Cantor's first proof in DETAILS

[Zuhair]

Apologies beforehand for this long proof, and for any possible errors,
typos, mistakes that most possibly would be there with such a long
draft. I'v written this with the intention to give what I think it to
be the complete story of Cantor's first proof. So the following is my
view of this proof, it came from reading on-line proofs other than the
original one, since I don't have the original article of Cantor.
References given below.

If a mistake in this proof is noticed, then please feel free to
outline it.

CANTORS FIRST PROOF OF UNCOUNTABILITY OF REALS
--------------------------------------------------------------------------------

Statement: There is no bijection between the set N of all naturals
and the set R of all reals.

Proof:
We prove that for every injection (x_n) from N to R, there
exist a real J such that J not in the range of (x_n).

Notation: for every x_i, i shall be called the place of x_i in (x_n),
while x is the value of x_i. Whenever mentioned in this article
symbols < , > , = and =/= are comparisons of the values of entries of
sequences mentioned, while the places of those entries shall be
compared by "lies before" , "lies after" , is the first entry, is the
last entry, in the same place, etc..

(x_n) is said to have the Intermediate Value property (IVP) iff
for every two entries x_i,x_j of (x_n) there exist an entry x_k
of (x_n) such that: x_i < x_k < x_j or x_i > x_k > x_j

If (x_n) don't possess IVP, then it is easy to find J.

If (x_n) possess IVP, then we construct sequences (a_n), (b_n)
in the following manner:

Let a_0 = x_0
Let b_0 be the first entry in (x_n) such that b_0 > a_0.
Let a_i+1 be the first entry in (x_n) such that a_i < a_i+1 < b_i.
Let b_i+1 be the first entry in (x_n) such that a_i+1 < b_i+1 < b_i.

We notice that for every i,j: j>i -> (a_j > a_i) & (b_j < b_i)
i.e. (a_i) is an increasing sequence, and (b_i) is a decreasing
sequence.

We also notice that for every i: a_i < b_i
since a_0 < b_0, and since by definition
for all i. a_i+1 < b_i+1, then by
induction:

for all i: a_i < b_i.

From that we have the following result:

Result 1: for all i. for all a_i. Exist a_i+1 (a_i < a_i+1).
Result 2: for all i. for all b_i. Exist b_i+1 (b_i+1 < b_i)

Result 3: for all i,j: a_i < b_j
Proof: either i=j, or i<j or i>j
if i=j then a_i < b_i & b_i = b_j so by identity a_i < b_j
if i<j then a_i < a_j & a_j < b_j then by transitivity a_i < b_j
if i>j then a_i < b_i & b_i < b_j then by transitivity a_i < b_j

Define (lies before): for all i,k. a_i lies before x_k in (x_n) iff
Exist m. a_i = x_m & m < k

Define (lies after): for all i,k. a_i lies after x_k in (x_n) iff
Exist m. a_i = x_m & k < m.

Similar definitions applies to b_i.

Result 4: for all i: b_i lies after a_i in (x_n)

Proof: b_0 lies after a_0 by definition.
for all i. i>0 -> a_i-1 < b_i < b_i-1
But also i>0 -> a_i-1 < a_i < b_i-1
Now if we suppose that b_i lies before a_i in (x_n)
then a_i will no longer be the first item in (x_n)
that is > a_i-1 and < b_i-1, unless b_i lies at
the same place of a_i in (x_n). which is impossible
since a_i < b_i and (x_n) is an injection.

Result 5: for all i: a_i+1 lies after b_i in (x_n)

Proof: a_1 lies after b_0 in (x_n).
i >0 -> a_i < b_i < b_i-1
i>0 -> a_i < a_i+1 < b_i-1
Now if a_i+1 lies before b_i in (x_n), then
b_i would no longer be the first item in (x_n)
that is > a_i and < b_i-1, unless a_i+1 and b_i
lie at the same place in (x_n), which is impossible
since a_i+1 < b_i and (x_n) is an injection.

Result 6: for all i: a_i+1 lies after a_i & b_i+1 lies after b_i
Since at each i. a_i+1 lies after b_i which lies after a_i
then by transitivity a_i+1 lies after a_i
Similarly b_i+1 lies after a_i+1 which lies after b_i.

Informally as i increase each a_i,b_i is coming from a deeper
and deeper place in (x_n).

Define (external): for all k. x_k external in (x_n) iff
x_k an item of (x_n) &
x_k not an item of (a_n) &
x_k not an item of (b_n).

Result 7: For every x_k. x_k external in (x_n) ->
Exist i. (a_i lies before x_k in (x_n) & a_i+1 lies after x_k in
(x_n))

Proof: x_0 (which is a_0) lies before x_k, and if the above
doesn't hold then for every a_i lying before x_k in (x_n)
a_i+1 would lie also before x_k in (x_n), since the place
of each a_i is a natural number and so is k, then this would
entail the existence of infinitely many naturals before k
which is absurd.

Define : a_i is last of (a_n) lying before x_k in (x_n) iff
(a_i lies before x_k in (x_n) & ~ (a_i+1 lies before x_k in (x_n)))

Result 8: For all k. x_k external in (x_n) ->
for all i. (a_i is last of (a_n) lying before x_k in (x_n) ->
b_i lies before x_k in (x_n) or b_i lies after x_k in (x_n))

Proof: properties of natural numbers, and definition of external.

Define (intervene): for all k,i. x_k intervene a_i,b_i in (x_n) iff
x_k is external in (x_n) &
a_i is last of (a_n) lying before x_k in (x_n) &
b_i lies after x_k in (x_n).

Define (passed): for all k,i. x_k passed a_i,b_i in (x_n) iff
x_k is external in (x_n) &
a_i is last of (a_n) lying before x_k in (x_n) &
b_i lies before x_k in (x_n).

Result 9: for all k. x_k is external in (x_n) ->
Exist i. x_k intervene a_i,b_i in (x_n) or x_k passed a_i,b_i in
(x_n).

Proof: Results 7.8 and definitions above.

Lemma 1: for all k,i. x_k intervene a_i+1,b_i+1 in (x_n) ->
x_k < a_i+1 or x_k > b_i

Proof: if not then x_k would be an item of (x_n) that is
> a_i+1 & < b_i and since it lies before b_i+1 in
(x_n) then this violates the definition of b_i+1.

Lemma 2: for all k,i. x_k passed a_i,b_i in (x_n) ->
x_k < a_i or x_k > b_i

Proof: if not then x_k would be an item of (x_n)
that is > a_i & < b_i and since it lies before a_i+1
in (x_n) then this violates the definition of a_i+1.


let L be the least upper bound on (a_n), that is

(for all i. a_i =< L) & for all X. (for all i. a_i =< X) -> L =< X.

Theorem 1. for all i. a_i =/= L

Proof: assume there exist t such that a_t = L
then a_t+1 > a_t, but from definition of L we
must have L >= a_t+1, and since L=a_t
thus we'll arrive at a_t >= a_t+1 > a_t
which is absurd.

Theorem 2. for all i. L < b_i

Proof: assume there exist r such that b_r =< L
then b_r+1 < L and b_r+1 >= a_i for all i
thus L is not the "least" upper bound of (a_n).
A contradiction.

Theorem 3: for all i,j: a_i < L < b_j

Proof: Definition of L and Theorem 1,2.


Theorem 4. for all i. x_i =/= L

Proof:
Suppose that x_k = L, then x_k is external in (x_n) (Th.1,2)

for all i If x_k intervene a_i+1,b_i+1 in (x_n), then
x_k < a_i+1 or x_k > b_i , But a_i+1 < L < b_i.
A contradiction.

If x_k intervene a_0,b_0, then x_k < a_0
But a_0 < L and L=x_k.
A contradiction.

If x_k passed a_i, b_i in (x_n), then
x_k < a_i or x_k > b_i, But a_i < L < b_i.
A contradiction.

Let J=L

QED

Corollary:
For every injection (x_n*) from N* to R, where
N* is bijective to N. Then (x_n*) misses a real
from its range.

Proof: Let (g(n*)) be a bijection from N* to N.
Define (x_n) as {y_n| Exist n*: y_n* in (x_n*) & g(n*) = n}
so range of (x_n) = range of (x_n*)
But domain of (x_n) is N.
So (x_n) is an injection from N to R.
Thus it misses a real. (above proof),
so (x_n*) misses a real too, since it has
the same range of (x_n).

QED

References:

[1] http://www.math.jhu.edu/~wright/Cantor_Pick_Phi.pdf


[2]http://www.proofwiki.org/wiki/Real_Numbers_are_Uncountable/
Cantor's_First_Proof


[3]http://en.wikipedia.org/wiki/Cantor's_first_uncountability_proof


Zuhair

[Zuhair]

A better statement would be:
for every two entries x_i, x_j of (x_n) where x_i < x_j
there exist an entry x_k of (x_n) such that: x_i < x_k < x_j.

Zuhair

[Charlie-Boo]

On Nov 12, 4:05 am, Zuhair <zaljo...@gmail.com> wrote:
> Apologies beforehand for this long proof, and for any possible errors,
> typos, mistakes that most possibly would be there with such a long
> draft. I'v written this with the intention to give what I think it to
> be the complete story of Cantor's first proof. So the following is my
> view of this proof, it came from reading on-line proofs other than the
> original one, since I don't have the original article of Cantor.
> References given below.
>
> If a mistake in this proof is noticed, then please feel free to
> outline it.
>
> CANTORS FIRST PROOF OF UNCOUNTABILITY OF REALS

> ---------------------------------------------------------------------------­-----

This is very sloppy. You need to give a high level explanation
first. Otherwise, people have to waste time going through formalisms
when the problem is with the logic and can be exposed by examining the
logic alone. You immediately go into your formalisms. This is like
trying to figure out someone's computer programs without the specs
(higher level) or documentation (redundant statements that serve only
to clarify by expressing something in a different way.)

C-B

[Jesse F. Hughes]

Charlie-Boo <shyma...@gmail.com> writes:

> This is very sloppy. You need to give a high level explanation
> first. Otherwise, people have to waste time going through formalisms
> when the problem is with the logic and can be exposed by examining the
> logic alone. You immediately go into your formalisms. This is like
> trying to figure out someone's computer programs without the specs
> (higher level) or documentation (redundant statements that serve only
> to clarify by expressing something in a different way.)

Zuhair expects that everyone interested in the details already knows the
proof, Charlie. Not an unreasonable assumption.

--
"I suggest to those who listen that they enjoy the world, whatever
their piece of it may be, as much as they can over the next few days,
as soon enough, it will pass away, thanks to people who call
themselves 'mathematicians'." -- JSH envisions geek Ragnarok

[Zuhair]

What the references are for then? you'll get the informal level of
this argument there.
To go explicate every bit and piece informally as well as formally
would be really taxing.

Zuhair

[LudovicoVan]

"Zuhair" <zalj...@gmail.com> wrote in message
news:86a85cce-2a84-4c9f...@o8g2000yqh.googlegroups.com...

> Let a_0 = x_0
> Let b_0 be the first entry in (x_n) such that b_0 > a_0.
> Let a_i+1 be the first entry in (x_n) such that a_i < a_i+1 < b_i.
> Let b_i+1 be the first entry in (x_n) such that a_i+1 < b_i+1 < b_i.

In Cantor's proof a_{i+1} and b_{i+1} are the two first entries encountered
(in any order) in (x_n) *after* the entries corresponding to a_i and b_i.
This does not seem to be the case with your proof, where it instead seems
that entries are just picked every time restarting from the beginning of
(x_n).

Could you clarify? I'd like to be sure before I proceed reading it...

-LV

[LudovicoVan]

"LudovicoVan" <ju...@diegidio.name> wrote in message
news:k7rehn$rci$1...@dont-email.me...

