Cantor's diagonal proof wrong?
Curt Welch
Nath is not something I specialize in (and I don't read this group
normally), but I've been looking at a few things lately and I've decided
that some very big mistakes have been made in math because people started
playing around the concept of infinity without realizing the trouble they
were creating for themselves.
When I was shown Cantor's diagonal proof that the number of reals was not
countable back in college, I thought it was a fascinating proof. It seemed
to uncover some great mystery about the nature of numbers that was not at
first obvious. It sounded very logical and I quickly embraced it as fact.
Lately however, I've come to see things very differently. I now belief the
proof is totally bogus. And the huge body of work built on top of the
concept is likewise, totally bogus.
But, like I said, I'm not a math expert by any means. So I'm posting the
idea here so you experts can have fun laughing at me.
The reason I came to these conclusions is because I've spent a lot of time
trying to uncover the mysteries of AI (artificial intelligence). i.e.,
trying to understand how to build a machine that is just as intelligent as
we are. And as I've worked on that, I've come to some understanding (also
unproven) about what "thinking" is all about. And this led me to question
some fundamental ideas in other fields, like math.
The problem with math, is that when you start playing with ideas like
infinity, you are making some basic assumptions on what an idea is. Yet,
no one knows how we think, or why we think. Yet, that hasn't stopped
mathematicians and philosophers in making endless assumptions in order to
try and understand the very nature of thought (and existence), and how it
applies to their field. However, I think they got a few things very wrong.
And it's my work on AI which caused me to question Cantor's diagonal proof
in the first place. And instead of accepting it as fact, as I did back in
college when it was shown to me, I looked at it again thinking it was
invalid. And when I look at like that, I see that it is.
Let me demonstrate.
I claim that there is only one type of infinity. That there are just as
many integers as there real numbers. (or more accurately, that the concept
of the size of an infinite set is a contradiction in itself).
So, let me create a mapping. I'll start with the mapping from the integers
to the reals in the range 0 to 0.99999....
I R
0 0.0000...
1 0.1000...
2 0.2000...
10 0.0100...
123 0.3210...
So you just reverse the digits in the integer to create the real. I claim
this mapping is one to one and covers all the reals in that range. For any
real you give me, I can easily give you the matching integer. Just reverse
the digits. This includes all the irrationals because in fact, you can't
give me an infinite irrational to work with. You can only give me an
algorthm for creating it. And any algorithm you give me, I can modify to
create the matching integer.
Let's look at this mapping with Cantor's diagonal proof. We construct a
real number by picking digits from the diagonal which is different from
each row in the table. Well, as it happens, the diagonal in this mapping
is all zeros, so we can pick a simple real like 0.1111... as the number
which can not be in the table. I'll call this number D. Cantor's proof
seems to show quite clearly that D is not in the table, because it can not
be located at any row of the table (for what seems to be obvious reasons).
Let me define D(n), as the first N digits of this "constructed" missing
real diagonal. The first N digits are 1, and the rest are 0.
So D(2) is 0.11 and D(5) is 0.11111 etc.
We see that D(5) can not be located in the first 5 rows of the mapping.
But, we also can easily prove that D(5) does show up at row 11111.
So, as we construct D(n), we see that even though it doesn't match any of
the rows up to the point we have reached, it is always further down in the
table. And because the table is infinite, we will always be able to find
it futher down in the table.
So, this proves that D(n) for all values of n, from 0 to infinity, is in
the table.
So, now we have a contradiction. Cantor's proof says that D(infinity) is
not in the table, yet D(n) for all values of n, including infinity, is in
the table from the above logic, which is just as clear and straight forward
as Cantor's. So how can both be true? If they are not both true, which is
one wrong and the other one not?
How can this be? I say, it's because there's a contradiction in Cantor's
proof, and the contradiction is not the one that everyone assumes - that
there are more reals than integers.
As another example, let me show that the number of integers are also
greater than the number of integers, using the logic of Cantor's proof.
Lets create a table of integers like this:
...000000
...000001
...000002
...000010
...000123
It's just a normal list of integers, but instead of following the normal
convention of leaving off the leading zeros (which we all know are implied
even if we don't write them) I include them in that table.
So lets use Cantor's logic on this table and see if we can construct a
number which is not in the table. We take the numbers from the diagonal,
and construct the number ...111111 just like we did above.
Since we construct this number by changing a digit from every row, we know,
by Cantor's logic, that the resulting number can not be in the table.
Therefore, with the wisdom of Cantor, I've proved that the number of
integers is greater than the number of integers. There are some integers
which are simply not in the list of all integers.
Ok, so if Cantor was wrong, why was he wrong?
The answer is one already well known to mathematicians. They just never
realized how it applied here. You can't use infinity as if it existed. It
doesn't exist. "infinity" is only a name for something which can not
exist.
The contradiction that Cantor put into his assumptions in the diagonal
proof, was that something of infinite size does exist. The number I call
D, the constructed diagonal which can not exist, he declares does it exist.
That's the contradiction in his assumption which causes the final
contradiction.
If you think it's ok to use infinity like it was real, it becomes possible
to prove anything by contradiction. I can easily for example prove that 1
= 0 by making the same mistake by playing with an infinit series of 1 - 1 +
1 - 1 + 1 ..., or by using 1/0 in a proof as if it were a number that
existed.
So, what I'm saying is that infinite sized sets don't exist at all, and
can't exist. And any time you start with an axiom which says "infinite
sized sets do exist", you have introduced an contradiction into your axioms
which guaranties contradictions in your results.
We don't have "the set of all integers". What we have is a counting
algorithm that can generate as many integers as you need for any
application.
It's perfectly valid to talk about what infinite algorithms do as they
approach infinity. But once you start to pretend they reach it, you have
stepped over the line into a world filled with contradiction, and a world
which has nothing to do with the universe we exist in.
This is because "ideas" are not "magic". They are the result of mechanical
computation. And mechanical computation takes time. So any time you talk
about computing an infinite sized set (like the set of all integers) you
have stepped outside the realm of reality and into a fantasy world full of
contradictions which you created by putting the contradiction of the
existence of infinity into your world. If you start an algorithm running
to create your infinite sized set, it will never finish, so any attempt to
talk about what you do after that is invalid. In Cantor's proof, he asked
us to construct an infinite sized real, and then check to see if it was in
one of the rows of the table. And as I showed above, as you construct it,
the number you have will always be in the table. No matter how long you
spend constructing D, the value you have will always be in the table.
If you "pretend" the job of construction does end (the program that can
never halt does halt), then you have put a contradiction into the system
that allows you to prove anything by contradiction. You can prove there
are more reals than integers, or that there are more integers than integers
(as I did above), or that 1 = 0. As long as you can slip that
contradiction in as an axiom (without anyone raising a penalty flag), you
can prove anything you want by contradiction - because you started with a
contradiction.
Much other important work, such as Gödel's, also fell prey to this same
mistake.
Oh, and if you want a mapping from the integers to all the reals, here's
one:
0 0.0
1 0.1
10 1.0
123 2.31
i.e., you take the integer and number the digits like this:
... D4 D3 D2 D1 D0
And you construct the real as: ... D3 D1 . D0 D2 D4 ...
You use the even digits to construct the real to the right of the decimal
point, and the odd digits to construct the real to the left of the decimal
point. So your integer which grew to infinity in one direction, now
creates a real which grows to infinity in two directions.
Now, I know most (if not all of you), will tell me I'm crazy. Many
probably won't even read my post. But if you think I'm crazy, tell me
this. How is my proof that the number of integers is greater than the
number of integers, any less valid than Cantor's proof that the number of
reals is greater than the number of integers? If you can't tell me that,
then why would you believe Cantor's proof is valid? If you can, I'd like
to hear about it.
Has any one else put forth this same argument (or others) that Cantor's
proof is invalid?
--
Curt Welch http://CurtWelch.Com/
cu...@kcwc.com http://NewsReader.Com/
Virgil
cu...@kcwc.com (Curt Welch) wrote:
> Here's something all of you should have some fun with.
>
> Nath is not something I specialize in (and I don't read this group
> normally), but I've been looking at a few things lately and I've decided
> that some very big mistakes have been made in math because people started
> playing around the concept of infinity without realizing the trouble they
> were creating for themselves.
>
> When I was shown Cantor's diagonal proof that the number of reals was not
> countable back in college, I thought it was a fascinating proof. It seemed
> to uncover some great mystery about the nature of numbers that was not at
> first obvious. It sounded very logical and I quickly embraced it as fact.
>
> Lately however, I've come to see things very differently. I now belief the
> proof is totally bogus. And the huge body of work built on top of the
> concept is likewise, totally bogus.
AS the "diagonal " proof was Cantor's SECOND proof of the uncountability
of the reals, and there have been several subsequent proofs, all of
which are totally independent of the "diagonal" construction, it would
not affect the validity of the theorem itself even if the "diagonal"
proof were to be found flawed.
For which reason, no sensible mathematician is the least worried that
such a flaw would in any way weaken the validity of the theorem itself.
José Carlos Santos
> Let me demonstrate.
>
> I claim that there is only one type of infinity. That there are just as
> many integers as there real numbers. (or more accurately, that the concept
> of the size of an infinite set is a contradiction in itself).
>
> So, let me create a mapping. I'll start with the mapping from the integers
> to the reals in the range 0 to 0.99999....
>
> I R
>
> 0 0.0000...
> 1 0.1000...
> 2 0.2000...
>
> 10 0.0100...
>
> 123 0.3210...
>
> So you just reverse the digits in the integer to create the real. I claim
> this mapping is one to one and covers all the reals in that range. For any
> real you give me, I can easily give you the matching integer. Just reverse
> the digits. This includes all the irrationals because in fact, you can't
> give me an infinite irrational to work with. You can only give me an
> algorthm for creating it. And any algorithm you give me, I can modify to
> create the matching integer.
Could you please tell me then which integer corresponds to the real
number 1/9 (or 0.11111111111111111111... if you prefer)?
Best regards,
Jose Carlos Santos
Manuel Petit
> So, let me create a mapping. I'll start with the mapping from the integers
> to the reals in the range 0 to 0.99999....
>
> I R
>
> 0 0.0000...
> 1 0.1000...
> 2 0.2000...
>
> 10 0.0100...
>
> 123 0.3210...
>
> So you just reverse the digits in the integer to create the real. I claim
> this mapping is one to one and covers all the reals in that range. For any
Well, you have to prove that claim.
> real you give me, I can easily give you the matching integer. Just reverse
> the digits. This includes all the irrationals because in fact, you can't
> give me an infinite irrational to work with. You can only give me an
> algorthm for creating it. And any algorithm you give me, I can modify to
> create the matching integer.
Ok start with:
e = 1 + 1/1! + 1/2! + 1/3! + ...
>
> Let's look at this mapping with Cantor's diagonal proof. We construct a
> real number by picking digits from the diagonal which is different from
> each row in the table. Well, as it happens, the diagonal in this mapping
> is all zeros, so we can pick a simple real like 0.1111... as the number
> which can not be in the table. I'll call this number D. Cantor's proof
> seems to show quite clearly that D is not in the table, because it can not
> be located at any row of the table (for what seems to be obvious reasons).
