties from the original set. The second lesson, which follows from
the first, is that when dealing with infinite sets it is *mandatory* to
define the characteristics of the particular infinite set we are dealing
with, and to verify that those characteristics are not accidentally
altered by some sequence of mathematical actions. The simplest example,
of course, is the often heard claim that adding a single element to an
infinite set, such as an additional "9" at the end of the infinite
sequence 0.9999 ..., leaves the set unchanged, but since this is
equivalent to claiming that the 1/4 point and the 1/2 point refer to the
same position, the truth is that such actions at least occasionally, and
perhaps always, *redefine* the original set. Or, from another point of
view, it can be said that such actions result in a *shift* from the
original set to a second set, one which has different characteristics
than the first set. Another example occurs when we take two points that
are next to each other, divide the "space" between them in two, and add
another point, a technique which is often used to "prove" that between
any two points, infinitely many more points exist (which if true, really
would prevent us from leaving the room, as the Achilles says). In this
case, the result is again a *redefinition* of the old set into a new set,
one which has twice the "density of points" of the old set. Needless to
say, obtaining facts from a new set (such as our position in a room) and
claiming that these facts apply to the old set results in a defective
argument, with invalid results.

Overall, therefore, we at least occasionally need a method for referring
to sets that allows us to see a finite number of both the first elements,
and the last elements. One method was shown above, namely:

Z+ = {1, 2, 3, ..., n, ..., (Inf - 1), Inf}

A method for describing repeating decimals like 0.99999 ... is:

0.999 ... Inf ... 999

In this example, the last three 9's are just that, the *last* three 9's
in this infinite sequence. However, I do not have much experience in
forming representative schemes, and in any case I am a firm believer that
several heads are better than one, so I am open to suggestions,
improvements, and corrections (although I will ask you to trust me for
the moment that the infinite set of 9's in this infinite decimal includes
*all* of the 9's, so that a reference to "the last three 9's" in this
decimal is indeed valid).

The reason I put so much emphasis on infinity being "all of something" is
probably best given by example. After I said that we tend to think of
infinity as "a large number," instead of what it really is, namely all of
something, David made the comment that: "You might think of it like that,


but I don't, and I doubt that you'd find a working mathematician who

does." However, later in the same post, after I said: "And once you have


a set containing the infinite set of integers, how many more integers

exist? None, because your set already contains *all* of them." David
responded with: "There is no largest integer. Whatever made you think
there was?"

My conclusion is that the awareness that infinity is not just a large
number, but all of something, is a little like our awareness that we are
going to die: When asked whether we will die, everyone responds yes, but
in the next moment we turn around and act as if we were going to live
forever. In other words, we usually forget just "how big" infinity really
is, and this often leads to further errors. However, once we realize that
infinity is all of something, the next step, as shown by the Dichotomy,
is to be very specific about just which set of infinity we happen to be
working with, because while we cross an infinite set of points the moment
we enter a room, we do not cross "the infinite set of halfway points from
the center to the door" until we actually reach the center.

This largely wraps up the basics (although it just mentions many of them,
without thoroughly defending them), but I might as well add the second
lesson from the Achilles while I'm at it. There are an infinite number of
halfway points from the center to the door, a fact which virtually no one
will dispute. However, at any *finite* distance from the door whatsoever,
we will have crossed only a finite number of halfway points, leaving an
infinite number of points still uncrossed. Since the next "object"
smaller than a finite distance is something that appears to have a length
of zero as seen from the finite realm -- an infinitesimal length -- the
unavoidable conclusion is that the vast majority of the halfway points,
an infinite number of them, in fact, exist an infinitesimal distance from
the door. Since this region contains infinitely many points, it is
infinitely larger than a point, and yet it is also infinitely smaller
than any finite length. This is actually the definition of an
"infinitesimal" (not to be confused with the infinitesimal realm, which
may or may not include the possibility of such objects), a concept that
has been very controversial for well over 2,000 years.

Without going into too many details, the ancient Greeks simply added a
postulate, known today as the Archimedian postulate (although Archimedes
himself said that it came from Eudoxus), that simply eliminates their
existence. This postulate states that for any two non-zero numbers a and
b, such that a < b, there exists a finite number n such that n * a > b.
In other words, this postulate states that any two non-zero numbers a and
b always have a finite, rather than an infinite, ratio between them. From
still another viewpoint, the Archimedian postulate states that “non-zero”
and “finite” are in fact the same thing, a claim which automatically
eliminates infinitesimals.

If infinitesimals do not exist, however, then our system of mathematics
must indeed be fatally flawed, just as Zeno said, because the Achilles
takes the most basic premises and standard methods of logic from
mathematics and uses them to *prove* that an infinite number of points
*must* exist within a region that is smaller than any finite length
whatsoever, the definition of an infinitesimal. Nor can we simply declare
that "it turns out that infinitesimals exist after all," because the
various proofs of Georg Cantor, barring some previously unseen flaw,
prove that infinitesimals do not exist.