> "Zuhair" <zalj...@gmail.com> wrote in message
> news:86a85cce-2a84-4c9f...@o8g2000yqh.googlegroups.com...
>
>> Let a_0 = x_0
>> Let b_0 be the first entry in (x_n) such that b_0 > a_0.
>> Let a_i+1 be the first entry in (x_n) such that a_i < a_i+1 < b_i.
>> Let b_i+1 be the first entry in (x_n) such that a_i+1 < b_i+1 < b_i.
>
> In Cantor's proof a_{i+1} and b_{i+1} are the two first entries
> encountered

That should read: a_{i+1} and b_{i+1} are the first two entries...

-LV

[Jesse F. Hughes]

Zuhair <zalj...@gmail.com> writes:

> What the references are for then? you'll get the informal level of
> this argument there.
> To go explicate every bit and piece informally as well as formally
> would be really taxing.

Charlie is an ass and a blowhard. I wouldn't take his comments too
seriously.

--
"It's one of the easiest tickets to true fame--not this silly stuff
where people cheer you for a few years and then forget about you--but
the kind of fame where school kids have to read your biography and do
reports on you." -- Another reason to support James S. Harris.

[Uirgil]

In article <k7rehn$rci$1...@dont-email.me>,

"LudovicoVan" <ju...@diegidio.name> wrote:

> "Zuhair" <zalj...@gmail.com> wrote in message
> news:86a85cce-2a84-4c9f...@o8g2000yqh.googlegroups.com...
>
> > Let a_0 = x_0
> > Let b_0 be the first entry in (x_n) such that b_0 > a_0.
> > Let a_i+1 be the first entry in (x_n) such that a_i < a_i+1 < b_i.
> > Let b_i+1 be the first entry in (x_n) such that a_i+1 < b_i+1 < b_i.
>
> In Cantor's proof a_{i+1} and b_{i+1} are the two first entries encountered
> (in any order) in (x_n) *after* the entries corresponding to a_i and b_i.
> This does not seem to be the case with your proof, where it instead seems
> that entries are just picked every time restarting from the beginning of
> (x_n).

It does not seem that way to those who are capable of reading what
Zuhair said.

>
> Could you clarify? I'd like to be sure before I proceed reading it...

What is not clear? Zuhair's plan clearly produces a nested sequence of
closed intervals I_i = [a_i,b_i] with each I_(n+1) a proper subinterval
of the interior of I_n.

[Uirgil]

In article <k7reln$sas$1...@dont-email.me>,

"LudovicoVan" <ju...@diegidio.name> wrote:

> "LudovicoVan" <ju...@diegidio.name> wrote in message
> news:k7rehn$rci$1...@dont-email.me...
> > "Zuhair" <zalj...@gmail.com> wrote in message
> > news:86a85cce-2a84-4c9f...@o8g2000yqh.googlegroups.com...
> >
> >> Let a_0 = x_0
> >> Let b_0 be the first entry in (x_n) such that b_0 > a_0.
> >> Let a_i+1 be the first entry in (x_n) such that a_i < a_i+1 < b_i.
> >> Let b_i+1 be the first entry in (x_n) such that a_i+1 < b_i+1 < b_i.
> >
> > In Cantor's proof a_{i+1} and b_{i+1} are the two first entries
> > encountered
>
> That should read: a_{i+1} and b_{i+1} are the first two entries...
>
> -LV

Zuhair's method works quite as well as Cantor's.

At least for those who can understand them.

[Shmuel Metz]

In <k7rehn$rci$1...@dont-email.me>, on 11/12/2012

at 06:18 PM, "LudovicoVan" <ju...@diegidio.name> said:

>In Cantor's proof a_{i+1} and b_{i+1} are the two first entries
>encountered (in any order) in (x_n) *after* the entries
>corresponding to a_i and b_i. This does not seem to be the case
>with your proof, where it instead seems that entries are just
>picked every time restarting from the beginning of (x_n).

Things equal to the same thing are equal to each other.

>Could you clarify?

Yes.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org

[Zuhair]

On Nov 12, 9:18 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message

Yes in this proof the entries are as you said will be picked every
time restarting from the beginning of (x_n) provided you follow the
rules stipulated in definition of the a's and b's. BUT you'll see as
you go down the proof that this will also lead to every a_i+1 and b_i
+1 coming after a_i , b_i in (x_n). (See Result 6 of this proof).

As regards the "order" of a_i and b_i, then if you meant by order the
quantitative comparison of their "values" then clearly in this proof
we MUST have a_i < b_i by definition. But if you meant by "order" the
places of them in (x_n), then the mere definition of the a's and b's
do not mention itself any place order restriction, but as I said still
by following those rules this will eventually lead to each b_i coming
after a_i in (x_n) [Result 4].

Also I agree with your notation that a_i+1 is better written as a_{i
+1}. But I guess it is understood like that anyway.

Zuhair

[LudovicoVan]

"Zuhair" <zalj...@gmail.com> wrote in message
news:86a85cce-2a84-4c9f...@o8g2000yqh.googlegroups.com...
<snip>

> Theorem 4. for all i. x_i =/= L
>

> <...>
>
> Let J=L
>
> QED

Same argument, same objection: as easily proven, the limit interval here
must be degenerate, that is it is the singleton (in interval notation)
[L;L]. So what you claim amounts to saying that the limit value L is not in
(x_n), but that is just incorrect in that if you consider the limit value,
then of course it does belong to (x_n), in the limit! More formally L =
lim_{n->oo} (a_n) = lim_{n->oo} (b_n) , then just consider an injection from
N* instead of N and you can even talk meaningfully about that "last value".

You should rather try and show the mistake in my objection instead of
proposing the same argument again and again. As I had put it there:

<< an omega-th end-point, a_oo, would necessarily be drawn from an omega-th
entry of the sequence! Formally, we have the following property:

A m : a_m e (x_n) & b_m e (x_n)

That works not only for n and m in N, but also for n and m in N*. >>

Objection to Cantor's First Proof
<https://groups.google.com/d/msg/sci.math/T2V4Jh7zzD8/wDM_wsyQZ0QJ>

(Note that mine is against Cantor's First Proof of which yours remains a
paraphrase, not a faithful reproduction.)

-LV

[Zuhair]

On Nov 13, 8:33 pm, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message

Honestly I couldn't understand your objection because it is not well
phrased (to me at least).
For instance what is the domain of (x_n), (a_n) , (b_n) in your
account? The first part of the argument (BEFORE the COROLLARY) that
I've presented here is stipulated clearly to be about (x_n) being an
injection from N to R, and also clearly it follows from the definition
of the a's and b's that (a_n) as well as (b_n) would have their
domains being N also, and N is stipulated clearly to be the set of all
natural numbers. So in this presentation there is NO omega-th end
point in (x_n) nor there is in (a_n) nor in (b_n) because to have an
Omega-th end point you need the domain of those functions to have
Omega (or any limit of all naturals) as an element, and N clearly
doesn't have Omega as an element, nor does it have any element of it
that has an omega-th position.

So your objection is (as far as I can tell) is not applicable to the
first part of the proof presented here (i.e. the part before the
corollary). Do you agree with that? I mean do you agree that the FIRST
part of the proof which is about injections from N to R succeed in
proving the existence of a real that is not in the range of all such
injections. Remember again N is fixed to be the set of all natural
numbers! so do you agree to this part of the proof?

Now about the idea of using N* instead of N as the domain of (x_n),
(a_n) and (b_n), where N* as far as I understand is some countable
infinite well ordered set that has an omega-th place like for example
N Union {oo} where oo is any limit of all natural numbers, an explicit
example would be N*=Omega+1={0,1,2,...,Omega} as defined by Von
Neumann.

Of course it is crystal clear that if we use N* as the domain of
(x_n), (a_n) and (b_n) then of course it can "Temporarily" elude
Cantor's argument (for that setting, but not for the setting where N
is the domain) this is clear, much as the diagonal put on top of the
original set does temporarily cast such impression, but unfortunately
still Cantor's argument Catches it, see the Corollary that I've
presented, it tackles the case where N* is the domain of (x_n) (a_n)
and (b_n), you'll see that we can define a new sequence (x'_n) that
have the same range as (x_n) but with the domain being N, and since
the domain of (x'_n) is N, then we'll simply apply the first part of
the proof on (x'_n) and elucidate some real that is not in its range,
and thus not in the range (x_n) (since (x_n) and (x'_n) have exactly
the same range) and thus even with this case (x'_n) which has the
extended domain N*, still would be proved to be missing a real (See
Corollary of this proof).

Zuhair

[Zuhair]

sorry for the last line, I meant,.... thus even with this case (x_n)
which
has the extended domain N*,.....

Zuhair

[Uirgil]

In article <k7u09t$tjd$1...@dont-email.me>,

"LudovicoVan" <ju...@diegidio.name> wrote:

> "Zuhair" <zalj...@gmail.com> wrote in message
> news:86a85cce-2a84-4c9f...@o8g2000yqh.googlegroups.com...
> <snip>
>
> > Theorem 4. for all i. x_i =/= L
> >
> > <...>
> >
> > Let J=L
> >
> > QED
>
> Same argument, same objection: as easily proven, the limit interval here
> must be degenerate, that is it is the singleton (in interval notation)
> [L;L]. So what you claim amounts to saying that the limit value L is not in
> (x_n), but that is just incorrect in that if you consider the limit value,
> then of course it does belong to (x_n), in the limit! More formally L =
> lim_{n->oo} (a_n) = lim_{n->oo} (b_n) , then just consider an injection from
> N* instead of N and you can even talk meaningfully about that "last value".

The limit of a strictly increasing sequence (the a_i) is NOT EVER a
member of the sequence.

The limit of a strictly decreasing sequence (The b_i) is NOT EVER a
member of the sequence.

No values which are bounded below by a strictly increasing sequence and
bounded above by a strictly decreasing sequence are members of either
seequence.

Thus proving that, given any sequence of values in R, there must be
values in R not appearing in that sequence.

>
> You should rather try and show the mistake in my objection instead of
> proposing the same argument again and again.

Until you can falsify the original argument, which you have not done,
there is no need to falsify your false argument against it.

As I had put it there:
>
> << an omega-th end-point, a_oo, would necessarily be drawn from an omega-th
> entry of the sequence! Formally, we have the following property:
>
> A m : a_m e (x_n) & b_m e (x_n)
>
> That works not only for n and m in N, but also for n and m in N*. >>

That assumes that one is forced to work with N*, whereas sensible people
work with N and have no problems.

>
> Objection to Cantor's First Proof
> <https://groups.google.com/d/msg/sci.math/T2V4Jh7zzD8/wDM_wsyQZ0QJ>
>
> (Note that mine is against Cantor's First Proof of which yours remains a
> paraphrase, not a faithful reproduction.)

Your alleged argument against the Cantor proof does not work against
either Cantor's proof, nor Zuhair's proof, nor my proof for that matter,
since your N* is irrelevant for all of them.

[LudovicoVan]

"Uirgil" <uir...@uirgil.ur> wrote in message
news:uirgil-91F13B....@BIGNEWS.USENETMONSTER.COM...

> No values which are bounded below by a strictly increasing sequence and
> bounded above by a strictly decreasing sequence are members of either
> seequence.
>
> Thus proving that, given any sequence of values in R, there must be
> values in R not appearing in that sequence.

I'll have a look at Zuhair's follow-up as soon as I manage, but let me for
now just point out that the above argument is obviously bogus: the rationals
too are dense (have the IVP as Zuhair has called it) and, by the very same
argument, we have proved that the rationals too are not countable... see?