>
> Let me define D(n), as the first N digits of this "constructed" missing
> real diagonal. The first N digits are 1, and the rest are 0.
>
> So D(2) is 0.11 and D(5) is 0.11111 etc.
>
> We see that D(5) can not be located in the first 5 rows of the mapping.
> But, we also can easily prove that D(5) does show up at row 11111.
>
> So, as we construct D(n), we see that even though it doesn't match any of
> the rows up to the point we have reached, it is always further down in the
> table. And because the table is infinite, we will always be able to find
> it futher down in the table.
>
> So, this proves that D(n) for all values of n, from 0 to infinity, is in
> the table.
You fooled yourself. Your are assuming that your table is a table of
reals, but in fact is a table of integers: remember you still have to
prove your mapping is one to one, until then you can only assume you
have a table of integers. Since it is a table of integers, no surprise
the diagonal element is in the table.
>
> So, now we have a contradiction. Cantor's proof says that D(infinity) is
> not in the table, yet D(n) for all values of n, including infinity, is in
> the table from the above logic, which is just as clear and straight forward
> as Cantor's. So how can both be true? If they are not both true, which is
> one wrong and the other one not?
>
> How can this be? I say, it's because there's a contradiction in Cantor's
> proof, and the contradiction is not the one that everyone assumes - that
> there are more reals than integers.
Your (flawed) reasoning went as follows:
* assume N == R [remember you claimed but did not prove]
* Cantor's proof says D(infinity) is not in the table
* Since you assume N == R, you build a table of naturals
and call it a table of reals.
* After building such table, you revel in finding
that D(infinity) is in the table.
* And you conclude that since D(infinity) is in fact in
*your* table, Cantor is wrong, and that you have proved
that N == R.
Again, all your problems start with your table and your (wrong)
assumption that your mapping is one to one and hence you built a table
of reals. Ex falso sequitur quodlibet.
manuel,
Tim Peters
integers with non-trivial infinite decimal representations]
...
> Has any one else put forth this same argument
Yup, many times, over many years.
> (or others)
Those too.
> that Cantor's proof is invalid?
If this isn't literally true, it's close enough <wink>: look at sci.math on
any day over the last decade, and you'll find the same basic argument in
some then-current thread.
Rather than repeat all this, how about going to
http://groups.google.com/groups?group=sci.math
and entering
Cantor diagonal
in the search box? There are close to 5,000 hits on that today. Broaden
the search to sci.logic and comp.theory to get more. Add words like
"wrong", "idiot", "invalid", "flawed", "refuted", "incorrect", "fallacy",
and "disproof" to get slightly different flavors of confusion.
Of course there's no end to this: simply arrange all the claimed
refutations of Cantor's argument in a list. Make a new refutation whose
i'th word is obtained by picking a random word other than the i'th word of
the i'th refutation in the list. The refutation so constructed will be a
refutation not on the list, and will make about as much mathematical sense
as any refutation on the list.
Now understanding clearly why *that* argument is full of beans is a good
start at understanding why Cantor's argument is not <wink>.
David C. Ullrich
>Here's something all of you should have some fun with.
>
>Nath is not something I specialize in (and I don't read this group
>normally), but I've been looking at a few things lately and I've decided
>that some very big mistakes have been made in math because people started
>playing around the concept of infinity without realizing the trouble they
>were creating for themselves.
It would make a lot more sense to decide that given that the proof
has been accepted for over a century, by every mathematician in the
world, you must be missing something.
>When I was shown Cantor's diagonal proof that the number of reals was not
>countable back in college, I thought it was a fascinating proof. It seemed
>to uncover some great mystery about the nature of numbers that was not at
>first obvious. It sounded very logical and I quickly embraced it as fact.
>
>Lately however, I've come to see things very differently. I now belief the
>proof is totally bogus. And the huge body of work built on top of the
>concept is likewise, totally bogus.
>
>But, like I said, I'm not a math expert by any means. So I'm posting the
>idea here so you experts can have fun laughing at me.
But what the heck, posts like this do have a purpose. Haha. Hahahaha.
>[...]
>
>Let me demonstrate.
>
>I claim that there is only one type of infinity. That there are just as
>many integers as there real numbers. (or more accurately, that the concept
>of the size of an infinite set is a contradiction in itself).
>
>So, let me create a mapping. I'll start with the mapping from the integers
>to the reals in the range 0 to 0.99999....
>
>I R
>
>0 0.0000...
>1 0.1000...
>2 0.2000...
>
>10 0.0100...
>
>123 0.3210...
>
>So you just reverse the digits in the integer to create the real. I claim
>this mapping is one to one and covers all the reals in that range. For any
>real you give me, I can easily give you the matching integer.
What integer maps to the real number 1/3?
>[...]
>
>It's perfectly valid to talk about what infinite algorithms do as they
>approach infinity. But once you start to pretend they reach it, you have
>stepped over the line into a world filled with contradiction, and a world
>which has nothing to do with the universe we exist in.
Uh, nothing in the proof has anything to do with "what infinite
algorithmms do when they reach infinity", whatever that means.
>[...]
>
>Now, I know most (if not all of you), will tell me I'm crazy.
I doubt that. Whether or not people call you crazy depends on
how you react to the simple demonstrations that you're simply
wrong about all this - too early to tell about that since you
haven't had a chance to reply to the replies yet.
>Many
>probably won't even read my post. But if you think I'm crazy, tell me
>this. How is my proof that the number of integers is greater than the
>number of integers, any less valid than Cantor's proof that the number of
>reals is greater than the number of integers?
Well, you simply state without proof that that mapping above maps
some integer onto any given real.
>If you can't tell me that,
>then why would you believe Cantor's proof is valid? If you can, I'd like
>to hear about it.
It's not true that every real is the image of some integer under that
mapping, for example 1/3 gets missed.
>Has any one else put forth this same argument (or others) that Cantor's
>proof is invalid?
Yes, several times a year on sci.math for the last decade or so.
************************
David C. Ullrich
Rasmus Villemoes
> So, let me create a mapping. I'll start with the mapping from the integers
> to the reals in the range 0 to 0.99999....
>
> I R
>
> 0 0.0000...
> 1 0.1000...
> 2 0.2000...
>
> 10 0.0100...
>
> 123 0.3210...
>
> So you just reverse the digits in the integer to create the real. I claim
> this mapping is one to one and covers all the reals in that range.
The same (or at least nearly the same) mapping was used in an arXiv
article some time ago. A few weeks later, the author posted another
article with a corollary to the above 'theorem' (that the mapping
is bijective). The corollary stated "Not every integer is finite".
--
Rasmus Villemoes
<http://home.imf.au.dk/burner/>
Curt Welch
> Could you please tell me then which integer corresponds to the real
> number 1/9 (or 0.11111111111111111111... if you prefer)?
....1111111
Why is it ok to write 0.111... but not ...11111 ?
My point is that 1/9 is in fact an algorithm for generating a real value.
It is not the real value itself. It's just a name we use to talk about the
real value which is 0.1111 repeating forever. And I can just as easily
define the integer of 1 repeating forver. The only reason we do not do
that is a matter of convention. It's not (so I claim) in violation of what
integers are.
If you start with 0, and continue to apply the +1 function to it, and
ignore all the values you come up with which does not have all 1's, you
find you have the exact same type of defintion that gives you 1/9 when you
generate a string of one's running to the right, instead of running to the
left.
Also, all integers have an implied infinite string of 0's running to left
(just like reals have am implied infinite string of 0's running to the
left and right). So when you write 123, you are really writing
....00000123.
Just change the implied 0, to an implied 1, and there you have it. The
integer pair for the real 1/9.
Torkel Franzen
> Why is it ok to write 0.111... but not ...11111 ?
You can indeed write "...11111", and also "fnoggle fnoggle" and
"froufy forbandle ignotoot!".
José Carlos Santos
>>Could you please tell me then which integer corresponds to the real
>>number 1/9 (or 0.11111111111111111111... if you prefer)?
>
>
> ....1111111
>
> Why is it ok to write 0.111... but not ...11111 ?
Because the sum 1/10 + 1/100 + 1/1000 + ... converges, whereas the sum
1 + 10 + 100 + ... doesn't.
> My point is that 1/9 is in fact an algorithm for generating a real value.
> It is not the real value itself.
Then you should define the meaning of "real value". Not that it really
matters, of course. What Cantor did was to prove that no bijection
exists between the set of all natural numbers and the set of real
numbers. Now what you seem to be doing consists (or so it seems) in
redefining the concept of real number (or "real value", as you call it)
getting something different. But then your concept of "set of real
numbers" will be different from Cantor's. So, when you state that there
is a bijection between the set of all natural numbers and *your* set of
real numbers, there is in fact no contradiction between you and Cantor,
since you are talking about different things.
Rasmus Villemoes
> José_Carlos_Santos <jcsa...@fc.up.pt> wrote:
>
>> Could you please tell me then which integer corresponds to the real
>> number 1/9 (or 0.11111111111111111111... if you prefer)?
>
> ....1111111
>
> Why is it ok to write 0.111... but not ...11111 ?
Because 0.111... has a well defined interpretation in terms of the
limit of the convergent series
sum_{i=1 to infinity} 1·10^{-i}
whereas the ...11111 is just nonsense.
> My point is that 1/9 is in fact an algorithm for generating a real
> value. It is not the real value itself.
Yes, it _is_ the real number "one ninth".
> It's just a name we use to talk about the real value which is 0.1111
> repeating forever. And I can just as easily define the integer of 1
> repeating forver. The only reason we do not do that is a matter of
> convention. It's not (so I claim) in violation of what integers are.
Then your notion of an integer is in violation with the generally
accepted notion. If you would post a system of axioms (similar to the
Peano axioms) for your "integers" in which ...11111 has a well defined
interpretion as an "integer", then we can continue the discussion.
> If you start with 0, and continue to apply the +1 function to it,
>and ignore all the values you come up with which does not have all
>1's, you find you have the exact same type of defintion that gives
>you 1/9 when you generate a string of one's running to the right,
>instead of running to the left.
Yes, there is an infinite family of integers consisting of "only 1's",
namely 1 (one), 11 (eleven), 111 (one hundred and eleven) etc., just
as there is an infinite family of reals between 0 and 1 whose decimal
representation starts with a number of 1's and has 0's elsewhere
(namely .1, .11, .111 etc.). The _difference_ is that the latter has a
real number (namely 1/9) as a limit, whereas the former has no limit
at all. (Well, it diverges in such a nice way that one sometimes says
it converges to infinity, but infinity is not an integer nor a real).
> Just change the implied 0, to an implied 1, and there you have it.
> The integer pair for the real 1/9.
No. What you have is an infinite string of 1's which do NOT represent
an integer.
Lee Rudolph
Interstingly enough, however, (and in concord with Yosenin-Volpin's
hyperfinistic doctrine), you *cannot* write "froufy forbandle ignotoot!"
*arbitrarily often*.
Lee Rudolph
Curt Welch
> AS the "diagonal " proof was Cantor's SECOND proof of the uncountability
> of the reals, and there have been several subsequent proofs, all of
> which are totally independent of the "diagonal" construction, it would
> not affect the validity of the theorem itself even if the "diagonal"
> proof were to be found flawed.