To summarize, infinite sets, although they do contain an infinite number
of elements, often and perhaps always contain a last element. However, a
particular infinite set is just that, a *particular* set, and adding even
a single element can change both the definition and the functionality of
a infinite set (such as adding the point 1/2 to the infinite set of
halfway points from the door to the 1/4 point). Also, objects equivalent
to an infinitesimal, something which is infinitely larger than a point
and yet smaller than any finite length (with an apparent length of zero
as seen from the finite realm), must exist. However, Cantor's work at
least appears to prove that both of these conclusions are false, that
adding a finite number of points to an infinite set does not change the
basic characteristics of the set, and that infinitesimals do not exist.

Phil


Dirk Van de moortel

unread,
Nov 30, 2002, 9:32:25 AM11/30/02
to

"Phil" <tu...@jump.net> wrote in message news:3DE87E22...@jump.net...

> Okay, this is getting too spread out, with 10 different "sub-threads"
> addressing various related concepts or problems all over the place, so I
> will attempt to consolidate a bit. On the other hand, some issues are so
> minor that they are best left where they are, so I will also make a few
> minor posts "at the point of origin."
>

[snip]

> To summarize, infinite sets, although they do contain an infinite number
> of elements, often and perhaps always contain a last element.

Will this be the title of your next book?

> However, a
> particular infinite set is just that, a *particular* set, and adding even
> a single element can change both the definition and the functionality of
> a infinite set (such as adding the point 1/2 to the infinite set of
> halfway points from the door to the 1/4 point).

Aaaarg!

> Also, objects equivalent
> to an infinitesimal, something which is infinitely larger than a point
> and yet smaller than any finite length (with an apparent length of zero
> as seen from the finite realm), must exist.

HEELP!

> However, Cantor's work at
> least appears to prove that both of these conclusions are false, that
> adding a finite number of points to an infinite set does not change the
> basic characteristics of the set, and that infinitesimals do not exist.

You definitely need a doctor.

Dirk Vdm


John Zinni

unread,
Nov 30, 2002, 10:21:11 AM11/30/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE87E22...@jump.net...

I agree with Dirk .... Aaaarg!

Could you POSSIBLE shroud these concepts in any more WORDS!!!

This is a technique used by another individual in this group.

Are you sure your name is not philX???

If I can find an hour or two to kill, I'll see if I can tease out what you
are actually trying to say here.

--
Cheers
John Zinni


Stephen Speicher

unread,
Nov 30, 2002, 12:59:11 PM11/30/02
to
On Sat, 30 Nov 2002, John Zinni wrote:

> "Phil" <tu...@jump.net> wrote in message news:3DE87E22...@jump.net...

[Mercifully snipped ...]

>
> I agree with Dirk .... Aaaarg!
>
> Could you POSSIBLE shroud these concepts in any more WORDS!!!
>

Phil makes up in volume what he lacks in content.

This is all preparation for his "book" setting the world straight
about relativity, being birthed for the last 12 years, I believe.

--
Stephen
s...@speicher.com

Ignorance is just a placeholder for knowledge.

Printed using 100% recycled electrons.
-----------------------------------------------------------

Dirk Van de moortel

unread,
Nov 30, 2002, 12:49:08 PM11/30/02
to

"John Zinni" <j_z...@sympatico.ca> wrote in message news:pI4G9.67800$e%.1776411@news20.bellglobal.com...

[snip]

> I agree with Dirk .... Aaaarg!
>
> Could you POSSIBLE shroud these concepts in any more WORDS!!!

Your and Phil's posts don't even show on my regular news
server: it kills everything above 15 Kb. I have to consult a
much slower one - reserved for binary and alt-groups to
see these posts.

> This is a technique used by another individual in this group.
>
> Are you sure your name is not philX???

:-))

>
> If I can find an hour or two to kill, I'll see if I can tease out what you
> are actually trying to say here.

And make him post even longer extracts of his book?
Please be a good sport and don't? Thanks ;-)

Dirk Vdm


Ken S. Tucker

unread,
Nov 30, 2002, 1:33:13 PM11/30/02
to
"John Zinni" <j_z...@sympatico.ca> wrote in message news:<A6CF9.67022$e%.1449117@news20.bellglobal.com>...

(Actually I posted yesterday, but somehow things got lost).
Following your lead, dN/dx = 0, so N =constant.
Thus -(oo)<N<(oo).
Infinity (oo) > constant. So INT (dN/dx)dx = N
= INT (dN/dx =0)dx =N = 0*INT dx = N = 0*x = N.

|Infinity| is always greater than a (>)|constant|.

John Zinni

unread,
Nov 30, 2002, 3:41:20 PM11/30/02
to
"Dirk Van de moortel" <dirkvand...@ThankS-NO-SperM.hotmail.com> wrote
in message news:oS6G9.11074$s17.229...@hestia.telenet-ops.be...

[snip]

> Your and Phil's posts don't even show on my regular news
> server: it kills everything above 15 Kb. I have to consult a
> much slower one - reserved for binary and alt-groups to
> see these posts.

Opps! Sorry 'bout that Dirk.

[snip]

> And make him post even longer extracts of his book?
> Please be a good sport and don't? Thanks ;-)

OK, I promise to let it die here, but not before straightening out one final
irritating point.

"Phil" <tu...@jump.net> wrote in message news:3DE87E22...@jump.net...