-LV

[Zuhair]

On Nov 13, 11:16 pm, Uirgil <uir...@uirgil.ur> wrote:
> In article <k7u09t$tj...@dont-email.me>,
>
>
>
>
>
>
>
>
>
>  "LudovicoVan" <ju...@diegidio.name> wrote:
> > "Zuhair" <zaljo...@gmail.com> wrote in message

I showed in the Corollary that even if he use N* as the domain of
(x_n), still we can prove there is a missing real from the range of
(x_n). So Cantor's argument or my rephrasing of it both can easily be
shown to be applicable to N* (any set having a bijection with N) as
well as N.

Zuhair

[LudovicoVan]

"Zuhair" <zalj...@gmail.com> wrote in message

news:3929e6b6-2932-401d...@y6g2000vbb.googlegroups.com...

> On Nov 13, 11:16 pm, Uirgil <uir...@uirgil.ur> wrote:

<snip>

>> Your alleged argument against the Cantor proof does not work against
>> either Cantor's proof, nor Zuhair's proof, nor my proof for that matter,
>> since your N* is irrelevant for all of them.
>
> I showed in the Corollary that even if he use N* as the domain of
> (x_n), still we can prove there is a missing real from the range of
> (x_n). So Cantor's argument or my rephrasing of it both can easily be
> shown to be applicable to N* (any set having a bijection with N) as
> well as N.

You are simply missing the point there: we don't need N* to disprove Cantor,
we need N* to go beyond it and the standard notion of countability. In
fact, that there is a bijection between N* and N is a bogus argument too, as
the matter is rather about different order types.

-LV

[Uirgil]

In article <k7uf0m$v1r$1...@dont-email.me>,

"LudovicoVan" <ju...@diegidio.name> wrote:

> "Zuhair" <zalj...@gmail.com> wrote in message
> news:3929e6b6-2932-401d...@y6g2000vbb.googlegroups.com...
> > On Nov 13, 11:16 pm, Uirgil <uir...@uirgil.ur> wrote:
> <snip>
>
> >> Your alleged argument against the Cantor proof does not work against
> >> either Cantor's proof, nor Zuhair's proof, nor my proof for that matter,
> >> since your N* is irrelevant for all of them.
> >
> > I showed in the Corollary that even if he use N* as the domain of
> > (x_n), still we can prove there is a missing real from the range of
> > (x_n). So Cantor's argument or my rephrasing of it both can easily be
> > shown to be applicable to N* (any set having a bijection with N) as
> > well as N.
>
> You are simply missing the point there: we don't need N* to disprove Cantor,
> we need N* to go beyond it and the standard notion of countability.

I have yet to see you produces a valid disproof of Cantor either with N
or with N*.

[Uirgil]

In article <k7udtq$np6$1...@dont-email.me>,

The difference being that a monotone but finitely bounded sequence of
rationals need not have a limit among the rationals but MUST have a
limit among the reals, a LUB or GLB.

Density is not enough distinguish between Q and R, but the GLB/ LUB
property is enough.

Any densely ordered interval of positive length having the GLB/LUB
property is uncountable.

[LudovicoVan]

"Uirgil" <uir...@uirgil.ur> wrote in message

news:uirgil-B3AA26....@BIGNEWS.USENETMONSTER.COM...

> In article <k7udtq$np6$1...@dont-email.me>,
> "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Uirgil" <uir...@uirgil.ur> wrote in message
>> news:uirgil-91F13B....@BIGNEWS.USENETMONSTER.COM...
>>
>> > No values which are bounded below by a strictly increasing sequence and
>> > bounded above by a strictly decreasing sequence are members of either
>> > seequence.
>> >
>> > Thus proving that, given any sequence of values in R, there must be
>> > values in R not appearing in that sequence.
>>
>> I'll have a look at Zuhair's follow-up as soon as I manage, but let me
>> for
>> now just point out that the above argument is obviously bogus: the
>> rationals
>> too are dense (have the IVP as Zuhair has called it) and, by the very
>> same
>> argument, we have proved that the rationals too are not countable... see?
>

> The difference being that a monotone but finitely bounded sequence of
> rationals need not have a limit among the rationals but MUST have a
> limit among the reals, a LUB or GLB.

Yes, it's the *completeness* property that is required. Anyway, as
anticipated, I'll have to come back to this when I have time: the devil is
in the details!

-LV

[Charlie-Boo]

On Nov 12, 9:18 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Charlie-Boo <shymath...@gmail.com> writes:
> > This is very sloppy.  You need to give a high level explanation
> > first.  Otherwise, people have to waste time going through formalisms
> > when the problem is with the logic and can be exposed by examining the
> > logic alone.  You immediately go into your formalisms.  This is like
> > trying to figure out someone's computer programs without the specs
> > (higher level) or documentation (redundant statements that serve only
> > to clarify by expressing something in a different way.)
>
> Zuhair expects that everyone interested in the details already knows the
> proof, Charlie.  Not an unreasonable assumption.

Reasonable if he gives a link to a high level description of a proof
that he claims that he is exactly expressing.

C-B

[Charlie-Boo]

The first link has no high level description, the second link is empty
and the 3rd one does not show a parallel proof at a high level.

I can hardly imagine a person writing a proof without first developing
it at a high level then working out the details. Like any good
programmer, one can include the high level in parallel with the
details. If you want to be a bad programmer, then you have raised the
cost of one contributing (analogous to mainintain someone else's
software) which is your decision. Good luck. :)

C-B

[Uirgil]

In article <k7uh30$d4p$1...@dont-email.me>,

The completeness property will still be there when you come back!

[Jesse F. Hughes]

Charlie-Boo <shyma...@gmail.com> writes:

> On Nov 12, 9:18 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Charlie-Boo <shymath...@gmail.com> writes:
>> > This is very sloppy.  You need to give a high level explanation
>> > first.  Otherwise, people have to waste time going through formalisms
>> > when the problem is with the logic and can be exposed by examining the
>> > logic alone.  You immediately go into your formalisms.  This is like
>> > trying to figure out someone's computer programs without the specs
>> > (higher level) or documentation (redundant statements that serve only
>> > to clarify by expressing something in a different way.)
>>
>> Zuhair expects that everyone interested in the details already knows the
>> proof, Charlie.  Not an unreasonable assumption.
>
> Reasonable if he gives a link to a high level description of a proof
> that he claims that he is exactly expressing.

Yeah, great point.

He should have put something like this in his post.

References:

[1] http://www.math.jhu.edu/~wright/Cantor_Pick_Phi.pdf


[2]http://www.proofwiki.org/wiki/Real_Numbers_are_Uncountable/
Cantor's_First_Proof


[3]http://en.wikipedia.org/wiki/Cantor's_first_uncountability_proof

Oh, wait.

--
Jesse F. Hughes

"Never jibe a catboat in a fresh breeze when you have ladies on
board." -- New York Times, Nov. 29, 1885

[Shmuel Metz]

In <k7u09t$tjd$1...@dont-email.me>, on 11/13/2012

at 05:05 PM, "LudovicoVan" <ju...@diegidio.name> said:

>You should rather try and show the mistake in my objection

How many times?

[Shmuel Metz]

In <k7udtq$np6$1...@dont-email.me>, on 11/13/2012

at 09:25 PM, "LudovicoVan" <ju...@diegidio.name> said:

>I'll have a look at Zuhair's follow-up as soon as I manage, but let
>me for now just point out that the above argument is obviously
>bogus: the rationals too are dense (have the IVP as Zuhair has
>called it) and, by the very same argument, we have proved that the
>rationals too are not countable

Not even close; the argument does not show that the limit is rational.

[Zuhair]

On Nov 14, 12:45 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message

Now I think I'm beginning to somewhat perhaps understand your
argument. I think (I'm not sure though) that what you want to say is
that when we are having arguments with "LIMITS" then we must design
the whole argument such that the Limit comes from the sequence, and if
this design was not made then the argument is inherently deficient as
far as the truth of inferences derived from it is concerned. So what
you are trying to say is that Cantor's argument began with incomplete
arsenal so it ended up with misleading inferences. You are making an
argument at TRUTH level of the matter, and yet it is concerned with
formal technicality as well, which is an argument beyond the strict
formal technicality.

Anyhow if I'm correct, this form of reasoning for it to stand the
quest, then there must be a clear line of justification for it. For
instance the argument about whether the reals are countable actually
means literally whether there is a bijection between the reals and N,
so N is at the heart of the subject. Now to go and say that
countability of the reals (which means bijectivity of reals to N) can
only be reached about by circumventing N and using another countable
infinite set N* as the domain for any sequence in an argument using
limits is really strange somehow.

What you are having is the following:

[1]When we use N as the domain of injections (x_n), (a_n) and (b_n),
then Cantors argument PROVES and SHOWS that there is a real that is
not in the range of those functions.

[2]When we use N* as the domain of injections (x_n), (a_n) and (b_n),
then Cantor's argument will seize from working in the same way to show
the missing real.

[3]However we also have the corollary that even when we use N* as the
domain of those functions, still we can by a single common well
defined way define another sequence with exactly the same range of
those functions but from domain N, and we can apply Cantor's argument
and SHOW a missing real in the rang of those functions!

Now you call [1] deficient, [2] apt to reality standards [3] bogus.

Why? because we used N in an argument that involves a higher order
concept that must use N* instead. (That's your reply).

But again: why? what is the higher order part of the argument that you
see it demanding circumventing the heart of the subject (which is N
really) to some N*.

Is it the definition of Limit.

But limit is defined in this argument as the least upper bound, and I
don't see in the definition of L that I wrote (which is the standard
by the way) anything that has to do with necessarily picking it up
from some Omega_th end point? that has no meaning at all, so why?

Should I adopt this rational of yours then I'd ask you: why not say
pick L from the -1_th starting point. i.e. choose your domain to be
{-1,0,1,2,3,...} since this clearly also preclude Cantor's argument
and you clearly can make L be the -1_th digit of (x_n) [Remember a_0
is x_0, so x_{-1} lies "before" a_0].

Or you'll say that {-1,0,2,3,...} is also a kind of high order
countable set?

Your argument is simply shunning one of the most important two sets in
this argument, that is N, and using some replacement, without any
clear justification.

Zuhair

[Uirgil]

In article
<6a63fbfd-f7e7-458f...@d17g2000vbv.googlegroups.com>,

It is worse, mathematically speaking, than merely strange, it is
nonsense.

>
> What you are having is the following:
>
> [1]When we use N as the domain of injections (x_n), (a_n) and (b_n),
> then Cantors argument PROVES and SHOWS that there is a real that is
> not in the range of those functions.
>
> [2]When we use N* as the domain of injections (x_n), (a_n) and (b_n),
> then Cantor's argument will seize from working in the same way to show
> the missing real.

?"Cease"?

Right! The fact that one can make the proof seem false by changing it
does not make the original proof false.

[Zuhair]

On Nov 14, 10:18 am, Uirgil <uir...@uirgil.ur> wrote:
> In article

> <6a63fbfd-f7e7-458f-af65-fae2c805c...@d17g2000vbv.googlegroups.com>,

Actually he is changing the requirement for a proof, i.e. what
constitutes a proof, to him he thinks (apparently) that the question
involves high order principles that cannot be met by using N as a
domain, But he didn't give any line of justification for that.

Zuhair

[Zuhair]

On Nov 14, 10:18 am, Uirgil <uir...@uirgil.ur> wrote:
> In article

> <6a63fbfd-f7e7-458f-af65-fae2c805c...@d17g2000vbv.googlegroups.com>,

Yes, Cease, i.e. stop, of course I'm speaking about stopping in the
sense of running the exact particulars of the argument per se, that's
why I said "...in the same way" for example when you use some N* which
has an omega_th position as the domain then for example Result 7
cannot be proven in exactly the same straightforwards way as it is
proved with N, to prove it you need to define it indirectly in terms
of bijections from N* to N ...., which is a long way. But ultimately
you will also succeed in finding a missing real as I pointed out. That
is merely a temporary conundrum with the argument that has no
significance to the reality of the matter, and has no philosophical
value whatsoever.