>
> For which reason, no sensible mathematician is the least worried that
> such a flaw would in any way weaken the validity of the theorem itself.
Well, first, the "diagonal proof" has spread to many different fields. And
if it is flawed, it would be good to understand how it's flawed.
Second, I suspect the flaw which I claim is in the diagonal proof is in all
the other proofs you speak of. I do not know all the other proofs and it
will take me some time to learn their language, and come up with counter
examples as I have done here to show their weakness. And to be honest, I
don't have enough interest in math to spend the time it would take to clean
up the mess that I believe has been created in the past 100 or so years.
So lets just start with the diagonal proof and see if we can understand
what's going on here. If you can convince me that there is nothing wrong
with the diagonal proof, then there is no point looking further. If the
diagonal proof is at risk, then clearly it would be wise to understand why.
The issue at the heart of it is that we define a lot of things in math with
algorithms. And if you specify an algorithm that runs forever, and never
halts, then it's not valid to talk about what happens after it halts. It's
the same mistake in logic one would make if they claimed that time and
space was infinite in this universe, and then talked about what happens
after a spaceship sent out from earth, reached the edge of the universe.
If you say it's infinite, then it does not have an end. To try an
manipulate an infinite set is to make the mistake of assuming that it does
have an end. You can't have it both ways. Either it's infinite, and you
can never construct it, or it's finite, and you can construct it.
The flaw in the diagonal proof is that it asks us to construct an infinite
sized object, and then asks us to apply rules against it (check to see if
it is in the table) that only applies to finite sized objects, and finite
sized tables. If you have a finite sized table, and construct a similar
diagonal, number, then you do in fact know that the number you constructed
is not in the table. But you can't use the same logic for infinite sized
tables and infinite sized diagonals. It's because the diagonal you
construct is always further down in the infinite table. And since you can
never finish constructing it, you can never prove that what you
constructed, is not still further down in the table.
This problem is well seen in algebra when you try to divide by zero.
Division is an algorithm for reversing multiplication which is an algorithm
for counting which itself is an algorithm for generating abstract names for
objects.
When you try to divide by zero, the division algorithm never halts. It can
not produce an answer because of that. We call this "condition" infinity.
And when it happens in an equation, it's well known the result becomes
undefined. If you instead treat the result as if it were a valid number
(i.e., that the division algorithm did in fact produce an answer for you),
you have introduced a contradiction into your equation. And once you have
a contradiction which you audience does not see as a contradiction, you can
prove just anything you want.
So how could this confusion have existed for so long? It's because of the
power of language. It fools us without us even knowing it is doing it to
us. And this power of language to pull the wool over our eyes is what I
found in searching for AI. Cantor didn't trick us. We tricked ourselves
by making the mistake of believing that something that "sounds right" in
words, must be right.
Curt Welch
> Again, all your problems start with your table and your (wrong)
> assumption that your mapping is one to one and hence you built a table
> of reals. Ex falso sequitur quodlibet.
That's a valid stance (though I belive not true).
However, I also used the same diagonal argument to show that a table of
integers didn't contain all the integers. Where is the flaw in that
argument?
If the same logic used in Cantor's proof can be used to show that a table
of integers does not contain all the integers, then most there not be a
problem somewhere? If so, where is it?
Curt Welch
Ok, I feared this might be the case and did not search very far before
posting.
My "strong" argument is that using Cantor's diagonal logic shows that the
table of integers doesn't contain all the integers. Can you just tell me
where the flaw is in that argument to prevent me from having to search
1000's of posts?
Curt Welch
> The same (or at least nearly the same) mapping was used in an arX
> article with a corollary to the above 'theorem' (that the mapping
> is bijective). The corollary stated "Not every integer is finite".
I don't understand the term "bijective", but I can still quickly conclude
that the mapping led to the conclusion that "Not every integer is finite".
And since we all "know" that every integer is finite, there must be a
problem somewhere. And the assumption, is that the problem is with the
mapping.
I would counter that argument by saying that no integers are finite. They
all have an infinite string of 0's in front of them which, purely by
convention and practice, we choose not to write. The belief that integers
are finite is mis-founded and in strict opposition to the way they are
defined.
Do we write the value 0 as " "? Do we write 100 as "1 "? So why is it
that we think the integers are finite?
What does "finite" mean to you (or to the author of the article which wrote
"Not every integer is finite")?
Jon Haugsand
> Nath is not something I specialize in (and I don't read this group
> normally),
Obviously.
> but I've been looking at a few things lately and I've decided
> that some very big mistakes have been made in math because people started
> playing around the concept of infinity without realizing the trouble they
> were creating for themselves.
Don't you find just a tiny piece of pathetic arrogance here?
--
Jon Haugsand
Dept. of Informatics, Univ. of Oslo, Norway, mailto:jon...@ifi.uio.no
http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92
ste...@nomail.com
: Rasmus Villemoes <burner...@imf.au.dk> wrote:
:> The same (or at least nearly the same) mapping was used in an arX
:> article some time ago. A few weeks later, the author posted another
:> article with a corollary to the above 'theorem' (that the mapping
:> is bijective). The corollary stated "Not every integer is finite".
: I don't understand the term "bijective", but I can still quickly conclude
: that the mapping led to the conclusion that "Not every integer is finite".
: And since we all "know" that every integer is finite, there must be a
: problem somewhere. And the assumption, is that the problem is with the
: mapping.
: I would counter that argument by saying that no integers are finite. They
: all have an infinite string of 0's in front of them which, purely by
: convention and practice, we choose not to write. The belief that integers
: are finite is mis-founded and in strict opposition to the way they are
: defined.
No. Integers are finite. That is a direct conclusion of how
integers are defined. Every integer is finite.
: Do we write the value 0 as " "? Do we write 100 as "1 "? So why is it
: that we think the integers are finite?
Because that is how they are defined.
: What does "finite" mean to you (or to the author of the article which wrote
: "Not every integer is finite")?
A finite value is bounded by some integer.
Stephen
Curt Welch
> * Curt Welch
> > Nath is not something I specialize in (and I don't read this group
> > normally),
>
> Obviously.
>
> > but I've been looking at a few things lately and I've decided
> > that some very big mistakes have been made in math because people
> > started playing around the concept of infinity without realizing the
> > trouble they were creating for themselves.
>
> Don't you find just a tiny piece of pathetic arrogance here?
No, not at all. It's a huge heaping hunk of pathetic arrogance.
Extraordinary claims require extraordinary proof. So I offered one. I
used Cantor's logic to prove that a table of integers does not contain all
the integers. I'm still waiting for someone to explain the obvious
contradiction.
Curt Welch
Ok, that's valid. We do not have a common foundation to argue from.
But lets just ignore these side issues and look at the
strongest part of my argument where I use Cantor's logic to prove that a
table of integers does not contain all the integers. How can you show my
argument is invalid and at the same time, keep Cantor's logic about the
relationship between the integers and reals as valid?
Curt Welch
> So, when you state that there
> is a bijection between the set of all natural numbers and *your* set of
> real numbers, there is in fact no contradiction between you and Cantor,
> since you are talking about different things.
Ok, I see your position. I'm making assumptions that you do not accept.
I'd need to convence you that my assumptions follow from some of your basic
beliefs. And I don't have the power to do that right now because I don't
understand all your basic beliefs and don't know your full langauge.
But, still, I need someome to show me the error of my logic in my proof
that the table of integers does not contain all the integers. That should
not require us to build a common foundation about reals and integers to
argue from. It only requires that we have a common foundation about
integers, and the logic used in Cantor's diagonal proof. We can leave the
definition of reals out of the argument.
Torkel Franzen
> How can you show my
> argument is invalid and at the same time, keep Cantor's logic about the
> relationship between the integers and reals as valid?
If you're a genuine crank, it is of course quite impossible to show
you anything. If you're not, you'll find out about these things for
yourself.
Chairman of the David Hilbert Appreciation Society
> Here's something all of you should have some fun with.
[...]
A quick note on how my comments relate to your notation:
You're refering to the set {0, 1, 2, 3, ...} as the integers,
but this set is known as the naturals and is denoted by N. The
integers would include negative values as well
{..., -2, -1, 0, 1, 2, ...}, and is denoted by Z.
I'll refer to the set {0, 1, 2, ...} as N. I'll also refer
to the rationals as Q. Keep these changes in mind to minimize
confusion.
[...]
> I claim that there is only one type of infinity. That there are just as
> many integers as there real numbers. (or more accurately, that the concept
> of the size of an infinite set is a contradiction in itself).
>
> So, let me create a mapping. I'll start with the mapping from the integers
> to the reals in the range 0 to 0.99999....
>
> I R
>
> 0 0.0000...
> 1 0.1000...
> 2 0.2000...
>
> 10 0.0100...
>
> 123 0.3210...
You'll notice that each natural on the left corresponds to a finite
length decimal representation on the right however not all real
numbers can be represented by this method.
[...]
> Let's look at this mapping with Cantor's diagonal proof. We construct a
> real number by picking digits from the diagonal which is different from
> each row in the table. Well, as it happens, the diagonal in this mapping
> is all zeros[...]
Exactly. Because every number on the right side of your list
is a rational. You're putting N into one-to-one correspondence
with a subset of Q, not R.
[...]
> As another example, let me show that the number of integers are also
> greater than the number of integers, using the logic of Cantor's proof.
>
> Lets create a table of integers like this:
>
> ...000000
> ...000001
> ...000002
>
> ...000010
>
> ...000123
>
> It's just a normal list of integers, but instead of following the normal
> convention of leaving off the leading zeros (which we all know are implied
> even if we don't write them) I include them in that table.
>
> So lets use Cantor's logic on this table and see if we can construct a
> number which is not in the table. We take the numbers from the diagonal,
> and construct the number ...111111 just like we did above.
How many digits does this diagonal string have?
It has infinitely many digits.
We agree that it's not on the list, however it's not a natural
number and so we can't conclude that it represents a natural
number that _isn't on the list_.
The key difference between Cantor's diagonal proof with the reals
and your attempted use of that proof strategy with integers is
that the integers have finite length representations and the
reals don't.
When we diagonalize we get an infinite length string...
Think about this please, don't just react.
[...]
> Much other important work, such as Gödel's, also fell prey to this same
> mistake.
Let's just deal with Cantor first.
> Oh, and if you want a mapping from the integers to all the reals, here's
> one:
>
> 0 0.0
> 1 0.1
> 10 1.0
> 123 2.31
>
> i.e., you take the integer and number the digits like this:
>
> ... D4 D3 D2 D1 D0
>
> And you construct the real as: ... D3 D1 . D0 D2 D4 ...
Your mistake is common. The problem that most people seem to
have with Cantor's well known diagonal proof is that they
don't know what the real numbers are. I'm not joking about this.
This is especially so for people who aren't mathematicians
yet have a technical undergrad background which gives them
exposure to Cantor's diagonal proof, without a proper development
of the real numbers. Following graduation the student has great
confidence in his technical abilities, but doesn't actually know
what the real numbers are.
His exposure to the properties of real numbers is based on
informally taught subjects like Algebra, Trig, and Calc where
every variable and every answer to every problem is shown on
the tiny calculator screen as a finite length decimal expansion.