> Here "Inf" stands for "infinity," to avoid some of the confusion caused
> when I used "I" to represent infinity (since "I" normally refers to an
> integer, not infinity).

In the study of mathematics at any level higher than the secondary school
level, double-strike I represents the set of Pure Imaginary numbers,
double-strike Z represents the set of Integers.

(OK, so I'm a little anal-retentive. I blame my undergrad years.)

--
Cheers
John Zinni


Dirk Van de moortel

unread,
Nov 30, 2002, 5:16:34 PM11/30/02
to

"John Zinni" <j_z...@sympatico.ca> wrote in message news:Ao9G9.8855$cx4.1...@news20.bellglobal.com...

> "Dirk Van de moortel" <dirkvand...@ThankS-NO-SperM.hotmail.com> wrote
> in message news:oS6G9.11074$s17.229...@hestia.telenet-ops.be...
>
> [snip]
>
> > Your and Phil's posts don't even show on my regular news
> > server: it kills everything above 15 Kb. I have to consult a
> > much slower one - reserved for binary and alt-groups to
> > see these posts.
>
> Opps! Sorry 'bout that Dirk.

Ha, this one is *much* better already :-))

>
> [snip]
>
> > And make him post even longer extracts of his book?
> > Please be a good sport and don't? Thanks ;-)
>
> OK, I promise to let it die here, but not before straightening out one final
> irritating point.
>
> "Phil" <tu...@jump.net> wrote in message news:3DE87E22...@jump.net...
> > Here "Inf" stands for "infinity," to avoid some of the confusion caused
> > when I used "I" to represent infinity (since "I" normally refers to an
> > integer, not infinity).
>
> In the study of mathematics at any level higher than the secondary school
> level, double-strike I represents the set of Pure Imaginary numbers,
> double-strike Z represents the set of Integers.
>
> (OK, so I'm a little anal-retentive. I blame my undergrad years.)

Good one!

Night,
Dirk Vdm


Phil

unread,
Nov 30, 2002, 8:47:23 PM11/30/02
to
Many of our current beliefs concerning infinity and infinite sets come
from Georg Cantor's "Mengenlehre" (theory of sets or classes, especially
the infinite sets). In the Mengenlehre, Cantor was able to prove
incredible things, such as, there are just as many positive even numbers
as positive integers, that the sqr(infinity) = infinity, and the
existence of different classes of infinity, the "transfinite sets."
However, all of these conclusions rest on a single assumption, the belief
that once a set has been defined, it cannot possibly be redefined by
actions such as adding to the set, dividing the set by two, or the
matching scheme used to link the elements of one set with the element of
another set. The purpose of this post will be to examine that assumption.

One of the first potential problems is that the belief that any operation
can be performed on an infinite set without changing the number of
elements in the set *may* just be another way of stating that all
infinite sets (at least all denumerable sets) have the same number of
elements, which is what Cantor was supposedly proving. Obviously, if
Cantor "proved" that all sets have the same number of elements by first
*assuming* that all sets have the same number of elements, we have a
problem.

I will briefly restate the situation from my earlier post. Cantor's proof
that there are just as many positive even numbers as positive integers
requires a *valid* one-to-one mapping method or bijection. Given the
denumerable set D and the set of positive even numbers E, Cantor gave:

D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }
| | | | | | | | | |
E = { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... }
Figure 1

Now, there is no question about whether a valid bijection exists for the
two sets in Figure 1, because there is. The question is whether, in the
process of *defining* the relationship between these two sets, Cantor
also redefined D. An alternative mapping shows that this may indeed be
the case, although this alternative does not by itself *prove* that a
redefinition occurred.

D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }
| | | | |
E = { 2, 4, 6, 8, 10, ... }
Figure 2

There is also no question that for the sets of Figure 2, a bijection does
not exist, even though each and every element in the set of even numbers
has again been linked to a unique element in the set of integers. In
fact, we can legitimately conclude that for the sets *shown in Figure 2*,
D has exactly twice as many elements as E. However, if we are to admit
the possibility that the mapping method of Figure 1 may have redefined
the sets, then that possibility also exists for Figure 2, so that the
conclusion that D = 2 * E does not necessarily apply to *the* denumerable
set, and *the* set of even numbers.

An important point is that in both cases, the *initial definitions* for D
and E were exactly the same. It was the method of mapping that produced
different results. There are (at least) three reasons for believing that
in Figure 1, the method of mapping the elements *redefined* D as being
something other than the denumerable set, specifically, as being an
infinite set of positive integers which is exactly half the size of the
denumerable set.

First, although the numerical value of elements does not determine the
number of elements in a set, there are situations where conclusions can
be drawn from those values. In the case of Figure 1, each element from D
is mapped to an element from E that has a value twice as large as the
element from D. For all *finite* sets of integers and even numbers, the
value of the even elements is exactly twice the value of the integers to
which they are mapped. If we are to believe that the mapping process of
Figure 1 does *not* redefine D as being 1/2 the denumerable set, then
this ratio must somehow "mystically vanish" for infinite sets.