Zuhair

[Uirgil]

In article
<192db05a-9b98-4df0...@n5g2000vbk.googlegroups.com>,

You wrote "seize". I was merely asking if you meant "cease".

[Ross A. Finlayson]

On Monday, November 12, 2012 10:20:40 AM UTC-8, LudovicoVan wrote:
> "LudovicoVan" <ju...@diegidio.name> wrote in message
>
> news:k7rehn$rci$1...@dont-email.me...
>

> > "Zuhair" <zalj...@gmail.com> wrote in message
>

> > news:86a85cce-2a84-4c9f...@o8g2000yqh.googlegroups.com...

>
> >
>
> >> Let a_0 = x_0
>
> >> Let b_0 be the first entry in (x_n) such that b_0 > a_0.
>
> >> Let a_i+1 be the first entry in (x_n) such that a_i < a_i+1 < b_i.
>
> >> Let b_i+1 be the first entry in (x_n) such that a_i+1 < b_i+1 < b_i.
>
> >
>

> > In Cantor's proof a_{i+1} and b_{i+1} are the two first entries
>
> > encountered
>
>
>

> That should read: a_{i+1} and b_{i+1} are the first two entries...
>
>
>
> -LV

>
>
>
> > (in any order) in (x_n) *after* the entries corresponding to a_i and b_i.
>
> > This does not seem to be the case with your proof, where it instead seems
>
> > that entries are just picked every time restarting from the beginning of
>
> > (x_n).
>
> >
>
> > Could you clarify? I'd like to be sure before I proceed reading it...

With EF the natural/unit equivalency function, a_0 is EF(0) and b_0 is EF(1), the sequence completes without the following conclusion.

Here, of couse that is in the projectively extended real numbers, as complete ordered field and an ordered ring, with rather restricted transfer principle, a la Bishop and Cheng, or Finlayson.

Regards,

Ross Finlayson

[Zuhair]

The third link speaks about the whole proof at high level. Also the
first link give you a summary of this proof, and the second link
(which you just need to copy and past to addresses bar) shows another
formal proof (not high level), and gives you a parallel way of
understanding matters. Definitely if you have read those references
you'll get the high level you need for this proof.

Zuhair

[Ross A. Finlayson]

http://www.tiki-lounge.com/~raf/finlayson_injectrationals.pdf

[LudovicoVan]

"Zuhair" <zalj...@gmail.com> wrote in message

news:6a63fbfd-f7e7-458f...@d17g2000vbv.googlegroups.com...

> On Nov 14, 12:45 am, "LudovicoVan" <ju...@diegidio.name> wrote:

<snip>

>> You are simply missing the point there: we don't need N* to disprove
>> Cantor,
>> we need N* to go beyond it and the standard notion of countability. In
>> fact, that there is a bijection between N* and N is a bogus argument too,
>> as
>> the matter is rather about different order types.
>

> Now I think I'm beginning to somewhat perhaps understand your
> argument.

That's cool, maybe in another while you'll actually get what the argument
was.

-LV

[LudovicoVan]

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote in message
news:c2ac7c88-5567-408d...@googlegroups.com...

> <http://www.tiki-lounge.com/~raf/finlayson_injectrationals.pdf>

I'll see if I can understand it: for now, thanks for sharing.

-LV

[Uirgil]

In article <k84u6j$u4e$1...@dont-email.me>,

Since the entire issue is about the "standard notion of countability",
we should settle everything about that before trying to go beyond it.

So at present any argument "beyond" the "standard notion of
countability" is totally irrelevant to this thread.

[Zuhair]

On Nov 16, 11:40 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message

You don't have any argument, you just have an unbacked assertion that
actually springs from your ignorance the matter.

Zuhair

[LudovicoVan]

"Zuhair" <zalj...@gmail.com> wrote in message

news:84d0fa32-32bf-4c3f...@y8g2000yqy.googlegroups.com...

> On Nov 16, 11:40 am, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Zuhair" <zaljo...@gmail.com> wrote in message
>> news:6a63fbfd-f7e7-458f...@d17g2000vbv.googlegroups.com...>
>> On Nov 14, 12:45 am, "LudovicoVan" <ju...@diegidio.name> wrote:
>> <snip>
>>
>> >> You are simply missing the point there: we don't need N* to disprove
>> >> Cantor,
>> >> we need N* to go beyond it and the standard notion of countability.
>> >> In
>> >> fact, that there is a bijection between N* and N is a bogus argument
>> >> too,
>> >> as
>> >> the matter is rather about different order types.
>>
>> > Now I think I'm beginning to somewhat perhaps understand your
>> > argument.
>>
>> That's cool, maybe in another while you'll actually get what the argument
>> was.
>

> You don't have any argument, you just have an unbacked assertion that
> actually springs from your ignorance the matter.

You started by asking me and were given two links with which to play: you
first have failed to say anything useful, now you are just a inconsistent
liar.

-LV

[Ben Bacarisse]

A first step is to remove the indexes from the irrationals. In a
argument about the supposed countability of the irrationals, to refer to
them with indexes (e.g. p_i) looks like begging the question.

As it happens, I don't think the indexes do anything but add a layer of
confusion. I think you can rename the various quantities without
altering the meaning, i.e. rather than talk about irrationals p_i and
p_h just use p and r (q is taken).

--
Ben.

[Ross A. Finlayson]

On Nov 16, 8:12 am, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
> "LudovicoVan" <ju...@diegidio.name> writes:

> > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote in message

It's constructive, that.

Regards,

Ross Finlayson

[Ben Bacarisse]

> It's constructive, that.

The key set, Q_<i (which would then be called Q_<p) is empty though.

--
Ben.

[Ross A. Finlayson]

On Nov 16, 7:50 pm, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:

Note the emphasized bit.

Thanks,

Ross Finlayson

[Ben Bacarisse]

"Ross A. Finlayson" <ross.fi...@gmail.com> writes:

> Note the emphasized bit.

I did. That the set in question is empty simply implies that for every
rational q < p, there is an irrational r < p with r > q. That the set
in question is *not* empty would give any member of it the property of
being a rational least upper bound of all the irrationals less than p.

--
Ben.

[Ross A. Finlayson]

The rationals are dense in the reals. So are the irrationals, nowhere-continuous, everywhere-dense, whose complement in the reals is each other: given those properties, they're indistinguishable.

There's a contradiction either way - where the construction of the proof emphasizes one way, in vacuo, it's a plain claim.

Then, there are reasonable definitions about our continuum of real numbers, to establish the standard and here extra the standard, where, the proof-theoretic constructs of the standard, do admit their own incompleteness.

The conscientious mathematician is interested in the limits of the standard.

Yes, the classical is perfect in the meso-scale, and as we know, there's more to it than that, for the grandest and most sublime of scales.

Empty, it's as well a contradiction.

Regards,

Ross Finlayson

[Ben Bacarisse]

> The rationals are dense in the reals. So are the irrationals,
> nowhere-continuous, everywhere-dense, whose complement in the reals is
> each other: given those properties, they're indistinguishable.
>
> There's a contradiction either way - where the construction of the
> proof emphasizes one way, in vacuo, it's a plain claim.

It's possible the limit intersection is not defined. I'd want a
topologist to review that since my intuition about an uncountably
infinite intersection of open sets of rationals is probably not
reliable.

But, one way or the other, you can't say that the intersection contains
a rational -- it's ether undefined or it's empty.

> Then, there are reasonable definitions about our continuum of real
> numbers, to establish the standard and here extra the standard, where,
> the proof-theoretic constructs of the standard, do admit their own
> incompleteness.
>
> The conscientious mathematician is interested in the limits of the standard.
>
> Yes, the classical is perfect in the meso-scale, and as we know,
> there's more to it than that, for the grandest and most sublime of
> scales.
>
> Empty, it's as well a contradiction.

I don't understand any of that, but please don't go to any effort to
help me understand.

--
Ben.

[Ross A. Finlayson]

Each is non-empty.

Replace "irrationals", with "rationals", and vice-versa. The three proof-points hold. Is it clear that, as indistinguishable, each applies?

I don't "make" you understand - these are plain statements, if you don't recognize their content, I'm happy to address relevant points as to the matter, but I won't make or compel your understanding, as belief. I expect an interested reader to establish for themselves, their belief, here in as to whether: the rationals' topological properties of density, in the reals, define constructively these properties. Then, in the consideration of those properties and that the rationals' complement in the reals share the very same features, they don't just share but mutually define those properties.

Then, where the rationals and irrationals are indistinguishable, for only these properties, where I'll agree there's no true vacuum in this space of mathematical objects but in vacuo: conscientious as a mathematician, the results so follow.

Then, here we share a belief system that for example the complete ordered field (give or take division by zero, and modern mereological/boundary and meromorphological/analytical methods have it so defined), the rationals in their construction, and so on follow from standard topological definitions that aren't necesarily so defined as those of Cauchy and Weierstrass. Of course there are well-known results that their definitions are equivalent that people believe, correspondingly with an example like Nelson's IST being co-consistent with ZFC. Then, for something as simple in definition as continuity in real analysis, and how the rationals by themselves meet some of these definitions as used to found analysis, it is well-known, and not widely discussed, there are reservations of real analysts and topologists. Of course, here this sharing of the belief system is by no means imposed: reasonable constructions of the numbers, here with their necessary features for results generally accepted as true.

"It is so that ... else ...": emphasis mine, of a fact.

Contrapositive: constructive.

And warm regards,

Ross Finlayson

[Ben Bacarisse]

Too much Google damage, I can't face correcting it again...

That may be true, I don't know. If there is symmetry (and it's not
obvious to me because the irrationals are not topologically the same as
the rationals) then both would be empty or undefined, rather than
non-empty.

> I don't "make" you understand - these are plain statements, if you
> don't recognize their content, I'm happy to address relevant points as
> to the matter, but I won't make or compel your understanding, as
> belief.

No one is asking you to. I simply disagree with you. If it's not where
I stated, where is the error?

<snip>

> "It is so that ... else ...": emphasis mine, of a fact.
>
> Contrapositive: constructive.

Yes, I did understand the form of the statement. Maybe you can
re-phrase the contradiction that follows if the set it empty? As
written I don't see any connection between the two parts.

There is a contradiction if the set is not empty, so the choice seems to
be between the limit being undefined and the set being empty.

--
Ben.

[Ross A. Finlayson]

On Saturday, November 17, 2012 5:38:01 PM UTC-8, Ben Bacarisse wrote:
> Too much Google damage, I can't face correcting it again...

I concur Google and its groups is a quite poor interface here to USENET.

Basically this set is of an interval, that it is non-empty, and non-degenerate (not point width): it contains rationals.

Again, blind to whether they're rationals or irrationals, simply due their properties of being everywhere-dense in the reals, and nowhere-continuous, the same result follows.

There exists for each one of the other, it carries through induction.

Then, about the "choice" in as to whether there exist here between any two rationals infinitely many irrationals, there are between any two points in the standard reals infinitely many of each. (Here, "standard" is noted where of course I described the reals via a space-filling continuum that has its points in a row, as well establishing well known primary results of the standard, extra the standard.) There's the consideration in as to: whether or not the value is "undefined" to preserve all semantics and results of all extensions of the standard, here the properties so ascribed hold, in the immediate development. Then, that "undefinition", is a definition, an axiom, and etcetera, and here, not an axiom in this development.