It's my belief that the repetition of a student seeing "any real
number" expressed as finite string on an LCD display combined
with his informal exposure to the subject results in his developing
a powerfully felt conviction that any/all real numbers are finite
length decimal expansions.
Finite length decimal expansions are a subset of rational numbers,
they are countable. Cantor essentially proved this in 1873.
The fact that Cantor proved a result which you intuitively expect
should give you greater confidence in his work.
[...]
> Has any one else put forth this same argument (or others) that Cantor's
> proof is invalid?
Yes. This happens often in sci.math. People mistakenly think
that the set of real numbers is the set of all finite decimal
representations. By showing a one-to-one correspondence between
N and all finite decimal representations you're rediscovering
a 125 year old result.
--
Replace Roman numerals with digits to reply by email
Curt Welch
So far, everyone has taken issue with my defintion of "reals" or with the
idea that my mapping from reals to integers is not complete. And that's
fine. It is not easy for me to argue my position there because I don't
have the correct foundation.
But how can you show me (ok, a student of your's who is not a crank), where
the flaw is in the logic I used to show that the table of integers does not
contain all the integers?
It's clear from the defintion of the table, that it does contain all the
integers. Yet, when we apply the same logic which Cantor's proof used, we
are able to contruct an integer which is not in the table.
Is your position that the integer we construct is not an integer? So
therefor it's not surprising that it is missing from the table?
Ah, I think that argument would fit into the arguments I've seen put forth
in this thead so far. If that is the position, then I see I won't be able
to stop here. I'll have to go deeper. I'll have to find a way to show the
contradiction exists in some set of axioms. Damn.
Ross A. Finlayson
Hi,
I'm writing to belabor "the binary case is sufficient and necessary."
I'm reminded of my request about belaborment, which was about
communication and confusion issues, Virgil. Why do you think the
antidiagonal argument is flawed?
In the binary case, there is one specific anti-diagonal.
Consider an arbitrary base. Any method you use to generate some
antidiagonal will affect more than one location in the expansion as a
binary number. In that way, it might reset one of the previous
locations that would have been different, thus that the antidiagonal
would not be different at that location. That's an implication that a
number represented in a different base is a different number, and
stranger things are known to occur. That is perhaps just an artifact
of the algorithm.
That's similar to the argument that any number is representable in any
radix (base). The point is being that if there is some list, to
generate the list in a base greater than three, where three is as well
shown useless as a base to definitely generate an antidiagonal, and
construct an antidiagonal in some way that it is not rational so it
couldn't be dually represented.
Add a leading zero to each element of the list, then only in a
specific case is the antidiagonal an element in the range.
You refer to other arguments about the naturals and the reals, so do
I. With regards to the nested intervals, they are not constructed,
with EF. Then again, my line of reasoning easily uses what you would
not term standard real numbers.
The rationals are dense in the real numbers.
Curt, you might want to learn about Skolem. Skolem extended the work
of Loewenheim to show that everything is countable. People handwave
about that and they're quite nonsensical in their ludicrous nonsense,
because the extensions are no different than the set. What that means
is they say that they have a set of integers that maps to a powerset
of integers, but in a receding slippery slope type of way that still
claims the opposite true. That's why they call that quandary Skolem's
paradox.
http://groups.google.com/groups?q=Skolem+Cantor
If you accept that the powerset result does not always hold true,
then, both Skolem's and Cantor's "paradoxes" dissolve, where Cantor's
is that a set of all sets would be its own powerset, and would map to
itself with the identity function. Without transfinite cardinals, for
everyone, measure theory needs some few new foundations, or rather,
just rephrased foundations, with perhaps some meaningful results, and,
that is about it, and all of transfinite cardinal mathematics is its
own little subfield where you axiomatize that so, just so all the work
put into transfinite cardinals was not a total waste of time, like a
pickled three-headed sheep.
Curt, what's the point, man? Do you want to map the reals to the
integers? What good is mapping the reals to the integers? Do you
think calculus is easier to understand if dx is a llittle
infinitesimal coefficient and when you sum the product of the function
and dx over the range that you get the integral? Even if that was too
slippery for general use, the limit being a safety feature of sorts,
and all the calculus was done using limit, wouldn't that be better?
Me, I was just offended that somebody claimed infinite sets weren't
equivalent. Now I feel better about it, because I've proved a few
things about that to people.
Do Zeno's paradoxes prevent Achilles from catching the tortoise? No,
they don't. Does Skolem's paradox prevent there being uncountable
sets? It does. You've probably heard of the "paradoxical" barber,
there are no paradoxes and so that barber does not exist anywhere,
because everybody in that town is shaved by the barber unless (if and
only if they don't) they shave themselves, everybody shaves, nobody
ever leaves town, and the well-meaning barber, who as an expert
probably shaves himself, also is the barber shaving himself. So, the
barber shaves himself and anybody else who doesn't shave themself.
Take two infinite sets. If there is a way that for each you can
select an element of each set and remove it from that set, do that.
That's a terse constructive proof that infinite sets are equivalent.
Cantor's results have meaning, they in a way force certain conclusions
about the nature of binary logic, because of that one element that is
unmapped, call it the antidiagonal or something, infinity rolls right
back over to zero like an odometer.
That gets into that any set X is an ordinal, and that the order type,
and successor, and X+1, and the powerset, are all the same thing.
When you're talking about mapping the naturals to the reals, there's
probably actually some useful formulas or "functions" that be used to
derive mathematical results that are not otherwise immediately
apparent. Here's a mapping between the natural numbers (0, 1, 2, ...,
non-negative integers) and the unit interval of the reals ( R[0,1],
every real number between zero and one inclusive): the natural/unit
equivalency function, EF. It's simple, order the reals from least to
greatest or greatest to least, and then map zero from the integer to
least or greatest, and then, in order, pair elements of those sets.
The binary antidiagonal does not exist on the range or is dually
represented, or you can add leading zeroes, and non-standard real
numbers, which are very much real numbers, are used thus that results
about mapping the naturals and reals do not apply. So anyways,
integrate EF and the result is equal to one, where you might think it
would be equal to one half, because you'd figure it would be just like
f(x)=x from zero to one.
That has to do with how points on the real number line are defined in
terms of preceding and following points on that same line, and that
points on a continuous line are in a sense one-sided, where that side
is in the direction of the ray's passage on that line, as the reals
are ordered thus that for two different real numbers one is lesser and
one is greater, or oppositely one is greater and one is lesser. When
the number is by itself then it has two sides and twice the weight,
because two different straight lines can pass through it.
You may as well consider a different method for sweeping through those
points, such as a spiral of sorts or alternatively taking the next
indefinite real element on the lesser and greater side. Again, that
leads to models of non-standard reals, which are real numbers.
Anyways, Curt, some people are very attached to their notions of
cardinality and the uncountable, they think it's very sophisticated
and urbane. A lot of work has been done based upon the simple notion
that f(x) = x+1 doesn't equal x+1. Most don't give a damn either way.
You can say that half of the integers are even, and that half of the
integers are positive, and that a given fraction of the integers are
primes or perfect squares, without the necessity of the transfinite
cardinals. As well, it is shown that a proper subset of a set has
less elements than the superset. There are more rationals than
integers, and more reals than irrationals or rationals. A powerset
has more elements than the set, in a sense, that's not the problem.
Curt, 1+1=2, and 2+2=4. Can't you leave the Cantorians their paradise
and well enough alone? Biblically, Adam and Eve were cast from
Paradise after they partook of the tree of knowledge. If they hadn't,
they'd still be there and that would be the end of the story.
Warm regards,
Ross F.
Daniel W. Johnson
> But how can you show me (ok, a student of your's who is not a crank), where
> the flaw is in the logic I used to show that the table of integers does not
> contain all the integers?
It's at the point where you called "....1111111" an integer.
--
Daniel W. Johnson
pano...@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
Curt Welch
<mathgee...@hotmail.com> wrote:
> Curt Welch wrote:
> When we diagonalize we get an infinite length string...
>
> Think about this please, don't just react.
Yeah, as I just posted before I read your post, I saw that argument coming.
Or, from another perspective, what everyone else was saying sank in before
I read your post.
> Your mistake is common. The problem that most people seem to
> have with Cantor's well known diagonal proof is that they
> don't know what the real numbers are.
Yes, that's it. I currently have a different view of what real numbers are
than what everyone else is using. (as well as a different view of what
integers or natural numbers are). To continue this investigation, I have
to master the justification mathematicians use to define both.
When I first started seeing this contradiction develop, I was torn between
trying to understand if math was based on a contradiction, or if it was
just a matter of choice that the axioms used to define math created a
contradiction that was easy to avoid of you simply choose different axioms.
After some debates many months ago (not here), I concluded that it was just
a matter of choice of which axioms you picked to build your definition of
math on. But I thought this "proof" that not all integers are in the table
of integers made it clear there was a fundamental contradiction in the
axioms. But it does not.
Or, if it does, I don't know enough about how axioms are used to define
math to argue the point. So before I can finish answering my own question
about whether math is based on a contradiction, or not, I have to master a
deeper understanding of math.
The reason I've looked into this is because various arguments such as
Gödel's incompleteness theorem spills over into arguments about building
intelligent machines. And some would claim there is an important
relationship there that seems to indicate there are limits to what machines
can do - some even take it so far to believe that it "proves" you can not
build a conscious machine.
My belief is that you can build a conscious machine - that we, in all our
glory, are nothing more than a complex biological machine. If this is
true, then everything we create, like all our ideas about math, are nothing
more than the product of our behavior - which is nothing more than the
actions of a complex physical machine.
At the heart of all this is the issue of how the mental world ties to the
physical world. And that touches on how the ideas developed in all the
fields of reason, like math, relate to the physical world.
The fact that it's reasonable in math, to talk about the size of one
infinite set being larger than the size of another infinite set, doesn't
seem to tie correctly to the physical world.
So if the world of math is nothing more than something happening in the
physical world, why would the two worlds not seem to "fit together" better?
This is the ultimate question I'm exploring.
I believe the answer is that the "world of math" which we have created, is
fundamentally different from the physical world. We have used the power of
language to create a world of math, which can not exist in the physical
world.
I think hidden in here is a subtle, but important, distinction between the
ideas of "existence" and "describing existence" which allows us to define
some really crazy things. We know for example it's trivial to use language
to tell a lie - to make up a story which we know is not real. But, just
because a story is not real, doesn't mean it can't be true at some place,
and some time, in the universe.
But, is there a point, where you use language to create a story, which can
never be true in this universe? I think there is, and I think the world of
math has done just that. They have used language to create an imaginary
world, and at some point, they cross the line from possible, to impossible,
in the physical universe. And I think that point happens when we pretend
that infinite sized things, like the real value 1/9, can be constructed in
zero time, and that we can then manipulate this value as if it existed.
It's the confusion between manipulating the word which represents an
object, vs. manipulating the object itself. Once you start using words to
describe other words (which is what math is all about), the distinction
between words and objects gets lost. And at that point, I think we start
to define things with words, that can never exist as objects in the
physical universe.
So, this could mean, that the idea of different sized infinite sets is an
idea that can not exist for real in the physical universe, even though we
have no problem using language to create an imaginary world where these
things do exist.