Second, take a *finite* set X consisting of the first 20 integers, and
derive two sets from X, namely Df, containing all the integers in X, and
Ef, containing all the even numbers in X. Next, define a mapping scheme
similar to Figure 1, including the normally *unspoken*, or *assumed*
requirement that the *last* element of Ef be linked to the *last* element
of Df. The result is shown in Figure 3.

Df = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
| | | | | | | | | |
Ef = { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 }
Figure 3

Here we see that the mapping method used in Figure 1, including any
normally unstated assumptions, results in a process that *redefines* Df
as being half its original size, with 10 elements instead of the 20 it
was originally defined as having. Given that this process redefines Df
for any finite set X as being half its original size, the only way that D
in Figure 1 can actually be the denumerable set is if this process also
"mystically vanishes" when X becomes infinite.

Third, it is possible to modify the set of even numbers in a way that
results in the denumerable set, but which leaves infinitely many elements
of this modified set unmapped with set D. In Figure 4, E is again the set
of even positive numbers, D is the set that was initially defined as
being the denumerable set, and O is the set of odd positive numbers.

D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }
| | | | | | | | | |
E = { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... }
O = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ... }
Figure 4

As before, a bijection definitely exists between D and E, proving that
they have the same number of elements. The question is still whether D is
actually the denumerable set, or whether, in the process of executing
this mapping scheme, D was accidentally redefined as being a new set, one
which is half the size of the denumerable set. Set O contains the odd
numbers, and is obviously the same size as E. Adding O to E results in a
new set D2, which has two obvious features. First, it does not have a
bijection with D, just as shown in Figure 2. Second, the fact that it
contains *all* of the even and odd numbers means that it *must* be the
denumerable set.

At this point, defenders of Cantor might want to claim that "Yes, D2 is
the denumerable set, but so is D, because Cantor's proofs show that all
such sets are the same size." This would be incorrect, however, because
Cantor's claims rest entirely on two points: first, that D and E have the
same number of elements (which is true), and second, that the mapping
scheme used to establish the bijection does *not* accidentally redefine D
as being a new set, one which is half the size of the denumerable set.
Nowhere does Cantor, or anyone else for that matter, *prove* that no
accidental redefinition occurred. It is an assumption which cannot be
used to *prove* that D "really is" the same size as D2. Furthermore, it
is almost certainly a false assumption, because D2, since it contains
*all* the even and odd numbers (neither E nor O are accidentally
redefined by the mapping scheme), really is the denumerable set, and as
Figure 4 shows, it does *not* have a bijection with D.

A similar situation exists with line segments. The usual claim is that
all finite line segments contain exactly the same number of points, a
claim which is backed up with the following Figure.

|
| /
|/
|
/|
|/ |
/ |
/| |
/ | |
/__|__|__
A B C
Figure 5

In this proof, lines are drawn from point A through segment B and segment
C, each line intersecting each segment at a single point. An important
aspect of this situation is that for any two points on segment C
whatsoever, even points right next to each other, lines drawn from those
points to point A will also intersect segment B at two points. This must
be true, because if the two lines intersected segment B at a single
point, they would intersect each other twice, at segment B and point A: a
geometric impossibility. This means that for any point on segment C
whatsoever, we can find a single, unique, corresponding point on segment
B, simply by drawing a line from segment C through segment B to point A.
Since there is a corresponding point
on segment B for every point on segment A, it necessarily follows that
there are just as many points on segment B as there are on segment A,
even though segment A is twice as long as segment B.

The two line segments in Figure 5 do indeed have exactly the same number
of points, but as Figure 6 shows, a different method of linking the
points on segment B with the points on segment C can easily lead to a
different conclusion. In this figure, lines once again link the points on
segments B and C, but this time, the lines are parallel.

|
|
|
|
|_____|
|_____|
|_____|
|_____|
B C
Figure 6

In this example, every point in segment B has a corresponding point in
segment C, but the reverse is obviously not true, which means that the
two segments do not have the same number of points. In this example,
segment B obviously has only half the points of segment C.

Once again, we see that simply using a different method for aligning the
elements of two infinite sets leads to completely different conclusions
concerning their sizes. The first thing to note is that using lines to
link or match the points in one line segment with the points in another
line segment is in fact a perfectly valid method for comparing the number
of points in the two line segments. In other words, the conclusions about
each of the particular line segments in Figures 5 and 6 are correct. The
reason that Figures 5 and 6 lead to different conclusions is because,
once again, the method used to link the points on one segment with the
points on the other segment has the potential to *redefine* the
characteristics of the segments. In Figure 6, which uses parallel lines
to link the two segments, the distance between the lines remains constant
as they move from the shorter segment to the longer segment. As a result,
the points of intersection on the two segments are always an equal
distance apart, regardless of which parallel lines are used to link the
two segments. In Figure 5, however, the lines linking the two segments
are not parallel, which causes the points of intersection on the longer
segment to be farther apart than the points of intersection on the
shorter segment. In fact, for any two intersecting lines whatsoever, the
points of intersection are always exactly twice as far apart on the
longer segment as they are on the shorter segment, suggesting that the
points on the shorter segment have a greater *density* than the points on
the longer segment.