Then of course in mathematical disagreement, it would be unreasonable to then have a contradiction to the construction of the rationals and reals in their cardinalities, without a supporting framework establishing as well for other features of these systems of numbers, that Platonists have exist regardless of our definitions, to be accepted as a standard feature. Thus, I went about establishing primary results in foundations to:

1) keep the integral calculus without cardinals necessary for a measure theory
2) describe how EF the natural/unit equivalency function maps from naturals to the unit interval, uniquely
3) with regards to ordinals and their powersets, display of simple features about how their inexhaustibility supercedes restriction of their comprehension

and so on, in a forward development of foundations, for application.

Application is key: real analysis as applied doesn't need transfinite cardinals, and there has been research that would establish applications of transfinite cardinals.

The Universe is everything, and all the relations among all its things is infinite, and being total, would be its own powerset, this just to be an example of a reasonable thought experiment that infinite sets are equivalent, here the universal set.

So, then back to that: given two sets everywhere-dense and nowhere-continuous in the reals, who are each other's complement in the reals, thus whose union is the reals: given those two sets, there is via simple induction the results as above.

I'll happily agree with you on general principles: the Platonist numbers care not one whit our opinion.

Warm regards,

Ross Finlayson

[Ben Bacarisse]

"Ross A. Finlayson" <ross.fi...@gmail.com> writes:

<snip>

> Basically this set is of an interval, that it is non-empty, and
> non-degenerate (not point width): it contains rationals.

We've both stated our positions and the argument is not advancing. You
wrote a lot after this, but I could not connect it to the document I was
commenting on. It purports to show that the rationals and the
irrationals have the same cardinality. If you posted it for a critique,
I gave it a shot, but if I did not find the error, where is it? And if
you are sure of your reasoning why is it not published?

<snip>
--
Ben.

[Ross A. Finlayson]

Ben, et alia,

I'll compliment you for this: maintaining trust in the standard and looking for what's beyond it. That's a sign of an enlightened mind, as is to look beyond the standard, for the real. And, Goedel admits or confirms that there are mathematics extra the standard, or it would be complete.

This article has never been submitted to a journal for publishing, though it's in AMS article format. Conveniently it fits on one page. The source gathers dust on some PC or disk, simple enough to recreate, in TeX, or here AMS TeX. I'd leave out the QED boxes except the last, though each marks a result extra the standard. USENET is a poor communication method, but it's publishing, each post with its own ID.

Now, the emphasized point is that for each irrational there are the rationals less than it. Then for those, as each q_h is less than q_i, here as it's defined, for the rationals less than the irrationals less than q_i, their union doesn't include all the irrationals less than q_i, because it only includes the rationals less than q_i. Then for that there's a p_h. Else: exists q_h > p_h or some p_h = p_i, contradiction.

Then as to whether that's a degenerate interval but not point width: basically that's a definition as so defined with "Let ...." It's not empty because it is of rationals and there are irrationals between those rationals.

That the contradiction would otherwise follow, here with trichotomy of the reals that exactly one of r_a < r_b, r_a = r_b, or r_a > r_b, trichotomy, is the establishment of this item: the non-empty set containg q_h, and an irrational.

Quod erat demonstrandum.

Regards,

Ross Finlayson

[Ross A. Finlayson]

Of course, you can simply note there is no p such that there isn't a q s.t. p < q < q_i, but, that q is not q_h, and of course, there exists another p s.t. q < p < q_i.

With warm and pleasant regards,

Ross Finlayson

[Ben Bacarisse]

"Ross A. Finlayson" <ross.fi...@gmail.com> writes:

> On Sunday, November 18, 2012 4:32:40 PM UTC-8, Ben Bacarisse wrote:
>> "Ross A. Finlayson" <ross.fi...@gmail.com> writes:
>>
>> <snip>
>>
>> > Basically this set is of an interval, that it is non-empty, and
>> > non-degenerate (not point width): it contains rationals.
>>
>> We've both stated our positions and the argument is not advancing. You
>> wrote a lot after this, but I could not connect it to the document I was
>> commenting on. It purports to show that the rationals and the
>> irrationals have the same cardinality. If you posted it for a critique,
>> I gave it a shot, but if I did not find the error, where is it? And if
>> you are sure of your reasoning why is it not published?
>
<snip>

> This article has never been submitted to a journal for publishing,

But you don't say why. You seem convinced by it and it's dynamite.

> Now, the emphasized point is that for each irrational there are the
> rationals less than it. Then for those, as each q_h is less than q_i,
> here as it's defined, for the rationals less than the irrationals less
> than q_i, their union doesn't include all the irrationals less than
> q_i, because it only includes the rationals less than q_i. Then for
> that there's a p_h. Else: exists q_h > p_h or some p_h = p_i,
> contradiction.

Your subscripts are getting the way again. For an irrational p you
define Q(p) as the set of all rationals < p. You then remove from that
set all the rationals less than all the irrationals less than p:

L(p) = Q(p) \ Union{r < p} Q(r) for irrational r

You claim that L(p) = {} leads to a contradiction but I can't see how.
I simply can't see how your "else" clause follows. Perhaps it's the
subscripts. Maybe writing it like this will make it clearer when you
explain it.

On the other hand, it seems clear to me that having any rational w in
L(p) does lead to a contradiction. It would follow that w < p (because
it must come from Q(p)) but also that there was no Q(r) that included
it. I.e. that no irrational r > w but still less than p exists, and
that's not the case.

I think your point is that L(p) is not empty because for every r < p
(i.e. for every member of the union) there is always a rational w with
r < w < q, and therefore a member of Q(p) that is not removed. But
that's not true in the limit, at least so I believe. If this is wrong,
then it means, surely, that the union is undefined.

We are both convinced (though I am less than certain due the uncountable
nature of the union) which I why I don't think there can be progress
unless someone else chips in.

<snip>
--
Ben.

[Ross A. Finlayson]

Matter: particle and wave.

The set of all subsets of the universe is itself.

Measure theory uses countable, not uncountable, additivity to establish results.

Infinite sets are infinite.

The universe is infinite or reference frames would be finite.

The closer science looks at the atom the smaller it is, the farther we look to space the bigger it is.

The rationals and irrationals are disjoint, and between any of them, from the complete ordered field, there are more of them. Ditto the irrationals and rationals.

Discovering applications of transfinite cardinals would be important news for many, and much work has gone into that research, without placement into the applied.

These are basically reasonings in as to why to consider the region about the rational, free of rationals, still with irrationals, and vice versa, because they're not the same.

Selecting each greatest element of that intersection and from that, in the limit, each next, derives an ordering of elements dense in the reals in their natural order.

Defined? It exists. And so does the square root of negative one.

Regards,

Ross Finlayson

[Ross A. Finlayson]

On Nov 20, 8:08 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:

> On Sunday, November 18, 2012 9:15:22 PM UTC-8, Ben Bacarisse wrote:

> > "Ross A. Finlayson" <ross.finlay...@gmail.com> writes:
>
> > > On Sunday, November 18, 2012 4:32:40 PM UTC-8, Ben Bacarisse wrote:
>

So, we know from modern particle physics that the particle, is both
particle, and wave. Now, where these are the closest things in
reality to modeling the continuum of space, mathematically, then where
is it discovered that the mathematical infinitesimal reflects and
defines our most fundamental substrate of reality?

One might point to string theory, with its general notions that the
strings are as many orders of magnitudes finer than the atom as the
atom is to our meter, here in allusion to the fluxions of each fluent
as Newton so described the infinitesimals in the reals, but why is it
ignored the obvious parallel to the atom, the post-Democritan atom, as
being both particle and wave?

The real numbers are continuous in their continuum, infinitesimals as
they are are in them and of them. And, there would be the infinity
and infinities, too.

Look farther, it's bigger, look closer, it's finer: our very real and
most modern experiments in physics don't just hint but advise
scientifically that the features of the most fundamental laws, most
easily described as generally applicable mathematically, define truths
of the real natural continuum beyond what our standard and modern
mathematics does. As well and simply logically, we find from our
regular theories as incomplete that there are true facts about the
objects of our theories, not decided by our standard theories, as is
generally attributed to Goedel.

Then, we do have the remarkable results in the infinities (though he
would aver to there being no infinitesimals as a pox on mathematics)
of Cantor then Russell and so formulated for its great applicability
generally in the finite and in infinite ordinals, well-ordered, of
Zermelo and Fraenkel (who advises that ZF isn't the end-all or be-
all). Compelling as they are, the conscientious mathematician also
finding the real natural continuum as compelling, mathematically,
works to discover how they are at once one and the other.

Then, here with a simple demurral to the dually-self-infraconsistent
dialetheic of the natural formulation of an axiomless system of
natural deduction, with conservation and symmetry of truth, the
technical philosopher satisfies the conscientious mathematician's
rigor, in the all, from the none, the all. Results follow in the
conciliation of our standard, and our emergent, discoveries of truth,
here in mathematics, logic, and technically, philosophy.

An entire new realm of application is simply waiting there to be
discovered. An entire new realm of application was simply waiting
there to be discovered.

Good luck with that, right then, warm regards,

Ross Finlayson

[Virgil]

In article
<8e72f34b-4acb-4e8d...@i7g2000pbf.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> So, we know from modern particle physics that the particle, is both
> particle, and wave.

What we do know is that those things we sometimes regard as being
small-and-particle-like things have some behaviors that are wave-like.

What those "things"REALLY are, we do not know.

And most of the time, don't much care, as long as our descriptions of
how we expect them to behave match our observations of how they do
behave!
--

[Ross A. Finlayson]

On Nov 25, 12:53 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <8e72f34b-4acb-4e8d-9797-f3b217e4e...@i7g2000pbf.googlegroups.com>,

>  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
> > So, we know from modern particle physics that the particle, is both
> > particle, and wave.
>
> What we do know is that those things we sometimes regard as being
> small-and-particle-like things have some behaviors that are wave-like.
>
> What those "things"REALLY are, we do not know.
>
> And most of the time, don't much care, as long as our descriptions of
> how we expect them to behave match our observations of how they do
> behave!
> --

Well that's simple, you're not a conscientious mathematician, who
cares.

Heh, you describe exactly the fallacy of argumentum ad ignorantium.

What those things REALLY are, we care. And, you don't here offer
anything about it.

Quit trodding on my coat-tails and get off it. And: get a job, get
off my lawn, and mow it, you hen-pecking, busybody, biddy.

Matter: particle and wave: at least. And, the physics of particles
is the closest thing there is to physically model the behavior of
numbers, in the real: on the metal.

Thanks, I understand that's quite high level and abstract thinking.
Now go print yourself a certificate of merit.

Regards,

Ross Finlayson

[Graham Cooper]

On Nov 26, 7:40 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>

wrote:
> On Nov 25, 12:53 pm, Virgil <vir...@ligriv.com> wrote:
>
> > In article
> > <8e72f34b-4acb-4e8d-9797-f3b217e4e...@i7g2000pbf.googlegroups.com>,
> >  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
> > > So, we know from modern particle physics that the particle, is both
> > > particle, and wave.
>
> > What we do know is that those things we sometimes regard as being
> > small-and-particle-like things have some behaviors that are wave-like.
>
> > What those "things"REALLY are, we do not know.
>
> > And most of the time, don't much care, as long as our descriptions of
> > how we expect them to behave match our observations of how they do
> > behave!
> >
>

> Well that's simple, you're not a conscientious mathematician, who
> cares.
>

Virgil could pass for a mathematician. He doesn't use deceit to win
points like most Cantorians, he'll try to put forward what he believes
is the truth, that there are MORE THAN 1,2,3...INFINITY points
between any 2 points.

It's just that his (MAINSTREAM) stance is imaginary and pieced
together which forces him to argue from the hip, and duck and weave
like they all do!