So, my focus, is trying to understand where language allows us to leave
physical realty behind, and exactly what happens when we do that. But I
see I need to learn more about the language of math before I can better
understand what we have defined here.
Curt Welch
> Curt, what's the point, man?
I've addressed that to some extend in a previous post now. My interst
comes from my exploring the ideas of AI. I'm not trying to "fix" math, I'm
trying to understand what has happened.
My interest is in understanding the relationship between what we can do
with language, and what exists in the physical world. My interest is to
try and understand if there is a clear point where we violate some
important principle and end up describing something with language, that can
never exist in our universe.
I can make up a story about a blue book on my desk. There is no blue book
on my desk, so I've just used language to describe something that does not
exist. However, just because it does not happen to exist does not mean it
is impossible for it to exist. I can describe a blue book on my desk which
is talking to me. That's something we see in the cartoons all the time.
And as far as we all know, such a thing does not exist in real life
anywhere in the universe. But it could exist for all we know.
But, is there some way to use language where we cross over from
very-unlikely, to flat out impossible? I think there might be. I'm trying
to understand if that point exists and how to describe it.
I'm trying to understand if some fields of math might have wondered off
into the "flat out impossible" land. And if they have, what it means for
those fields of reason and how they relate to the fields of reason which
have not left the land of the possible.
> Anyways, Curt, some people are very attached to their notions ...
That's key. People use language to justify what they believe. They seldom
if ever, really understand why they believe what they believe, yet, if they
can construct elaborate language to justify it, it makes them feel good, so
they do it. We all work this way. And that's part of the danger. The
language we use to justify everything exists simply because it makes us
feel good. Separating truth, from "good feelings" is much harder to do
than most people understand because in the end, none of cares as much about
truth as we do about feeling good. (but that's another argument for another
group).
You used a lot of language about math in your post which I do not
understand. I have a lot of work to do before I could discuss those issues
with you. But I did find your post interesting.
fishfry
cu...@kcwc.com (Curt Welch) wrote:
> r...@tiki-lounge.com (Ross A. Finlayson) wrote:
>
> > Curt, what's the point, man?
>
> I've addressed that to some extend in a previous post now. My interst
> comes from my exploring the ideas of AI. I'm not trying to "fix" math, I'm
> trying to understand what has happened.
>
> My interest is in understanding the relationship between what we can do
> with language, and what exists in the physical world. My interest is to
> try and understand if there is a clear point where we violate some
> important principle and end up describing something with language, that can
> never exist in our universe.
>
> I can make up a story about a blue book on my desk. There is no blue book
> on my desk, so I've just used language to describe something that does not
> exist. However, just because it does not happen to exist does not mean it
> is impossible for it to exist. I can describe a blue book on my desk which
> is talking to me. That's something we see in the cartoons all the time.
> And as far as we all know, such a thing does not exist in real life
> anywhere in the universe. But it could exist for all we know.
>
> But, is there some way to use language where we cross over from
> very-unlikely, to flat out impossible? I think there might be. I'm trying
> to understand if that point exists and how to describe it.
>
> I'm trying to understand if some fields of math might have wondered off
> into the "flat out impossible" land.
You seem to be under the impression that math is required to restrict
itself to objects that can exist in the physical world. That is not
true. There are a finite number of atoms in the physical universe. There
are no infinite sets in the physical universe. Yet we can conceive of
infinite sets such as the natural numbers.
If all you're interested in is the physical world, then you don't need
infinite sets, let alone the real numbers.
Tapio
"José Carlos Santos" <jcsa...@fc.up.pt> wrote in message
news:2vpjm9F...@uni-berlin.de...
> Curt Welch wrote:
>
>>>Could you please tell me then which integer corresponds to the real
>>>number 1/9 (or 0.11111111111111111111... if you prefer)?
>>
>>
>> ....1111111
>>
>> Why is it ok to write 0.111... but not ...11111 ?
>
> Because the sum 1/10 + 1/100 + 1/1000 + ... converges, whereas the sum
> 1 + 10 + 100 + ... doesn't.
It depends on the point of reference. If your point of reference is is the
standard decimal dot, then the sum really does not converge. So far you are
right.
But if you choose another point of reference namely omega, which is defined
in this context as a number that is greater´than any integer, then the limit
of ...111111 equals to (1/9)*omega.
Satisfied?
I pointed couple of years ago in this newsgroup, that infinite integers and
finite integers have without inversion (which discussed in this tread) a
direct bijection with reals as the point of reference is on the right hand
side od decimal numbers. See the thread why reals are welläordere and
infinite integers.
Do not want to repeat it here.
Cantor was concentrated in finite integers (all natural numbers) and reals
and there is no bijection as long as you cannot accept that infinite string
can be finite between omega and zero.
Tapio
Tapio
"Rasmus Villemoes" <burner...@imf.au.dk> wrote in message
news:u0ly8h4...@legolas.imf.au.dk...
> cu...@kcwc.com (Curt Welch) writes:
>
>> José_Carlos_Santos <jcsa...@fc.up.pt> wrote:
>>
>>> Could you please tell me then which integer corresponds to the real
>>> number 1/9 (or 0.11111111111111111111... if you prefer)?
>>
>> ....1111111
>>
>> Why is it ok to write 0.111... but not ...11111 ?
>
> Because 0.111... has a well defined interpretation in terms of the
> limit of the convergent series
>
> sum_{i=1 to infinity} 1·10^{-i}
>
> whereas the ...11111 is just nonsense.
>
>> My point is that 1/9 is in fact an algorithm for generating a real
>> value. It is not the real value itself.
>
> Yes, it _is_ the real number "one ninth".
>
>> It's just a name we use to talk about the real value which is 0.1111
>> repeating forever. And I can just as easily define the integer of 1
>> repeating forver. The only reason we do not do that is a matter of
>> convention. It's not (so I claim) in violation of what integers are.
>
> Then your notion of an integer is in violation with the generally
> accepted notion. If you would post a system of axioms (similar to the
> Peano axioms) for your "integers" in which ...11111 has a well defined
> interpretion as an "integer", then we can continue the discussion.
I have defined infinite integers as a sum that is in this peculiar case sum
(n 0 --->oo) 1*10^n. I has no limit as the point of reference is the
standard point of reference. But is has limit that equals to (1/9)*omega.
Are you ready to continue discussion?
Tapio
Curt Welch
> Curt Welch <cu...@kcwc.com> wrote:
>
> > But how can you show me (ok, a student of your's who is not a crank),
> > where the flaw is in the logic I used to show that the table of
> > integers does not contain all the integers?
>
> It's at the point where you called "....1111111" an integer.
Yeah, I see that now. I need to learn more about the formal definitions
mathematicians use to define reals and integers.
I see part of it already from what has been written. If you think of
integers and reals as points on a line, then 0.1111.. seems to approach a
single location on the line, where as ..11111 does not approach a single
location on the line. So there's an obvious intuitive feeling that
.1111... is a single point where as "...1111" is not a single point.
However, until you reach an infinite number of digits, neither represents a
single point. Both represent constantly changing points. And though the
distance between the points is growing smaller in once case, and larger in
the other, you still at all times have an infinite number of points between
each changing point in both series.
So though I can sense the intuitive desire to see ..111 as not being an
integer, I don't yet understand how this is formally defined in mathematics
without putting a contradiction into the system. So I have to go learn all
that work of formal logic and axiomatic stuff which I know very little
about before I can even discuss this further.
Virgil
cu...@kcwc.com (Curt Welch) wrote:
> Virgil <ITSnetNOTcom#vir...@COMCAST.com> wrote:
> > AS the "diagonal " proof was Cantor's SECOND proof of the uncountability
> > of the reals, and there have been several subsequent proofs, all of
> > which are totally independent of the "diagonal" construction, it would
> > not affect the validity of the theorem itself even if the "diagonal"
> > proof were to be found flawed.
> >
> > For which reason, no sensible mathematician is the least worried that
> > such a flaw would in any way weaken the validity of the theorem itself.
>
> Well, first, the "diagonal proof" has spread to many different fields. And
> if it is flawed, it would be good to understand how it's flawed.
Name some of those "other fields".
>
> Second, I suspect the flaw which I claim is in the diagonal proof is in all
> the other proofs you speak of.
The "flaw" you suggest (and which others have suggested before you)
depends on using decimal (or other basal) representation of real
numbers. Cantor's first proof of the theorem does not, but depends on
the least-upper-bound/greatest-lower-bound property of the reals.
> I do not know all the other proofs and it
> will take me some time to learn their language, and come up with counter
> examples as I have done here to show their weakness.
You presume, out of your self-confessed ignorance, that they all have
weaknesses.
> And to be honest, I
> don't have enough interest in math to spend the time it would take to clean
> up the mess that I believe has been created in the past 100 or so years.
That mess is more in your imagination than in actuality. How is it that
none of those who have taken the time to "learn the language" have ever
found what is so obvious to one who has never taken that time?
This strikes me as another case of "fools rush in...".
The "decimal" representation of real numbers correlates real numbers
with infinite sequences of digits with one decimal point following a
finite number of digits. There are countably many reals (the rationals)
which have dual representation, but that need no be a consideration here.
One posits a mapping from the natural numbers, N = {1,2,3,...} to the
reals, R, say f:N -> R: n -> f(n).
If one can show that any such mapping must fail to cover all of R, then
one has shown that the cardinality of R is greater than that of N
(meaning, by Cantor's definition, that there is an injective function
from N to R but no surjective function from N to R, or equivalently, no
injective function from R to N).
We specify a numeral, say x, as follows:
(1) the decimal point precedes all non-zero digits.
(2) If the nth digit of f(n) is 2 then the nth digit of x is 3,
otherwise the nth digit of x is 2.
The numeral, x, contains no zeros or nines following the decimal point,
so has a unique representation, and differs from any decimal with unique
representation in at least one decimal place.
This numeral, x, specifies a real number not in the image of f.
If you, or anyone, chooses to claim that they have a mapping from N ONTO
R, you must show that the above rule fails to produce a real number.
Note that rejection of being able to define x means also that there
cannot be any functions from any infinite set to any infinite set in the
first place, so that the majority of analysis, including all of
calculus, would be down the tubes.
Jesse F. Hughes
> I do not know all the other proofs and it will take me some time to
> learn their language, and come up with counter examples as I have
> done here to show their weakness. And to be honest, I don't have
> enough interest in math to spend the time it would take to clean up
> the mess that I believe has been created in the past 100 or so
> years.
The world sighs at the missed opportunity.
--
"That's all the legacy I ever wanted, to have people remember me like
a shooting star streaking across their Life sky, illuminating, for
just one moment, unparalleled beauty unique to itself."
-- Weblogs are a particularly humble medium, unique to themselves.
Virgil
cu...@kcwc.com (Curt Welch) wrote:
> Manuel Petit <fre...@freston.org> wrote:
> > Again, all your problems start with your table and your (wrong)
> > assumption that your mapping is one to one and hence you built a table
> > of reals. Ex falso sequitur quodlibet.
>
> That's a valid stance (though I belive not true).
>
> However, I also used the same diagonal argument to show that a table of
> integers didn't contain all the integers. Where is the flaw in that
> argument?