The subject of the density of points on the continuum is normally never
addressed, and for good reason. Outside of stating that there are
infinitely many points on any finite segment, which in turn means that
the points are infinitely dense, there is no way to specify what the
density of points on a segment actually is. Nevertheless, a “quantity of
points” does exist in each unit length, and since density is simply a
“quantity of something” per unit length, the idea of a density of points
on the continuum, a “continuum density,” is a valid concept. The only
problem is that for a single line or line segment it is also largely
meaningless, since there is no way to specify what the continuum density
for a particular line segment actually is.

It only takes a single glance at Figures 5 and 6, however, to know that
when two lines or line segments are involved, this is not the case. Here
we can describe the continuum density of one line segment relative to the
continuum density of the other line segment. In the case of Figure 5, the
continuum density of the shorter segment is exactly twice as great as the
continuum density of the longer segment. The fact that the continuum
density of the shorter segment is twice as great as the longer segment is
normally an unconscious and unstated aspect of the proof that “all
segments have the same number of points.” We will now restate the proof
that all line segments have the same number of points, this time
including the normally unstated portion in parenthesis: Take two line
segments, segment B, one unit long, and segment C, two units long.
(Segment B, although only half as long as segment C, has a continuum
density twice as great as segment C, giving it twice as many points per
unit length.) By drawing lines from point A through segment B and C, we
see that every single point on segment C corresponds to a single, unique
point on segment B, which in turn proves that there are just as many
points on segment B as there are on segment C.

Of course, this conclusion certainly is true, but given that the
continuum density of segment B is twice as great as segment C, it is very
difficult to get excited about the fact that a unit length of segment B
has the same number of points as two unit lengths of segment C. In fact,
our reaction almost cannot help but change from a “Gee, that's amazing”
to an infinitely more sarcastic, “Gee, do you think?” The basic trick, of
course, is that linking two segments with lines which are not parallel,
which do not remain the same distance apart, unavoidably *defines* the
two segments as having different continuum densities. The inherent
structure of this “proof” is such that the continuum densities will
automatically appear in exactly the proportions needed to
give both segments exactly the same number of points, regardless of their
initial relative lengths.

This example makes it even more obvious that the fundamental assumption
behind all of Cantor's work, namely that methods for matching the
elements of one set with the elements of another set cannot possibly
affect the characteristics of either set, is false. We can either reject
this assumption, in which case we no longer need to believe that there
are as many even numbers as integers, or that when we walk into a room
and stop at the 1/8 point, that we are really at the 1/2 point, or that
processes which apply to finite sets of any size "mystically vanish" when
we shift to infinite sets, or that all segments have the same number of
points, even when the continuum densities of those segments are *exactly
the same*, or we can accept it as a premise, and accept that a large
percentage of the laws of mathematics mystically become invalid when
dealing with infinite sets. To restate: the assumption that sets are
unaffected by any mapping method may be equivalent to the assumption that
all infinite sets are the same size, which is what Cantor was supposedly
proving.

There really is no question about whether Cantor's hidden assumption
concerning the effect of mapping methods winds up violating the other
premises of mathematics -- it does -- but since these violations all
occur "at infinity" (with the possible exception of the line segments),
it may actually be possible to accept it without violating laws in the
finite realm. To me this is a joke, but I am sure that some will be
reluctant to part with the current beliefs, simply because they are the
current beliefs. If there is one aspect of Cantor's assumption that may
fail under all circumstances, however, it may be the fact that it is
unavoidably incompatible with infinitesimals -- those objects which are
infinitely larger than a point, and yet infinitely smaller than any
finite length -- because the Achilles, using very basic premises,
unavoidably leads to the conclusion that an infinite number of the
halfway points from the center of the room to the door lie in a region
that is infinitesimally close to the door. The subject of infinitesimals,
along with the flaws in Cantor's proof concerning transfinite sets, will
be discussed in the next post.

Phil


Stephen Speicher

unread,
Nov 30, 2002, 9:56:06 PM11/30/02
to
On Sun, 1 Dec 2002, Phil wrote:

[Monumental snip ...]

> The subject of infinitesimals,
> along with the flaws in Cantor's proof concerning transfinite sets, will
> be discussed in the next post.
>

I can't wait. I have set aside a hard disk especially for it.

John Zinni

unread,
Dec 1, 2002, 1:13:02 AM12/1/02
to
"Phil" <tu...@jump.net> wrote in message news:3DE96A55...@jump.net...

> Many of our current beliefs concerning infinity and infinite sets come

[BIG snip]

(fingers in ears) La .. La .. La .. Can't hear you!!!

--
Cheers
John Zinni


Dirk Van de moortel

unread,
Dec 1, 2002, 5:10:18 AM12/1/02
to

"Phil" <tu...@jump.net> wrote in message news:3DE96A55...@jump.net...
> Many of our current beliefs concerning infinity and infinite sets come
> from Georg Cantor's "Mengenlehre" (theory of sets or classes, especially
> the infinite sets). In the Mengenlehre, Cantor was able to prove
> incredible things, such as, there are just as many positive even numbers
> as positive integers, that the sqr(infinity) = infinity, and the
> existence of different classes of infinity, the "transfinite sets."
> However, all of these conclusions rest on a single assumption, the belief
> that once a set has been defined, it cannot possibly be redefined by
> actions such as adding to the set, dividing the set by two, or the
> matching scheme used to link the elements of one set with the element of
> another set. The purpose of this post will be to examine that assumption.