Herc

--
S: if stops(S) gosub S
G. GREENE: this proves stops() must be un-computable!
SCI.LOGIC

[Jesse F. Hughes]

"Ross A. Finlayson" <ross.fi...@gmail.com> writes:

> On Nov 25, 12:53 pm, Virgil <vir...@ligriv.com> wrote:
>> In article
>> <8e72f34b-4acb-4e8d-9797-f3b217e4e...@i7g2000pbf.googlegroups.com>,
>>  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>>
>> > So, we know from modern particle physics that the particle, is both
>> > particle, and wave.
>>
>> What we do know is that those things we sometimes regard as being
>> small-and-particle-like things have some behaviors that are wave-like.
>>
>> What those "things"REALLY are, we do not know.
>>
>> And most of the time, don't much care, as long as our descriptions of
>> how we expect them to behave match our observations of how they do
>> behave!
>> --
>
>
> Well that's simple, you're not a conscientious mathematician, who
> cares.
>
> Heh, you describe exactly the fallacy of argumentum ad ignorantium.

Let's add that fallacy to the enormous list of things Russell doesn't
understand.

--
Jesse F. Hughes
Playin' dismal hollers for abysmal dollars,
Those were the days, best I can recall.
-- Austin Lounge Lizards, "Rocky Byways"

[Ross A. Finlayson]

On Nov 25, 1:58 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

It is that: arguing that because the extends of knowledge, here of
the "true" nature of fundamental particles, that those are boundaries
to reason and knowledge, is false, in that way. Virgil's fallacy is
that the particles don't speak of numbers, as the numbers speak of
particles, here in argumentum ad ignorantium.

Here it is indeed that: particles being each particle and wave, is
what we know, here from theory and experiment. Then in terms of using
mathematics for mathematical physics, there's a strong expectation and
for some a belief that they would absolutely coincide in their
structure and formation, the objects of discourse, and fact (as is so
verified in our best scientific theories of the natural physics).

So, the universe would be its own collection of its own subsets. Does
it not exist? Are there results of transfinite cardinals in the
applied not due transfinite ordinals? Are the natural integers
exhaustible? Are not the rationals and irrationals each dense in the
reals?

Yes it does, no there aren't, no they aren't, and yes they are.

Regards,

Ross Finlayson

[Ross A. Finlayson]

No, the conscientious mathematician doesn't just adhere to and
elaborate the mundane, but here acknowledges there is more to
mathematics than we yet have, and strives for truth, here mathematical
truth, as it is. As a follower of mathematics, Virgil is of the timid
sort, always using argument established by others, thus to lend
credence to his opinion regardless of his tactics, vis-a-vis,
establishing original thought, here of course in a framework of
mathematics.

The conscientious mathematician doesn't just curate and dust.

Regards,

Ross Finlayson

[Virgil]

In article
<ab0e7583-f437-4fbb...@y5g2000pbi.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> On Nov 25, 12:53 pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <8e72f34b-4acb-4e8d-9797-f3b217e4e...@i7g2000pbf.googlegroups.com>,
> >  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
> >
> > > So, we know from modern particle physics that the particle, is both
> > > particle, and wave.
> >
> > What we do know is that those things we sometimes regard as being
> > small-and-particle-like things have some behaviors that are wave-like.
> >
> > What those "things"REALLY are, we do not know.
> >
> > And most of the time, don't much care, as long as our descriptions of
> > how we expect them to behave match our observations of how they do
> > behave!
> > --
>
>
> Well that's simple, you're not a conscientious mathematician, who
> cares.
>
> Heh, you describe exactly the fallacy of argumentum ad ignorantium.
>
> What those things REALLY are, we care. And, you don't here offer
> anything about it.

Unless knowing what those things "REALLY ARE" is essential to knowing
how to describe their behavior, which at least for quantum theory does
not seem to be the case, most of science is quite satisfied by merely
knowing how to describe their behavior.

>
> Quit trodding on my coat-tails and get off it.

Didn't now you could even afford a coat, much less one with tails long
enough to tread on.

You must look quite odd wearing one like that.

> And: get a job, get
> off my lawn, and mow it, you hen-pecking, busybody, biddy.

Tut-tut!

Temper! Temper!
--

[Jesse F. Hughes]

"Ross A. Finlayson" <ross.fi...@gmail.com> writes:

> On Nov 25, 1:58 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Let's add that fallacy to the enormous list of things Russell doesn't
>> understand.
>>
>> --
>> Jesse F. Hughes
>> Playin' dismal hollers for abysmal dollars,
>>   Those were the days, best I can recall.
>>             -- Austin Lounge Lizards, "Rocky Byways"
>
>
> It is that: arguing that because the extends of knowledge, here of
> the "true" nature of fundamental particles, that those are boundaries
> to reason and knowledge, is false, in that way. Virgil's fallacy is
> that the particles don't speak of numbers, as the numbers speak of
> particles, here in argumentum ad ignorantium.

You really are remarkably illucid today, but whatever it is that you're
squeaking about has fuck-all to do with argumentum ad ignorantium.

> Here it is indeed that: particles being each particle and wave, is
> what we know, here from theory and experiment. Then in terms of using
> mathematics for mathematical physics, there's a strong expectation and
> for some a belief that they would absolutely coincide in their
> structure and formation, the objects of discourse, and fact (as is so
> verified in our best scientific theories of the natural physics).
>
> So, the universe would be its own collection of its own subsets. Does
> it not exist? Are there results of transfinite cardinals in the
> applied not due transfinite ordinals? Are the natural integers
> exhaustible? Are not the rationals and irrationals each dense in the
> reals?
>
> Yes it does, no there aren't, no they aren't, and yes they are.
>

--
Jesse F. Hughes
"All Chinese are Confucianists when successful, and Taoists when they
are failures."
-- Lin Yutang, /My Country and My People/

[Graham Cooper]

On Nov 26, 7:53 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> It's just that his (MAINSTREAM) stance is imaginary and pieced

> together which forces him to argue from the hip, and duck and weave..

....

shoot from the hip, and duck and weave..

A sufficient strategy for a mob to scare newcomers with little effort
or expertise.

Herc

[Ross A. Finlayson]

On Nov 25, 4:05 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <ab0e7583-f437-4fbb-8a55-71422c06c...@y5g2000pbi.googlegroups.com>,

Heh, come back with a contribution to science. And learn to maintain
decorum in discourse, ....

Yes, contributions to science and mathematical knowledge would be
highly appreciated. And I'm all for the diligent practice of career,
with a general, and egalitiarian, respect. (Get a job.)

Gallery, et alia, please don't let my rebuffing of Virgil's off-topic,
ad hominem lack of mathematical commentary come across as anything but
warmly genial - I find strength and joy in the search for mathematical
truth, not the sweeping away of cobwebs, damning spiders. Most find
me pleasant in conversation, and there's much of it.

Regards,

Ross Finlayson

[Virgil]

In article
<538ec933-cf60-4e88...@jj5g2000pbc.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

Ross cannot afford to let himself understand that what he regards as
being "warmly genial" is in fact him futile attempt not to be a putzer.
--

[Virgil]

In article
<a9ed3b7f-20d5-49c9...@u4g2000pbo.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

Ross may be still in residing "here in argumentum ad ignorantium", but
none of his critics, of which there are many, have joined him here.

>
> Here it is indeed that: particles being each particle and wave, is
> what we know, here from theory and experiment.

Actually we do not know any such thing. What we do know is that the
things, whatever they may actually be, that we call electrons, amang
other things, seem to have some particle-like properties and some
wave-like properties. And as long as we can describe and predict their
behavior sufficiently accurately, we don't much care what they really
"are".

Some garbage snipped!
--

[Virgil]

In article
<bdf4255c-7c27-4c2b...@v6g2000pbb.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> No, the conscientious mathematician doesn't just adhere to and
> elaborate the mundane, but here acknowledges there is more to
> mathematics than we yet have, and strives for truth, here mathematical
> truth, as it is. As a follower of mathematics, Virgil is of the timid
> sort, always using argument established by others, thus to lend
> credence to his opinion regardless of his tactics, vis-a-vis,
> establishing original thought, here of course in a framework of
> mathematics.

A great majority of "original thought" is garbage, and must be filtered
through what is already established in order separate out the dross.
When one filters the dross out of Ross' "original thoughts" there is far
to often nothing left at all.

>
> The conscientious mathematician doesn't just curate and dust.

But must curate and dust too!
--

[Ross A. Finlayson]

On Nov 25, 6:16 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <bdf4255c-7c27-4c2b-a78c-62a2e38ec...@v6g2000pbb.googlegroups.com>,

Huh, that's not very funny. There is no try, only, do.

Virgil's opinion on original thought: "garbage". He claims to know
much about it.

Get out of me and Cantor's way.

EF, the equivalency function, is like no other function, and as a
result of development around the modern foundations, sees Cantor's
results as establishing its uniqueness and that of its compositions
(as countable line segments). Its range is a line from zero to one.
Draw that.

So yes, for the conscientious mathematician, it could be no less than
meeting the requirements of mathematical rigor and here, evident truth
in the pure and practical utility in the applied. This mathematics,
as museum, is living. Muse on that.

And, particle/wave duality is observed in nature, already, with the
most precise of our physical instruments, and simply with readily
available materiel. Is your mind so closed that you can't draw the
obvious parallel between the best and most modern of our physics and a
natural continuum in our natural universe of naturally mathematical
objects? Others lack that failing. And yes, they find it
interesting, and reasonable. It's pretty simple, this development in
numbers, when I can explain the development to people of all stripes,
and I befriend people of all walks of life.

And, it doesn't care one whit any of our opinion: really, what is.
So, Hancher, you hot-air buffoon, come back with an advancement for
science, and get out of its way.

[Virgil]

In article
<87566fe7-ec39-46e1...@n2g2000pbp.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> On Nov 25, 6:16 pm, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <bdf4255c-7c27-4c2b-a78c-62a2e38ec...@v6g2000pbb.googlegroups.com>,
> >  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
> >
> > > No, the conscientious mathematician doesn't just adhere to and
> > > elaborate the mundane, but here acknowledges there is more to
> > > mathematics than we yet have, and strives for truth, here mathematical
> > > truth, as it is.  As a follower of mathematics, Virgil is of the timid
> > > sort, always using argument established by others, thus to lend
> > > credence to his opinion regardless of his tactics, vis-a-vis,
> > > establishing original thought, here of course in a framework of
> > > mathematics.
> >
> > A great majority  of "original thought" is garbage, and must be filtered
> > through what is already established in order separate out the dross.
> > When one filters the dross out of Ross' "original thoughts" there is far
> > to often nothing left at all.
> >
> >
> >
> > > The conscientious mathematician doesn't just curate and dust.
> >
> > But must curate and dust too!
> > --
>
>
> Huh, that's not very funny.

Was not meant to be. Those who think that what haas gone before must
always be replaced by new things, as Ross seems to, seams not to
understand that culture, even in mathematics, advances by accumulation.

The mathematics of Euclid and Pythogoras is still of value even today,
and those, like Ross, who would throw it out with their bathwater
because it is not new, will lose it all.

>
> Virgil's opinion on original thought: "garbage". He claims to know
> much about it.

New mathematics is fine as long as one does not have to throw out a
couple of millennia of old math to use it.

>
> Get out of me and Cantor's way.

I agree with Cantor's mathematics but have yet to see anything by Ross
that qualifies as math, old or new.

>
> EF, the equivalency function, is like no other function

Unlike actual functions, it is nonsense, not really a function at all.

I defy Ross to give his alleged "equivalency function" a mathematically
acceptable definition of his alleged "equivalency function"here which
demonstrates that it has either any equivalency or any functionality.