>
> If the same logic used in Cantor's proof can be used to show that a table
> of integers does not contain all the integers, then most there not be a
> problem somewhere? If so, where is it?
It is in your not seeing what is important, "a" table isn't enough, it
must be "every" table to be enough.
The Cantor diagonal proof (as does his prior proof) shows that EVERY
table of reals (image of the naturals for some function) is missing
some reals.
To show that ONE table of integers is missing some integers does not
show that EVERY table of integers must be missing some integers, and it
is the "EVERY table" which is essential to the issue.
Virgil
cu...@kcwc.com (Curt Welch) wrote:
Does EVERY table of integers omit some integers?
What about the table f(n) = n?
Jesse F. Hughes
> The issue at the heart of it is that we define a lot of things in
> math with algorithms. And if you specify an algorithm that runs
> forever, and never halts, then it's not valid to talk about what
> happens after it halts. It's the same mistake in logic one would
> make if they claimed that time and space was infinite in this
> universe, and then talked about what happens after a spaceship sent
> out from earth, reached the edge of the universe.
Yeah, what a logic error that would be. Clearly logic dictates that
infinite things have no edges.
--
Jesse F. Hughes
"To [mathematicians] amateur mathematicians are worse than scum, and
scarier than nuclear bombs."
-- James S. Harris on mathematicians' phobias
Jesse F. Hughes
> The reason I've looked into this is because various arguments such
> as Gödel's incompleteness theorem spills over into arguments about
> building intelligent machines. And some would claim there is an
> important relationship there that seems to indicate there are limits
> to what machines can do - some even take it so far to believe that
> it "proves" you can not build a conscious machine.
People abuse Geodel's theorems and so in retaliation you choose to
revamp mathematics to excise them dangerous diagonal arguments?
Huh.
--
Jesse F. Hughes
Quincy (age 3 1/2, looking at a picture): Are these people Canadians?
Me: Uh, no, they're Australian Aborigines.
Quincy: Do they fight Canadians?
Virgil
cu...@kcwc.com (Curt Welch) wrote:
> Rasmus Villemoes <burner...@imf.au.dk> wrote:
> > The same (or at least nearly the same) mapping was used in an arX
> > article some time ago. A few weeks later, the author posted another
> > article with a corollary to the above 'theorem' (that the mapping
> > is bijective). The corollary stated "Not every integer is finite".
>
> I don't understand the term "bijective", but I can still quickly conclude
> that the mapping led to the conclusion that "Not every integer is finite".
> And since we all "know" that every integer is finite, there must be a
> problem somewhere. And the assumption, is that the problem is with the
> mapping.
>
> I would counter that argument by saying that no integers are finite. They
> all have an infinite string of 0's in front of them which, purely by
> convention and practice, we choose not to write.
There is an essentail distinction between a number and the numeral by
which it is represented. An infinite numeral can represent a finite
number, e.g., 0.333... represents 1/3.
David Kastrup
> pano...@iquest.net (Daniel W. Johnson) wrote:
>> Curt Welch <cu...@kcwc.com> wrote:
>>
>> > But how can you show me (ok, a student of your's who is not a crank),
>> > where the flaw is in the logic I used to show that the table of
>> > integers does not contain all the integers?
>>
>> It's at the point where you called "....1111111" an integer.
>
> Yeah, I see that now. I need to learn more about the formal
> definitions mathematicians use to define reals and integers.
Basically, an integer is always a successor of another integer, and
you can repeat that process until you reach 0. How would you ever be
able to get rid of the ultimate first 1 in "...1111111"?
In contrast, the last 1 in "0.1111111..." does not make a difference
in the reals.
> So though I can sense the intuitive desire to see ..111 as not being
> an integer, I don't yet understand how this is formally defined in
> mathematics without putting a contradiction into the system. So I
> have to go learn all that work of formal logic and axiomatic stuff
> which I know very little about before I can even discuss this
> further.
You might be interested in p-adics as well. Those have repeating
decimals to the left of the decimal point (actually, nothing to the
right of them) and are written in some prime base p. p-adics form a
ring where every number with a nonzero last digit has an inverse. For
example, in 5-adics, 1/3 is equal to ...131313132. Multiply by 3, and
it carries all the way to the left to evaluate to 1: with p-adics, the
"leftmost" digit is as unimportant as the rightmost digit of real
numbers.
But they obey different laws than ordinary integers. In particular,
their existence is dependent on the number base, which is neither the
case for integers nor for reals.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
Virgil
cu...@kcwc.com (Curt Welch) wrote:
The "contradiction" is that you cannot prove that *every* table of
integers is missing some integers whereas Cantor has proved that *every*
table of reals is missing some reals (in fact missing more than are
actually tabulated).
Curt Welch
> If all you're interested in is the physical world, then you don't need
> infinite sets, let alone the real numbers.
My interest is not in the physical world alone, it's in the relationship
between the physical and mental worlds. To solve AI, we have to completely
understand that relationship.
At the same time, even if the number of atoms or the mass of the universe
is fixed, time is still infinite - almost by defintion. And a machine
which produces an infinite set of behavior is constructing an infinite set.
The notion of an infinite set is not completely disconnected from the
physcial universe.
Lee Rudolph
>cu...@kcwc.com (Curt Welch) writes:
>
>> The issue at the heart of it is that we define a lot of things in
>> math with algorithms. And if you specify an algorithm that runs
>> forever, and never halts, then it's not valid to talk about what
>> happens after it halts. It's the same mistake in logic one would
>> make if they claimed that time and space was infinite in this
>> universe, and then talked about what happens after a spaceship sent
>> out from earth, reached the edge of the universe.
>
>Yeah, what a logic error that would be. Clearly logic dictates that
>infinite things have no edges.
Good thing Occam's Razor is Ultrafinite(TM)!
Lee Rudolph
Virgil
cu...@kcwc.com (Curt Welch) wrote:
> Torkel Franzen <tor...@sm.luth.se> wrote:
> > cu...@kcwc.com (Curt Welch) writes:
> >
> > > Why is it ok to write 0.111... but not ...11111 ?
> >
> > You can indeed write "...11111", and also "fnoggle fnoggle" and
> > "froufy forbandle ignotoot!".
>
> Ok, that's valid. We do not have a common foundation to argue from.
>
> But lets just ignore these side issues and look at the
> strongest part of my argument where I use Cantor's logic to prove that a
> table of integers does not contain all the integers. How can you show my
> argument is invalid and at the same time, keep Cantor's logic about the
> relationship between the integers and reals as valid?
The "contradiction" is that you cannot prove that *every* table of
Virgil
cu...@kcwc.com (Curt Welch) wrote:
The "contradiction" is that you cannot prove that *every* table of
Virgil
cu...@kcwc.com (Curt Welch) wrote:
> Torkel Franzen <tor...@sm.luth.se> wrote:
> > cu...@kcwc.com (Curt Welch) writes:
> >
> > > How can you show my
> > > argument is invalid and at the same time, keep Cantor's logic about the
> > > relationship between the integers and reals as valid?
> >
> > If you're a genuine crank, it is of course quite impossible to show
> > you anything. If you're not, you'll find out about these things for
> > yourself.
>
> So far, everyone has taken issue with my defintion of "reals" or with the
> idea that my mapping from reals to integers is not complete. And that's
> fine. It is not easy for me to argue my position there because I don't
> have the correct foundation.
>
> But how can you show me (ok, a student of your's who is not a crank), where
> the flaw is in the logic I used to show that the table of integers does not
> contain all the integers?
The "contradiction" is that you cannot prove that *every* table of
integers is missing some integers whereas Cantor has proved that *every*
table of reals is missing some reals (in fact missing more than are
actually tabulated).
>
Virgil
cu...@kcwc.com (Curt Welch) wrote:
There is no such thing in the physical world as even a natural number.
"One", "two", "three", etc. are all entirely conceptual, not physical.
If you insist on physicality, give up mathematics.
Justin Young
>mathematicians use to define reals and integers.
Let's just stop right there. I would like to recommend a book to you,
Axiomatic Set Theory by Patrick Suppes. You can buy it on Amazon for about
$10. If you work all the way through that book, you will have a good idea
what
such things as the integers, real numbers, etc, actually ARE in a formal
sense. It also has a good section on infinite sets. After you master said
material, you should no longer have an issue with Cantor's proof.
fishfry
cu...@kcwc.com (Curt Welch) wrote:
> fishfry <BLOCKSPA...@your-mailbox.com> wrote:
>
> > If all you're interested in is the physical world, then you don't need
> > infinite sets, let alone the real numbers.
>
> My interest is not in the physical world alone, it's in the relationship
> between the physical and mental worlds. To solve AI, we have to completely
> understand that relationship.
>
I don't know what you mean by "solve AI." Claude Shannon, the creator of
information theory, has an interesting argument that machine
intelligence is possible. He says, "We are machines, and we are
intelligent." It's hard to refute that. A human being is a collection of
a finite number of atoms, and we have self-awareness, consciousness, and
intelligence. Infinity is not required.
Curt Welch
> There is an essentail distinction between a number and the numeral by
> which it is represented. An infinite numeral can represent a finite
> number, e.g., 0.333... represents 1/3.
This is where I think all the intersting issues lie. In the use langauge.
In the fact that words referece other things. You use the characters in
this message to write 1/3 or you speak the word "one-third", or you write
0.3333.. and all these different signs refer in our minds to the same idea.
But that idea is in itself (I believe) nothing more than another word,
which refers to yet other ideas.
And I don't understand the full mess of reference math has created to
define itself. I need to understand it better.
Dave Seaman
> Torkel Franzen <tor...@sm.luth.se> writes:
>>cu...@kcwc.com (Curt Welch) writes:
>>
>>> Why is it ok to write 0.111... but not ...11111 ?
>>
>> You can indeed write "...11111", and also "fnoggle fnoggle" and
>>"froufy forbandle ignotoot!".
> Interstingly enough, however, (and in concord with Yosenin-Volpin's
> hyperfinistic doctrine), you *cannot* write "froufy forbandle ignotoot!"
> *arbitrarily often*.
When you say "often", do you mean "often" frequently, or do you mean "often"
only once?
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
Virgil
cu...@kcwc.com (Curt Welch) wrote:
If you think that mathematics should be any more physical than ideas,
you have missed the boat.
Dave Seaman
> José_Carlos_Santos <jcsa...@fc.up.pt> wrote:
>> So, when you state that there
>> is a bijection between the set of all natural numbers and *your* set of
>> real numbers, there is in fact no contradiction between you and Cantor,
>> since you are talking about different things.
> Ok, I see your position. I'm making assumptions that you do not accept.
> I'd need to convence you that my assumptions follow from some of your basic
> beliefs. And I don't have the power to do that right now because I don't
> understand all your basic beliefs and don't know your full langauge.
> But, still, I need someome to show me the error of my logic in my proof
> that the table of integers does not contain all the integers. That should
> not require us to build a common foundation about reals and integers to
> argue from. It only requires that we have a common foundation about
> integers, and the logic used in Cantor's diagonal proof. We can leave the
> definition of reals out of the argument.
It's been pointed out already that the flaw in your "proof" is at the
point where you claim ...11111 is an "integer."