The problem with you is that you know even less about mathematics
than you know about physics.

> One of the first potential problems is that the belief that any operation
> can be performed on an infinite set without changing the number of
> elements in the set *may* just be another way of stating that all
> infinite sets (at least all denumerable sets) have the same number of
> elements, which is what Cantor was supposedly proving. Obviously, if
> Cantor "proved" that all sets have the same number of elements by first
> *assuming* that all sets have the same number of elements, we have a
> problem.

And if he did *not* assume that, then *you* have a problem.

> I will briefly restate the situation from my earlier post. Cantor's proof
> that there are just as many positive even numbers as positive integers
> requires a *valid* one-to-one mapping method or bijection.

Take A={a,b,c} and B={1,2,3}.
Please give an example of an *invalid* bijection between A and B.
Please give an example of a *valid* bijection between A and B.


> Given the
> denumerable set D and the set of positive even numbers E, Cantor gave:
>
> D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }
> | | | | | | | | | |
> E = { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... }
> Figure 1
>
> Now, there is no question about whether a valid bijection exists for the
> two sets in Figure 1, because there is. The question is whether, in the
> process of *defining* the relationship between these two sets, Cantor
> also redefined D. An alternative mapping shows that this may indeed be
> the case, although this alternative does not by itself *prove* that a
> redefinition occurred.
>
> D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }
> | | | | |
> E = { 2, 4, 6, 8, 10, ... }
> Figure 2
>
> There is also no question that for the sets of Figure 2, a bijection does
> not exist, even though each and every element in the set of even numbers
> has again been linked to a unique element in the set of integers.

There is no question that a bijection *does* exist, since you gave
it in figure 1. The sets D and E are the same. The bijection of figure 1
does not go away when you draw a new figure with something that
is not even a bijection to begin with, since not all elements of D are
covered.
The existence of a bijection does not redefine the set.
The existence of a non-bijectional relation between the sets
does not redefine the sets.


Now, here is Cantor for Dummies.

Def 1:
By definition two sets have "the same number of elements" if
there is at least one bijection between the sets.

Def 2:
By definition a set is called "an infinite set" if it has "the same
number of elements" as a real subset of itself.

Theorem:
A set is called "an infinite set" if there is at least one bijection
between the set and a real subset of itself.

Can you prove the theorem?

Dirk Vdm


Phil

unread,
Dec 1, 2002, 5:13:43 PM12/1/02
to
Dirk! A response that I can actually work with! I'm impressed.

Dirk Van de moortel wrote:

> "Phil" <tu...@jump.net> wrote in message news:3DE96A55...@jump.net...
> > Many of our current beliefs concerning infinity and infinite sets come
> > from Georg Cantor's "Mengenlehre" (theory of sets or classes, especially
> > the infinite sets). In the Mengenlehre, Cantor was able to prove
> > incredible things, such as, there are just as many positive even numbers
> > as positive integers, that the sqr(infinity) = infinity, and the
> > existence of different classes of infinity, the "transfinite sets."
> > However, all of these conclusions rest on a single assumption, the belief
> > that once a set has been defined, it cannot possibly be redefined by
> > actions such as adding to the set, dividing the set by two, or the
> > matching scheme used to link the elements of one set with the element of
> > another set. The purpose of this post will be to examine that assumption.
>
> The problem with you is that you know even less about mathematics
> than you know about physics.
>
> > One of the first potential problems is that the belief that any operation
> > can be performed on an infinite set without changing the number of
> > elements in the set *may* just be another way of stating that all
> > infinite sets (at least all denumerable sets) have the same number of
> > elements, which is what Cantor was supposedly proving. Obviously, if
> > Cantor "proved" that all sets have the same number of elements by first
> > *assuming* that all sets have the same number of elements, we have a
> > problem.
>
> And if he did *not* assume that, then *you* have a problem.

Basically correct, although if Cantor's proofs rely on *anything* that is
neither an accepted premise of mathematics, nor a proven result which is based
on the accepted premises of mathematics, then it is a faulty proof, period.
That is one of the most basic facts of mathematical logic and reasoning. The
elements of any proof be be either accepted premises, or conclusions which
necessarily follow from accepted premises. I am claiming that Cantor's proofs
use an element -- the belief that sets cannot be accidentally redefined by any
mapping method -- which is not an accepted premise (except perhaps
unconsciously), nor can it be proven using the currently accepted premises of
mathematics. If this is correct, then the game is over, period. However, I then
go on to suggest the possibility that this element *may* be equivalent to the
premise that all (denumerable) infinite sets have exactly the same number of
elements, which of course would make Cantor's proofs an examples of circular
arguments.

> > I will briefly restate the situation from my earlier post. Cantor's proof
> > that there are just as many positive even numbers as positive integers
> > requires a *valid* one-to-one mapping method or bijection.
>
> Take A={a,b,c} and B={1,2,3}.
> Please give an example of an *invalid* bijection between A and B.
> Please give an example of a *valid* bijection between A and B.