In the past he has never done more with it that a lot of bizarre hand
waving and promises that it will square the circle and duplicate the
cube.
--

[Ross A. Finlayson]

On Nov 25, 9:37 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <87566fe7-ec39-46e1-b887-8200b2db8...@n2g2000pbp.googlegroups.com>,

No, I certainly hold as right as you do all our ancient and modern
mathematics. And moreso, they were once new. Now, I'm not saying you
should throw me off the boat because of a claim that the square root
of two is irrational, which it is. No, there are true facts about the
objects of discourse in standard theory, and Goedel proved that there
are true facts of the objects beyond the standard, in standard theory,
in incompleteness.

In fact, a large part of the reason for development of a modern, post-
standard, theory of geometry, sets, and numbers, is to reach for the
applied that is beyond the standard, including the standard, as
applied, and conciliation, in the pure, in mathematics.

For example, the integral calculus, which is readily and often
applied, uses countable additivity. Real analysts would readily make
use of and incorporate as much mathematics as could be applied, here,
there's countable additivity that defines measure useful real measure,
in the standard, in real analysis. Constructively, modern real and
discrete analysis and all that follows in the applied can be read out
from asymptotics instead of the infinite. (Of course, don't get me
wrong, I believe in the infinite, else, for example, we'd be in finite
reference frames and motion would be classical.)

An aside could aver to results of Vitali and Banach-Tarski, and how
these "paradoxes" are examples of what were once "paradoxes".

No, by no means do my theories of geometry, numbers, and sets discard
mathematics, only extending mathematics. For example there is much
use of transfinite ordinals. Indeed, it is a goal of any erstwhile
mathematician to discover, to extend, mathematics.

EF is simple and it's defined simply as a function, not-a-real-
function, standardly modeled by real functions. Dirac's delta and
Heaviside's are as so defined, as functions, not-real-functions,
standardly modeled by real functions. And, the definition of function
itself, here is modern and reflects over time the development of the
definition of what is a mathematical function. Then, in actually
extending the definition of what are the real numbers, in A theory, it
is directly defined, and applied.

There are hundreds of essays on it here.

So, thank you I don't have much use for "bizarre hand-waving", instead
would rather certainly that all parties here ascribe to the same
highest ideals of mathematical logic. And no I've never claimed to
trisect the angle in finite constructions nor other circle-squaring
notions, though, of course, in the asymptotic, constructions trisect
the angle. Perhaps you're confused with some other poster you badger.

Not all who wander are lost.

An erstwhile mathematician's career is to discover to extend
mathematics. It's every grad student's dream to extend mathematics,
into the pure, applied, and concrete. I'm just lucky with a simple
right-place, right-time, confluence of events, then with carrying that
out: discovery, to extend mathematics.

That you think new mathematics would see "throw[ing] out a couple of
millenia of old math" is ridiculous.

Now, that's funny.

Regards,

Ross Finlayson

[Virgil]

In article
<be566287-1de6-426b...@n2g2000pbp.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> EF is simple and it's defined simply as a function, not-a-real-
> function, standardly modeled by real functions. Dirac's delta and
> Heaviside's are as so defined, as functions, not-real-functions,
> standardly modeled by real functions. And, the definition of function
> itself, here is modern and reflects over time the development of the
> definition of what is a mathematical function. Then, in actually
> extending the definition of what are the real numbers, in A theory, it
> is directly defined, and applied.
>
> There are hundreds of essays on it here.

Then give a reference to some of them, preferably by someone other than
yourself.

In particular we need a mathematically satisfactorily definition of your
alleged EF, again preferably by someone other than yourself, which will
take it out of the realm of mythology.
--

[Ross A. Finlayson]

On Nov 25, 11:22 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <be566287-1de6-426b-a9d8-420bb9279...@n2g2000pbp.googlegroups.com>,

I wrote all that.

Regards,

Ross Finlayson

[Virgil]

In article
<ba2d403e-154a-46d2...@vy11g2000pbb.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

Did you?

I certainly do not ever recall seeing your alleged EF adequately
presented, and see now no references to where one might see it
presented, whether adequately or not.

And if you still will not provide a reference to it, a url, or something
through which anyone can access it to see it for him or her self, it is
as if no such thing ever existed.

Which in the absence of any evidence to the contrary, I will continue to
assume.
--

[Ross A. Finlayson]

On Nov 26, 12:03 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <ba2d403e-154a-46d2-9fc9-6e5ae92ed...@vy11g2000pbb.googlegroups.com>,

http://mathforum.org/kb/search!execute.jspa?forumID=13&objID=f13&forceEmptySearch=true&q=%22Equivalency+Function%22
at least hundreds of results

http://mathforum.org/kb/message.jspa?messageID=7888348 "Cantor-
Finlayson theory"

http://groups.google.com/group/sci.math/msg/af29323d694cf89e 1999 -
"Equivalency Function"

http://groups.google.com/group/sci.math/msg/ccb0941dc3421afd perhaps
the first mention

Do you know the old saw about "assume"?

My friends, or as I would so address you, the definition of EF is
written in some few lines: constantly monotonically increasing from
zero through one.

Regards,

Ross Finlayson

[Virgil]

In article
<b144a62b-11a5-4397...@6g2000pbh.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

Not one of which posts contains an original definition of what the
alleged "equivalency function" actually is, only a lot of crap by Ross
about how it is the greatest things since sliced bread.

http://groups.google.com/group/sci.math/msg/ccb0941dc3421afd

I find a citation from r 9/22/99 In which Ross states, what may well be
Ross' original "definition" of his alleged "Equivalency Function" which
as any mathematician can plainly see is not a function at all, and is
only equivalent to nonsense::

" Consider the function
f(x, d)= x/d
for x and d in N.  The domain of x is N from zero to d and the domain of
d is N as d goes to
infinity, d being greater than or equal to one.
I term this the Equivalency Function, and note it EF(x,d), also EF(x),
assuming d goes to
infinity."

>
http://groups.google.com/group/sci.math/msg/af29323d694cf89e 1999 -
"Equivalency Function"

>

> My friends, or as I would so address you, the definition of EF is
> written in some few lines: constantly monotonically increasing from
> zero through one.

Anyone who would call that mess a function, when it is either two
separate functions or infinitely many depending on which part of the
"definition" one is reading, is no mathematician.

Ant the only thing it demonstrates is Ross' total inability to think
mathematically.

I do not find any area of mathematics which would not be improved by its
total absence.

>
> Regards,
>
> Ross Finlayson
--

[Ross A. Finlayson]

On Nov 26, 11:33 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <b144a62b-11a5-4397-9c0d-ecd39e274...@6g2000pbh.googlegroups.com>,

> >http://mathforum.org/kb/search!execute.jspa?forumID=13&objID=f13&forc...

> > rch=true&q=%22Equivalency+Function%22
> > at least hundreds of results
>
> Not one of which posts contains an original definition of what the
> alleged "equivalency function" actually is, only a lot of crap by Ross
> about how it is the greatest things since sliced bread.
>
> http://groups.google.com/group/sci.math/msg/ccb0941dc3421afd
>
> I find a citation from r 9/22/99 In which Ross states, what may well be
> Ross' original "definition" of his alleged "Equivalency Function" which
> as any mathematician can plainly see is not a function at all, and is
> only equivalent to nonsense::
>
> " Consider the function
> f(x, d)= x/d
> for x and d in N. The domain of x is N from zero to d and the domain of
> d is N as d goes to
> infinity, d being greater than or equal to one.
> I term this the Equivalency Function, and note it EF(x,d), also EF(x),
> assuming d goes to
> infinity."
>

> http://groups.google.com/group/sci.math/msg/af29323d694cf89e1999 -

> "Equivalency Function"
>
>
>
> > My friends, or as I would so address you, the definition of EF is
> > written in some few lines:  constantly monotonically increasing from
> > zero through one.
>
> Anyone who would call that mess a function, when it is either two
> separate functions or infinitely many depending on which part of the
> "definition" one is reading, is no mathematician.
>
> Ant the only thing it demonstrates is Ross' total inability to think
> mathematically.
>
> I do not find any area of mathematics which would not be improved by its
> total absence.
>
>

I have a mathematics degree, at least the university affirms that I
think mathematically, unless they are regularly giving unmerited B.S.
degrees, which they don't. This is the regular modern undergraduate
curriculum with topology, algebra, real analysis, dynamical methods,
numerical methods, formal methods, and etcetera (National Merit
Finalist, Putnam). In computer programming I'm familiar with Java
since 1.0 and work Java and C++ at large, well-known corporations, for
bread, from chips and boards to the cloud, and have actually read
Knuth's TaoCP.

If you continue reading that thread from 1999 you'll find that it is
described what were termed "general" and "valid" EF being EF(n,d) and
EF(n).

http://groups.google.com/group/sci.math/msg/6c66dd07999bf4cf

Then, apparently I've spent almost fifteen years working up from that.

http://groups.google.com/group/sci.math/msg/dcaec5f3fd9d74c0
http://groups.google.com/groups/search?as_q=Dirac&as_epq=Equivalency+Function
http://groups.google.com/groups/search?as_q=Heaviside&as_epq=Equivalency+Function
http://groups.google.com/groups/search?as_q=Cantor&as_epq=Equivalency+Function
http://groups.google.com/groups/search?as_q=Finlayson&as_epq=Equivalency+Function

It matters not my opinion on EF as a function: for the mathematical
Platonist, the numbers have their own features, and we know that
standard, modern mathematics does not encompass them all, as Goedel
told us of its incompleteness.

So, and I agree that it's unlikely that a simple individual would put
forward mathematical progess, that's the only way it's ever been,
humble giants for Atlas.

So warm regards, my friends, and have a great day, as you would.

Regards,

Ross Finlayson

[Virgil]

In article
<ff6de177-a8db-461e...@ah9g2000pbd.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

I have three of them, so what?

Your EF is, at least as so far presented, of no mathematical interest or
impotance whatsoever.
--

[Ross A. Finlayson]

On Nov 27, 11:34 am, Virgil <vir...@ligriv.com> wrote:
> In article

> <ff6de177-a8db-461e-9048-b054aae71...@ah9g2000pbd.googlegroups.com>,

As a function, it has particular results in the framework of results
on uncountability of the reals, different than any other. And, it's
simply and standardly modeled by real functions.

That includes your quaint take on it.

That concludes your quaint take on it.

Then plain friends or whoever, well then it is with warm regards and a
general lack of affrontery to note that: for someone with a
continuing interest in mathematics, EF is great.

Now we'd all be interested in application of transfinite cardinals,
and we'd all like to know how the universe and everything is itself
and its own collection of all its subsets, then as to what particular
foundation (or, lack thereof) establishes mathematically that it is.
The twain don't meet.

Regards,

Ross Finlayson

[Virgil]

In article
<fb43d5d1-f3ad-4294...@y5g2000pbi.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> > Your EF is, at least as so far presented, of no mathematical interest or
> > impotance whatsoever.
> > --
>
>
> As a function, it has particular results in the framework of results
> on uncountability of the reals, different than any other.

Such results are more peculiar than particular, and are certainly in no
way useful either to issues of cardinality of the reals nor any part of
standard real analysis.

> And, it's
> simply and standardly modeled by real functions.

Whatever of it is at all useful can be better achieved without it.

>
> That includes your quaint take on it.

My "quaint take" is that there is nothing mathematically useful cpable
of being done with it that cannot better be done without it.

And Ross has certainly presented no mathematically sound evidences to
the contrary.

Nor can he!
--

[Ross A. Finlayson]

On Nov 27, 9:45 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <fb43d5d1-f3ad-4294-9641-d65ebfe2c...@y5g2000pbi.googlegroups.com>,

That is simple dispute.

No, deaf dumb blind monkey, it is what it is.