A "table", in the context of this proof, is a mapping defined on the
natural numbers. The identity map, given by f(n) = n, obviously covers
all of the naturals, meaning the range of f contains all the naturals.
What Cantor proved is that for any f: N -> R, there exists x in R such
that x is not in the range of f.
Dave Seaman
> Jon Haugsand <jon...@ifi.uio.no> wrote:
>> * Curt Welch
>> > Nath is not something I specialize in (and I don't read this group
>> > normally),
>>
>> Obviously.
>>
>> > but I've been looking at a few things lately and I've decided
>> > that some very big mistakes have been made in math because people
>> > started playing around the concept of infinity without realizing the
>> > trouble they were creating for themselves.
>>
>> Don't you find just a tiny piece of pathetic arrogance here?
> No, not at all. It's a huge heaping hunk of pathetic arrogance.
> Extraordinary claims require extraordinary proof. So I offered one. I
> used Cantor's logic to prove that a table of integers does not contain all
> the integers. I'm still waiting for someone to explain the obvious
> contradiction.
How many times are you going to repeat the same question without
bothering to read or respond to the many answers you have already
received?
One more time: the flaw in your argument is that ...11111 is not an
integer.
Now, stop claiming that no one has explained it to you. Perhaps you have
a different question, but that particular one has been answered many
times over.
robert j. kolker
Curt Welch wrote:
>
> Lately however, I've come to see things very differently. I now belief the
> proof is totally bogus. And the huge body of work built on top of the
> concept is likewise, totally bogus.
Forget the mysteries of A.I. Show which step in the diagnal proof is wrong.
If a list of decimal expansions cannot be put into 1-1 correspondence
with the integers, neither can the reals in the interval [0,1] which is
the point of the proof. The proof is acheived by showing that a
variation os the n-th digit of the n-th item of the list produces a
decimal expansion equal to a number in [0,1] but which is not on the
list which is a contradiction since we assume the list exhausted all
reals in the interval [0,1]. Why can't this diagonally varied sequence
of digits be in the list? Because it differs from the n-th item on the
list in the n-th place. If it were in the list it would be the K-th item
for some K, but its K-th digit would be unequal to its K-th digit which
is a contradiction.
Show why and where that is wrong.
Bob Kolker
Richard Tobin
Curt Welch <cu...@kcwc.com> wrote:
>Why is it ok to write 0.111... but not ...11111 ?
Because 0.111... is defined to be the limit of 0.1, 0.11, 0.111, ...
and ...11111 is not defined to be anything at all.
If you want to choose a definition for ...11111 that's ok, but make
sure you tell us all what it is before expecting us to discuss it.
-- Richard
Daniel W. Johnson
> However, until you reach an infinite number of digits, neither represents a
> single point.
What about 1/9? Is that a single point? And can you come up with a
decimal representation other than 0.1111... for it?
--
Daniel W. Johnson
pano...@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
Todd Trimble
>Let me demonstrate.
>
>I claim that there is only one type of infinity. That there are just as
>many integers as there real numbers. (or more accurately, that the concept
>of the size of an infinite set is a contradiction in itself).
>
So you claim that all infinite sets have the same cardinality?
Given a bijection f: X --> P(X) between X and its power set P(X),
what do you say about A = {x in X: x not in f(x)}? Since f is a
bijection, there exists y in X such that f(y) = A. Does y belong
to A? If so, then y is not in f(y) [cf. defn. of A], i.e., y is
not in A. Does y not belong to A? If that's the case, then it
is false that y is not in f(y); therefore, it's true that y is
in f(y), and thus y is in A. Either way we reach a contradiction;
therefore there is no bijection between X and P(X).
This is a version of the diagonal argument. Please point out
why it's "wrong".
Todd Trimble
robert j. kolker
Todd Trimble wrote:
>
> This is a version of the diagonal argument. Please point out
> why it's "wrong".
Curt does not understand how proofs work. You are pissing up a rope.
Bob Kolker
Victor Eijkhout
> My interest is not in the physical world alone, it's in the relationship
> between the physical and mental worlds. To solve AI, we have to completely
> understand that relationship.
Completely? I suggest you change fields.
V.
--
email: lastname at cs utk edu
homepage: www cs utk edu tilde lastname
Curt Welch
>
> > However, until you reach an infinite number of digits, neither
> > represents a single point.
>
> What about 1/9? Is that a single point? And can you come up with a
> decimal representation other than 0.1111... for it?
Well, this is where it gets interesting. If is clearly a single point if
you simply define it to be so. And that is exactly what seems to be common
practice. It's how I both was taught to think about the idea, and how I
have always talked about it in the past. All the irrationals as well as
the rationals exist as a precise, and single point on the real number line.
But now I'm starting to wonder if there may be value in looking at this
differently. Might it be valid to think of division of 1 by 9 as an
infinite processes (algorithm) that produces closer and closer
approximations, yet is always unable to produce the actual "point" on the
line?
When we write "1/9", are we making a reference to a single point on the
line, or are we making reference to the algorithm which produces closer and
closer approximations to the correct name for that point? Do all the
points really "exist" if you can not name them?
What do you gain or loose by talking like this? That is what I'm
interested in exploring.
The reason I started to explore these ideas is because I think the mind is
finite. It can only name a finite number of things. It can only
recognize, and respond, in a finite number of ways. It only has a finite
number of atoms, and neurons, to work with. It can not, do anything
"infinite".
To talk as if an infinite set can exist (the set of all points on the real
number line between 0 and 1 for example), can be related to believing the
mind can create an infinite set of ideas. But if the mind can not do it,
then the infinite set of ideas can not exist for us. Meaning, we can use
language to define the notion of an in infinite set easy enough, but the
idea we define, can't actually exist in this universe because it would
require a matching infinite mind. Ideas are physical, and have finite
physical limits.
Language gives us the power to talk about an imaginary universe where minds
exist that can manipulate infinite sets. But what is the advantage to
exploring the properties of that imaginary universe if that universe isn't
the one we exist in and doesn't have anything in common with the universe
we exist in?
The only thing that does seem infinite in our universe is space and time.
And maybe only time is really infinite. So we can define a processes,
which runs forever - like counting. But to assume it will ever finish
takes us out of our universe of existence and into a universe that does not
exist (at least for us).
I don't know if any of the above logic is valid, or even useful, though my
gut feeling is telling me it's very useful to learn to talk and think in
these ways. I'm only throwing it out as food for thought, and trying to
understand its implications.
Curt Welch
> To show that ONE table of integers is missing some integers does not
> show that EVERY table of integers must be missing some integers, and it
> is the "EVERY table" which is essential to the issue.
What I thought I was showing, is that by using the diagonal argument, I
could "prove" that a table which included every natural number, was
actually missing some natural numbers. If so, that creates a contradiction
where there shouldn't be a contradiction. But, my defintion of natural
numbers as it turns out didn't match what others use to define natural
numbers, so the contradiction which I saw wasn't a contradiction for them,
and it didn't show anything useful or new for them.
So now I need to further study how natual numbers and real numbers are
defined and to futher understand why people choose to define them like they
do.
Dave Seaman
> So now I need to further study how natual numbers and real numbers are
> defined and to futher understand why people choose to define them like they
> do.
<http://db.uwaterloo.ca/~alopez-o/math-faq/node5.html>
(Click on "What are numbers?")
Curt Welch
Yeah, that's the material that I know exists but that I also know I don't
know enough about to discuss this with you guys. Thanks for the
recommended reading.
What I have seen with a few seconds of looking at web pages on those
subjects is that the basic axioms used to define math from always seem to
include the belief that infinite sets are allowed to exist.
My thought is that once you do that, you have created problems in the world
of math that do not exist in this universe. My thought is that you should
be able to use a set of axioms that does not include infinite sets, yet
still, do all the math we do today.
That is, we define the natural numbers not as an infinite set, but as a
counting algorithm which can never complete. We define the existence of 0,
and the +1 function which when executed, creates the next natural number,
and also state that infinite recursion never completeness and is undefined.
We define a rule that says the creation of new objects always takes time.
So applying the +1 function an infinite number of times would never
complete.
But before I can map these "computational" based ideas into the world of
Axiomatic Set Theory, I have to learn the language of Axiomatic Set theory.
:)
I don't expect to have any problem understanding the validity of Cantor's
proof once I learn the language. I'm not trying to understand Cantor's
proof. I'm trying to understand how to define a world where Cantor's proof
is invalid - and if possible, what the value of such a world would be.
Well, actually, I know how to define the world. I've already done it in my
mind. What I don't understand is how useful (or useless) that world is yet
because I've not done it in a formal framework.
I believe a lot of my ideas parallels work done with computability and
Turing machines. But that's more that I don't know enough about yet.
Curt Welch
> There is no such thing in the physical world as even a natural number.
> "One", "two", "three", etc. are all entirely conceptual, not physical.
> If you insist on physicality, give up mathematics.
I am exploring things that you believe do not exist. And your outlook is
not uncommon in the world. It's by far most common view all of mankind
seems to like to share.
It's that very fact that makes me at times, believe I've found something
that has been missed for 100's of years. Matematics, by design, limits
it's focus to a scope which does not include the things I'm investigating.
And it's that limited scope which inherently causes people to not be able
to see the bigger picture.
You do not believe the "conceptual" world and the "physical" world are one
in the same. I do. And once you believe that, everything starts to get
very interesting, and everything starts to look very different.
Could everything I believe be wrong? Of course. Time will tell if this
belief turns out to be useful or just a silly waste of time.
ste...@nomail.com
: I don't expect to have any problem understanding the validity of Cantor's
: proof once I learn the language. I'm not trying to understand Cantor's
: proof. I'm trying to understand how to define a world where Cantor's proof
: is invalid - and if possible, what the value of such a world would be.
: Well, actually, I know how to define the world. I've already done it in my
: mind. What I don't understand is how useful (or useless) that world is yet
: because I've not done it in a formal framework.
: I believe a lot of my ideas parallels work done with computability and
: Turing machines. But that's more that I don't know enough about yet.
The proof of the unsolvability of the Halting Problem has a very
similar form to Cantor's proof. Computability theory has no
problem with infinite sets. Indeed it does not get interesting
until you start considering infinite sets.
Stephen
Curt Welch
> I don't know what you mean by "solve AI."
I use it in the general sense of being able to build a machine with all the
powers of cognition and behavior that humans have. I like to call that
"strong AI" but that term actually means something different already in the
AI argument.
> Claude Shannon, the creator of
> information theory, has an interesting argument that machine
> intelligence is possible. He says, "We are machines,
That's what you can not prove. It is something many of us accept on faith
where as other reject it on faith and that's where we stand. Neither side
can put forth a compelling argument which the other side will accept.
The ultimate proof which I will present is a machine which duplicates all
powers of human behavior and cognition. However, even that will not
quickly convince the other side. They will call the machine a zombie which
acts like a human, but is not conscious. Over time, that stance will
vanish from the culture.
> and we are
> intelligent." It's hard to refute that.
Actually it's easy to refute and hard to defend. If I say, "we are not
machines, we are conscious and machines are not, and never can be", how can
you prove me wrong? All you can really do is say "no, that's not true".
That AI debate is 50+ years old and no one has settled it.
> A human being is a collection of
> a finite number of atoms, and we have self-awareness, consciousness, and
> intelligence. Infinity is not required.