Since A and B have the same number of elements, there are no invalid bijections
between them. However, I can think of an invalid method of mapping the elements
from A to B, given that we want the mapping method to prove or disprove a
bijection:

A = { 1, 2, 3 }
| |
B = { a, b, c }

This mapping method could be incorrectly construed as showing that A and B do
not have the same cardinality, the same number of elements. The problem,
obviously, is that although an element from A is indeed left unmapped, there is
also an element from B that has been left unmapped.

Would you accept that if A and B have the same cardinality, then it is
impossible to produce a on-to-one mapping method that leaves x elements of A
unmapped, that does not also leave x elements of B unmapped?

> > Given the
> > denumerable set D and the set of positive even numbers E, Cantor gave:
> >
> > D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }
> > | | | | | | | | | |
> > E = { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... }
> > Figure 1
> >
> > Now, there is no question about whether a valid bijection exists for the
> > two sets in Figure 1, because there is. The question is whether, in the
> > process of *defining* the relationship between these two sets, Cantor
> > also redefined D. An alternative mapping shows that this may indeed be
> > the case, although this alternative does not by itself *prove* that a
> > redefinition occurred.
> >
> > D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }
> > | | | | |
> > E = { 2, 4, 6, 8, 10, ... }
> > Figure 2
> >
> > There is also no question that for the sets of Figure 2, a bijection does
> > not exist, even though each and every element in the set of even numbers
> > has again been linked to a unique element in the set of integers.
>
> There is no question that a bijection *does* exist, since you gave
> it in figure 1. The sets D and E are the same. The bijection of figure 1
> does not go away when you draw a new figure with something that
> is not even a bijection to begin with, since not all elements of D are
> covered.

There is a logical flaw in your reasoning. I am trying to demonstrate that the
*assumption* that the initial definition for D remains unaffected by the method
used to map the elements of E onto the elements of D is false. I am
*specifically* attacking this assumption. You are claiming that my attack is
false *because* the assumption is true. Well of course if assumption x is true,
then it is true! My question to you is, how do you *know* that D's definition
cannot be affected by the choice of a mapping method? I have given a
demonstration that it is not true. In order to prove my demonstration false,
you cannot simply *assume* that the assumption I am attacking is true. You have
to *at least* find proofs that D cannot be affected by the choice of a mapping
method -- and the fact that many mathematicians believe it is not a proof --
and preferably, find an actual flaw in my reasoning. Do you think I have a
valid point here? Is the simple assertion that D really, really is unaffected
by mapping methods (so there) a mathematically valid defense against an
argument which *demonstrates* that D *is* affected?

Earlier I asked if you believe that, given sets A and B which *do have* the
same cardinality, that if a one-to-one mapping method leaves x elements from A
unmapped, then there *must* be x elements from B unmapped. I'm going to assume
for the moment that the answer is yes. Look at the Figure 2 again. Can you
honestly say that a single element from E has not been mapped to an element
from D? I mean, 2 is linked to 2, 812 is linked to 812 -- there really is no
way for an element from E to be unmapped. And yet, elements from D *are*
unmapped. How is this possible if D and E have the same cardinality? This
should at least raise the question in your mind -- and it is only a question,
not a proof -- of whether mapping methods can in fact change a set's
definition, because there is no question but that sets D and E *as shown in
Figure 1* -- which uses a different mapping method -- really do have the same
cardinality.

Let's begin (if I have not already exceeded your patience) by showing Figure 1
again, but this time renaming the sets as E' and X. At this point we are not
assuming anything for sure, only leaving open the *possibility* that E and E',
and D and X', may or may not be the same. Cantor implicitly assumes that once D
has been defined as the denumerable set, it remains unaffected by any choice of
mapping methods between it and set E, the set of all positive even numbers. The
task here is to *try* to determine whether this assumption is in fact valid,
without first (and simply) assuming that it is is or isn't.

E' = { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... }
| | | | | | | | | |

X = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }

Figure 3

Let's take the mapping method which is *shown* in Figure 3, and try to put it
in words, instead.

"Given the set of all positive even numbers, E', find a set of integers, X,
such that there exists a valid, one-to-one mapping between every element of E',
and every element of X."

Now, can you honestly state that this statement does not *automatically define*
set X as having *exactly* the same number of elements as set E'? Suppose that,
in some previous discussion, set X had been defined as being the denumerable
set. Would that definition *still exist*, untouched and unaffected, even after
the statement above? If so, then you can continue to claim that I am an idiot,
and that Cantor's proofs, which all rely on the ASSumption that once X is
defined as the denumerable set, that subsequently redefining it as an infinite
set of integers which has exactly the same number of elements as E' has no
effect on it, are valid, that they follow all the rules of mathematical logic
and reasoning. But you will be wrong -- at least, on the second point.

The truth is that every mapping scheme *potentially* has its own inherent
definition, as shown by the statement above, and these inherent definitions
cannot be ignored. The mapping method I showed in Figure 2 (shown again below
for convenience) is an example of a mapping method between the denumerable set
and the even numbers which does *not* redefine the denumerable set. The
difference is that, when put into words, my method merely requires that every
element in set E be mapped in a one-to-one way with an element from D, with no
elements from E left unmapped. This does not *inherently* define D (or X) as
being a set with *exactly* the same number of elements as E.