It is what it is.

What it is.

Regards,

Ross Finlayson

[Virgil]

In article
<9a7d2fa5-933a-4669...@kt16g2000pbb.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

One notes the total absence of any mathematical content to Ross' posting.
--

[Ross A. Finlayson]

On Nov 27, 11:07 pm, Virgil <vir...@ligriv.com> wrote:
> In article

> <9a7d2fa5-933a-4669-9266-5627d204e...@kt16g2000pbb.googlegroups.com>,

No, Hancher, EF: it is what it is.

It is what it is.

So, go back to licking Muckenheim, here you just got beat.

And next time you have a gross overgeneralization that is clearly
fatuous, well, nobody'll be surprised. In fact it's somewhat expected
as the mode.

One notes that for what all you say there's nothing in it, you won't
shut up about it.

Then, a note for Cantor, you say you agree with him, does that include
of an Absolute Infinity? Or, do you just agree with whatever will be
least controversial mathematically to spew your spiteful attacks? We
love Cantor, for opening mathematics to more infinity, and he's not
done yet.

Note mathematical content.

Regards,

Ross Finlayson

[Virgil]

In article
<e2484fc9-f0b3-422a...@me7g2000pbb.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

And what that is is nothing of any use to anyone.

>
>
> So, go back to licking Muckenheim, here you just got beat.
>

> One notes that for what all you say there's nothing in it, you won't
> shut up about it.

Until you can demonstrate that there is anything in it,
which one notes that you have yet to do,
I will continue to hold that there is nothing in it.
--

[Marshall]

On Monday, November 26, 2012 11:33:02 PM UTC-8, Virgil wrote:
>
> I find a citation from r 9/22/99 In which Ross states, what may well be
> Ross' original "definition" of his alleged "Equivalency Function" which
> as any mathematician can plainly see is not a function at all, and is
> only equivalent to nonsense::
>
> " Consider the function
> f(x, d)= x/d
> for x and d in N. �The domain of x is N from zero to d and the domain of
> d is N as d goes to
> infinity, d being greater than or equal to one.
> I term this the Equivalency Function, and note it EF(x,d), also EF(x),
> assuming d goes to
> infinity."
> >
> http://groups.google.com/group/sci.math/msg/af29323d694cf89e 1999 -
> "Equivalency Function"

Um, so EF is a restriction of division?

The domain of x depends on the value of d. I don't recall having seen
that sort of thing before, but I guess I do know what that means.
But I can't figure out what the domain of d is. It sorta looks like the
domain of d depends on what d is, but what the heck would that mean?

And it's just a name, but what about EF has anything to do with
equivalency?


Marshall

[Ross A. Finlayson]

On Nov 28, 4:58 pm, Marshall <marshall.spi...@gmail.com> wrote:
> On Monday, November 26, 2012 11:33:02 PM UTC-8, Virgil wrote:
>
> > I find a citation from r 9/22/99 In which Ross states, what may well be
> > Ross' original "definition" of his alleged "Equivalency Function" which
> > as any mathematician can plainly see is not a function at all, and is
> > only equivalent to nonsense::
>
> > " Consider the function
> > f(x, d)= x/d
> > for x and d in N. The domain of x is N from zero to d and the domain of
> > d is N as d goes to
> > infinity, d being greater than or equal to one.
> > I term this the Equivalency Function, and note it EF(x,d), also EF(x),
> > assuming d goes to
> > infinity."
>

> >http://groups.google.com/group/sci.math/msg/af29323d694cf89e1999 -

> > "Equivalency Function"
>
> Um, so EF is a restriction of division?
>
> The domain of x depends on the value of d. I don't recall having seen
> that sort of thing before, but I guess I do know what that means.
> But I can't figure out what the domain of d is. It sorta looks like the
> domain of d depends on what d is, but what the heck would that mean?
>
> And it's just a name, but what about EF has anything to do with
> equivalency?
>
> Marshall

Mr. Spight, it's about the equivalency or equipollency or equipotency
of infinite sets.
EF(n) = n/d, d->oo, n->d.

Properties include:
EF(0) = 0
EF(d) = 1
EF(n) < EF(n+1)
The domain of the function is of those natural integers 0 <= n <= d.

It's very simple this. Then, not a real function, it's standardly
modeled by real functions:
EF(n,d) = n/d, d E N, n->d
with each having those same properties.

Then, the co-image is R[0,1] as is the range.

Regards,

Ross Finlayson

[Alan Smaill]

"Ross A. Finlayson" <ross.fi...@gmail.com> writes:

Is this a version of the natural density of a subset of the natural
numbers?

http://en.wikipedia.org/wiki/Natural_density

> Regards,
>
> Ross Finlayson

--
Alan Smaill

[Marshall]

On Thursday, November 29, 2012 7:18:59 AM UTC-8, Ross A. Finlayson wrote:
> On Nov 28, 4:58 pm, Marshall <marshall.spi...@gmail.com> wrote:
> > Um, so EF is a restriction of division?
> >
> > The domain of x depends on the value of d. I don't recall having seen
> > that sort of thing before, but I guess I do know what that means.
> > But I can't figure out what the domain of d is. It sorta looks like the
> > domain of d depends on what d is, but what the heck would that mean?
> >
> > And it's just a name, but what about EF has anything to do with
> > equivalency?
> >
> > Marshall
>
> Mr. Spight, it's about the equivalency or equipollency or equipotency
> of infinite sets.

Ok.

> EF(n) = n/d, d->oo, n->d.

I'm not very good with limits, so I'm not sure exactly what this means.
Are you saying that the value of EF is a limit? A double limit? Also
I'm not sure how n can approach d when n is a parameter of EF.

I guess also you're saying EF is *not* a restriction on division.

> Properties include:
> EF(0) = 0
> EF(d) = 1
> EF(n) < EF(n+1)
> The domain of the function is of those natural integers 0 <= n <= d.

Whoa, you lost me. Here, EF has only one parameter, and you show
1. EF(0) = 0
2. EF(d) = 1
but also
3. EF(n) < EF(n+1)


But it seems you have done some hidden binding of d that I'm not
clear about. Given 2. above, EF applied to any nonzero number yields
1, which contradicts your 3. above. What gives?

> It's very simple this. Then, not a real function, it's standardly
> modeled by real functions:
> EF(n,d) = n/d, d E N, n->d
> with each having those same properties.
>
> Then, the co-image is R[0,1] as is the range.

I'm not following.


Marshall

[Virgil]

In article
<a78b6e05-ad1a-4390...@ah9g2000pbd.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> EF(n) = n/d, d->oo, n->d.
>
> Properties include:
> EF(0) = 0
> EF(d) = 1
> EF(n) < EF(n+1)
> The domain of the function is of those natural integers 0 <= n <= d.
>
> It's very simple this. Then, not a real function, it's standardly
> modeled by real functions:
> EF(n,d) = n/d, d E N, n->d
> with each having those same properties.

Thus it is necessarily ambiguous and totally useless.
--

[Ross A. Finlayson]

On Nov 29, 7:35 am, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:

No, though half of the integers are even.

They're related concepts in establishing that, to the asymptotic, it
can be established the size of proper subsets of the integers,
relative to the integers. Half of the integers are even. It was
wrong to say that it is not so, that not "half of the integers are
even", and that in terms of other densities of proper subsets of the
integers, that they aren't comparable. Those numbers have those
properties. Then I was of the impression that Schnirelmann density
from our number theory and asymptotic density were the same.

Here having infinitely many elements between zero and one, given their
constant difference, is enough to have them be dense as in the reals.

Then density as a quantity and density as a property aren't the same
thing, yet, they both are to denote the propensity of elements within
others of theirs.

Half of the integers are even.

http://groups.google.com/groups/search?as_q=&as_epq=Half+of+the+integers+are+even

And when I noted that another had concurred with this with regards, to
sets, it was roundly derided until they were directed to his Ph.D. at
M.I.T.

Regards,

Ross Finlayson

[Ross A. Finlayson]

Basically this reads that d is unbounded and n ranges from zero
through d.

EF(n,d) is a family of functions, with d unbounded it's a particular
EF(n) with properties modeled by those standard real functions, in a
similar way as to how, for example, Dirac's delta or Heaviside's step
are so modeled.

Being dense in the reals, in its range, leads to a variety of
considerations of anywhere dense elements in their natural order.

The reals: well-ordered.

Regards,

Ross Finlayson

[Virgil]

In article
<fcaa9d75-0b4c-4ccf...@i7g2000pbf.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

A family of functions or its alleged limit function can hardly be "dense
in the reals" according to any standard topology or linear ordering, at
least in any proper mathematical sense.

But then nothing about Ross' "EF" makes any sort of mathematical sense.
>
> The reals: well-ordered.

Not by Ross. While by some considered theoretically possible, no one has
yet come up with any explicit well-ordering of the reals, and for Ross
to suggest that he has done it is the arrogance of abject ignorance.
>
--

[Virgil]

In article
<1dd42260-760a-4366...@i2g2000pbi.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> They're related concepts in establishing that, to the asymptotic, it
> can be established the size of proper subsets of the integers,
> relative to the integers. Half of the integers are even.

And there are exactly as many of them which are even as there are of
them altogether!

> Here having infinitely many elements between zero and one, given their
> constant difference, is enough to have them be dense as in the reals.

Again nonsense!
Any set of numbers having "constant difference" between successive
members will fail to be dense in the reals or even "as in the reals".

>
> Then density as a quantity and density as a property aren't the same
> thing, yet, they both are to denote the propensity of elements within
> others of theirs.

Would you restate that in coherent English. please?
--

[FredJeffries]

On Nov 26, 9:19 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>

> http://mathforum.org/kb/search!execute.jspa?forumID=13&objID=f13&forc...

> at least hundreds of results
>
> http://mathforum.org/kb/message.jspa?messageID=7888348"Cantor-
> Finlayson theory"
>
> http://groups.google.com/group/sci.math/msg/af29323d694cf89e1999 -
> "Equivalency Function"
>
> http://groups.google.com/group/sci.math/msg/ccb0941dc3421afdperhaps
> the first mention
>
> Do you know the old saw about "assume"?
>
> My friends, or as I would so address you, the definition of EF is
> written in some few lines:  constantly monotonically increasing from
> zero through one.

You've had this function for 13 years now and you STILL can't
calculate the area of a triangle with it.

[Ross A. Finlayson]

Fred Jeffries who I respect: I'd like to think that's in the context
of modeling Dirac's delta with triangles or radial basis functions,
but what's important to describe of EF as plotted is this: removing
all the space between the integers and plotting the elements in the
range it would look like f(x) = x from zero to one, half a square and
a triangle, but the F-Sigma Lebesgue integral of EF evaluates to one
not one half, now that's the surprise.

EF: CDF: of the uniform distribution of the natural integers.

Regards,

Ross Finlayson

[FredJeffries]

On Nov 30, 8:39 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
>

> > You've had this function for 13 years now and you STILL can't
> > calculate the area of a triangle with it.
>
> Fred Jeffries who I respect:  I'd like to think that's in the context
> of modeling Dirac's delta with triangles or radial basis functions,
> but what's important to describe of EF as plotted is this:  removing
> all the space between the integers and plotting the elements in the
> range it would look like f(x) = x from zero to one, half a square and
> a triangle, but the F-Sigma Lebesgue integral of EF evaluates to one
> not one half, now that's the surprise.
>
> EF:  CDF:  of the uniform distribution of the natural integers.
>

Sorry, I can't decipher the above two paragraphs. All I see is
13 years and 3 math degrees and still can't calculate the
area of a triangle

[Virgil]

In article
<4b0f0a80-b9fd-46e3...@6g2000pbh.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> I'd like to think

But you never do!
--