> D = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... }
> | | | | |
> E = { 2, 4, 6, 8, 10, ... }
> Figure 2

Remember, set theory, in the form that Cantor used (and helped develop) is not
that old (about 1870), and it is extremely unlikely that *everything* about
sets and set theory has already been figured out. In addition, Henry Poincare,
one of the most brilliant mathematicians of all time, said about Cantor's work,
"Later generations will regard Mengenlehre as a disease from which one has
recovered." As it happens, Poincare was right, because Cantor's work does
indeed include an invalid assumption,


> The existence of a bijection does not redefine the set.
> The existence of a non-bijectional relation between the sets
> does not redefine the sets.
>
> Now, here is Cantor for Dummies.
>
> Def 1:
> By definition two sets have "the same number of elements" if
> there is at least one bijection between the sets.

This definition is valid, meaning it does not contradict any of the other
premises and definitions of mathematics. The only thing you have to remember is
that a bijection only proves equal cardinality for the sets *as they actually
exist in the bijection.* The fact that set X may have been defined in a certain
way on some other day does not automatically prove that set x as it exists in
the bijection has the same definition.

> Def 2:
> By definition a set is called "an infinite set" if it has "the same
> number of elements" as a real subset of itself.

This definition relies on Cantor's conclusion that sets like D, 0.5 * D, and
log(D) all have exactly the same number of elements, a conclusion which in turn
relies on the assumption that mapping methods do *not* have the potential to
carry their own definitions, definitions which can redefine the denumerable set
as being, for example, "a set of integers with exactly the same number of
elements as the set of even numbers," or "a set of integers with a quantity of
elements equal to the log of the denumerable set." I really don't know how many
times I will have to demonstrate that this assumption is false before people
WAKE UP.

> Theorem:
> A set is called "an infinite set" if there is at least one bijection
> between the set and a real subset of itself.
>
> Can you prove the theorem?

I can in fact find a mapping method that will redefine the denumerable set as
an infinite set of integers which has exactly the same number of elements as
any set you like, such as the set of all positive even numbers, the log of the
denumerable set, etc. But just as first *defining* a line segment with a length
of 2 as having a continuum density half that of a line segment with a length of
1, and then "proving" that both segments have the same number of points ... who
cares?

Dirk, I'll be honest and state that I don't think much of your reasoning
ability. I don't think you really understand the difference between sequences
of thought that lead to conclusions which are *necessarily* true (meaning they
are in alignment with reality, whether that be the reality of the daily world
or the reality of a world based on mathematical premises), and sequences which
merely "sound good," i.e., are internally consistent, but which do not have any
alignment with reality. Furthermore, you're obnoxious as hell. You strike me as
a "true believer," who thinks that all the current beliefs are true, and that
anyone who questions them should be put under house arrest for the rest of his
life, as was (properly) done with Galileo.

However, you have a wife and two boys, I hope you lead a long a healthy life so
that you can take care of them for many years, and I don't think you are
*completely* stupid! Therefore, I will be at least a little surprised if you
can seriously examine what I am saying and still, in "true believer" fashion,
actually think that mapping methods do *not* carry their own inherent
definitions which can, and sometimes do, redefine (actually just "define") the
sets that they map.

Phil


Dirk Van de moortel

unread,
Dec 1, 2002, 5:33:54 PM12/1/02
to

"Phil" <tu...@jump.net> wrote in message news:3DEA89C2...@jump.net...

[snip]

> > > I will briefly restate the situation from my earlier post. Cantor's proof
> > > that there are just as many positive even numbers as positive integers
> > > requires a *valid* one-to-one mapping method or bijection.
> >
> > Take A={a,b,c} and B={1,2,3}.
> > Please give an example of an *invalid* bijection between A and B.
> > Please give an example of a *valid* bijection between A and B.
>
> Since A and B have the same number of elements, there are no invalid bijections
> between them. However, I can think of an invalid method of mapping the elements
> from A to B, given that we want the mapping method to prove or disprove a
> bijection:
>
> A = { 1, 2, 3 }
> | |
> B = { a, b, c }
>
> This mapping method could be incorrectly construed as showing that A and B do
> not have the same cardinality, the same number of elements. The problem,
> obviously, is that although an element from A is indeed left unmapped, there is
> also an element from B that has been left unmapped.
>
> Would you accept that if A and B have the same cardinality, then it is
> impossible to produce a on-to-one mapping method that leaves x elements of A
> unmapped, that does not also leave x elements of B unmapped?

Def 1:


By definition two sets have "the same number of elements" if
there is at least one bijection between the sets.

The set M = { (1,a), (2,b), (3,c) } is a bijection between A and B.
So by definition A and B have "the same number of elements".

The existence of the bijection M is independent of whatever kind
of relation you write next: you can write down sets like
N = { (2,a), (3,b) }
or
P = { (2,a), (3,a), (1,c) }
or
Q = A x B
as much as you like and say about them what you like, but nothing
will make the set M and its existence go away, so nothing will
invalidate the proof that A and B have the same number of elements.

Do you understand this?

Do you also understand that there is no such thing as an
"invalid bijection"?

Dirk Vdm

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