Yeah, that is how I think. And I take it a step further and say infinity
is not even possible. But this is clearly not how everyone looks at this.
Curt Welch
> How many times are you going to repeat the same question without
> bothering to read or respond to the many answers you have already
> received?
I've been reading and responding non stop for many hours today. I'm sorry,
but I can't go any faster. And I'm sure the group thinks I've posted way
to much as it is. The thead will die soon, don't worry.
Curt Welch
> How many times are you going to repeat the same question without
> bothering to read or respond to the many answers you have already
> received?
Your funny. I had already posted 3 replies addmiting that I understood the
answers by the time you wrote this. :)
Do not get confused by the fact I post a lot and many times don't read all
the replies ahead of time. It's midnight my time and I'm just now
responding to the article you wrote 6 hours ago. There are 8 articles in
the thread that come after this that I have not yet been able to read
including two of your's. I'm doing my best to get through them all. I've
been reading and posting for about 8 hours today already. Give me time. :)
Curt Welch
Yeah, I thought it was just as "obvious" that it was defined as much as the
number "123" was defined. But I see these thigns are defined differently
than I thought. So I'm going to study more about just how these things
have been defined.
The reason I thought it was obvious is that the number 123 to me is defined
as 123 preceeded with an infinite number of leading zeros. If you are
allowed to define 123 that way, then why would it be wrong to define a new
number which is the same as 123, except where you change all the zeros to
ones? It seemed obvious and valid to me. But apparently, my "obvious"
logic does not mesh with the way it is defined in the greater community.
So I have some work to do to understand both how it is defined and what the
value is of defining it like that is.
The obvious difference (as has been pointed out to me) is that 123 with
leading zeros can be thought of as a series which converges on a single
number. 123 with leading ones is a series which does not converge on a
single number. But why is "convergence" a required part of what a natural
number is? That is what I need to further understand because it conflicts
with what I was thinking.
Acid Pooh
> fishfry <BLOCKSPA...@your-mailbox.com> wrote:
>
> > If all you're interested in is the physical world, then you don't need
> > infinite sets, let alone the real numbers.
>
> between the physical and mental worlds. To solve AI, we have to completely
> understand that relationship.
If those are necessary conditions for solving AI, you're screwed.
'cid 'ooh
Curt Welch
He's just trying to understand what I do understand. I think that's a
legitimate pursuit in any conversation. :)
Chairman of the David Hilbert Appreciation Society
> Chairman of the David Hilbert Appreciation Society
> <mathgee...@hotmail.com> wrote:
[...]
> The fact that it's reasonable in math, to talk about the size of one
> infinite set being larger than the size of another infinite set, doesn't
> seem to tie correctly to the physical world.
I think the current attitude among most mathematicians is that
mathematics doesn't need to have anything to do with the physical
world.
[...]
> But, is there a point, where you use language to create a story, which can
> never be true in this universe? I think there is, and I think the world of
> math has done just that.
There is a distinction between provability and truth. I think that
it is philosophy (not strictly mathematics) that deals with notions
of truth. Someone may want to correct me on that.
> They have used language to create an imaginary
> world, and at some point, they cross the line from possible, to impossible,
> in the physical universe.
It seems that you think mathematics is a fantasy game which has
nothing to do with the physical universe and therefore is broken
somehow. For mathematics to be not broken, you think, it must retain
a connection with the physical world, or something like that.
The following quote from Shapiro in _Thinking About Mathematics_,
responds to the sentiment above more eloquently than anything
I could think of offhand.
"...one might argue that if mathematics gave serious pursuit only
to those branches known to have applications in natural science,
we would not have much of the mathematics we have today, nor
would be have all of the /science/ we have today the history
of science is full of cases where branches of 'pure' mathematics
eventually found application in science. In other words, the
overall goals of scientific enterprise have been well served by
mathematicians pursuing their own disciplines with their own
methodology"
So, if you're only interested in the physical world or natural
science you've been well served by mathematicians doing
what they do, however wrong or incomprehensible it may
seem to you.
[...]
-
Replace Roman numerals with digits to reply by email
Curt Welch
> On 15 Nov 2004 03:47:37 GMT, Curt Welch wrote:
>
> > So now I need to further study how natual numbers and real numbers are
> > defined and to futher understand why people choose to define them like
> > they do.
>
> <http://db.uwaterloo.ca/~alopez-o/math-faq/node5.html>
> (Click on "What are numbers?")
Thanks, but that's still too advanced for me. It's using what seems to be
some very basic set theory nomenclature which I am forced to guess at it's
meaning.
Like I've said many times now, I have my work cut out for me. :)
Curt Welch
:)
That's exactly why AI is not an easy problem. It is the solution to how
the two worlds connect. If you can not explain the connection, you can not
solve AI. It's unproven if AI is a problem that we (humans) have the power
to solve. I do happen to belive we have the power to solve it, (or else it
would be kinda stupid for me to spend time on it), but I also know it is
just a leap of faith on my part to believe that.
Curt Welch
<mathgee...@hotmail.com> wrote:
> Curt Welch wrote:
> > Chairman of the David Hilbert Appreciation Society
> > <mathgee...@hotmail.com> wrote:
>
> [...]
>
> > The fact that it's reasonable in math, to talk about the size of one
> > infinite set being larger than the size of another infinite set,
> > doesn't seem to tie correctly to the physical world.
>
> I think the current attitude among most mathematicians is that
> mathematics doesn't need to have anything to do with the physical
> world.
Yes, that is exactly how I think most mathematicians think. And I believe
that has allowed them to create a disconnect between the world of
mathematical concepts and the physical world. I suspect most don't see why
this is even an issue in my mind.
Yes, I agree completely with that point of view. It would be totally
foolish to discontinue the pursuit of pure mathematics just because we had
not yet found a scientific use for it. Scientific or practical use of the
mathematical tools developed have for the most part, only come after they
were developed in the study of pure mathematics.
However, I believe that the world of ideas, and concepts, and math, are not
disconnected from the physical world. I believe they are one and the same.
I believe we simply just don't fully understand the connection yet. To
understand that connection requires we understand what the brain does, and
how it works.
At some point in time, I believe we will close that gap by gaining the
required understanding. When that happens, it should tells us some very
interesting things about the limits of our reality and our ability to
think, and create ideas, and use ideas, and our ability to understand the
universe we exist in.
Once we understand exactly how the physical worlds and the mental worlds
are connected, then that will act as a bridge between the two worlds that
currently does not exist. It will allow knowledge to flow from our base of
understanding about the physical universe, into the knowledge about our
mental universe, and vice versa.
Currently, the world of mathematics seemes to be built the best foundation
that could be defined, yet it was probably a somewhat arbitrary foundation
that seemed fitting for the job (there was nothing else to build it from).
But what if, once we understand the connection between thought and the
physical world, we find a very different looking foundation? What if we
built a new type of mathematics defined from the realities of the physical
foundation that the brain is actually built from? Would it be any
different from what we already have created in mathematics? I suspect it
would be, and using the ideas I already have about how the physical worlds
and mental worlds connect, I'm poking around with this idea to see if the
foundation of modern mathematics seems to mesh with the physical reality I
believe that thought is built on.
What I'm seeing, is that there seems to be a disconnect - especially in
this use of ideas of infinity. But clearly, it has been shown I don't
understand the foundations of modern mathematics. So until I do, I can't
figure out if there is anything useful to be done here or not.
It's also very likely that by the time I understand, and try to "adjust"
the foundation, that it will make no change whatsoever in any fields of
mathematics. I will simply have used different words to define the same
functionally equivalent foundation.
Virgil
cu...@kcwc.com (Curt Welch) wrote:
The natural numbers are usually the starting point of arithmetic, and
the standard basis for them is usually taken as something like the Peano
axioms, which make no reference to the numerals by which they are
represented.
See, for example, http://en.wikipedia.org/wiki/Peano_axioms
The numeral representation for the naturals comes later, and starts with
0 or 1, depending on whose version of the Peano axioms you use, and as
the natural numbers themselves increase builds up gradually the number
of digits used, but never to more that a finite number of digits for a
finite natural number.
Virgil
cu...@kcwc.com (Curt Welch) wrote:
> Chairman of the David Hilbert Appreciation Society
> <mathgee...@hotmail.com> wrote:
> > Curt Welch wrote:
> > > Chairman of the David Hilbert Appreciation Society
> > > <mathgee...@hotmail.com> wrote:
> >
> > [...]
> >
> > > The fact that it's reasonable in math, to talk about the size of one
> > > infinite set being larger than the size of another infinite set,
> > > doesn't seem to tie correctly to the physical world.
> >
> > I think the current attitude among most mathematicians is that
> > mathematics doesn't need to have anything to do with the physical
> > world.
>
> Yes, that is exactly how I think most mathematicians think. And I believe
> that has allowed them to create a disconnect between the world of
> mathematical concepts and the physical world. I suspect most don't see why
> this is even an issue in my mind.
The point is that any connection between mathematics and the physical
world is dependent on the assumptions one makes about the nature of the
physical world, and those assumptions are, of necessity, outside of
mathematics.
So that before a mathematician will agree to your application of
mathematics to the physical world, he will require you to state clearly
and unambiguously what you are assuming about the physical World, at
least insofar as is relevant to your application of mathematics to that
world.
This is remarkably difficult to do, even for scientists who must do it
on a regular basis.
Jesse F. Hughes
> Extraordinary claims require extraordinary proof. So I offered
> one.
I agree. It was an extraordinary proof.
--
"I don't know why I live in a world with so many supposed
mathematicians who are all so dumb AND rude. Why oh why couldn't
someone like Gauss or Dedekind still be around? Shoot, I'd even take
someone like Hardy at this point." -- James S Harris compromises
David C. Ullrich
>José_Carlos_Santos <jcsa...@fc.up.pt> wrote:
>
>> Could you please tell me then which integer corresponds to the real
>> number 1/9 (or 0.11111111111111111111... if you prefer)?
>
> ....1111111
>
>Why is it ok to write 0.111... but not ...11111 ?
It's ok to _write_ ...11111 . But ...11111 is not an _integer_.
>My point is that 1/9 is in fact an algorithm for generating a real value.
No, your point _was_ that that list gives a mapping that maps the
integers onto the reals.
Yesterday you speculated that people would call you crazy, and I
pointed out that that depends on how you reply to explanations
that you're wrong. The above is a very simple explanation of
why you're simply wrong here, and you're not saying "oops".
Instead below you attempt to redefine what the word "integer"
mneans.
So: You're crazy. There, happy now?
>It is not the real value itself. It's just a name we use to talk about the
>real value which is 0.1111 repeating forever. And I can just as easily
>define the integer of 1 repeating forver. The only reason we do not do
>that is a matter of convention. It's not (so I claim) in violation of what
>integers are.
>
>If you start with 0, and continue to apply the +1 function to it, and
>ignore all the values you come up with which does not have all 1's, you
>find you have the exact same type of defintion that gives you 1/9 when you
>generate a string of one's running to the right, instead of running to the
>left.
>
>Also, all integers have an implied infinite string of 0's running to left
>(just like reals have am implied infinite string of 0's running to the
>left and right). So when you write 123, you are really writing
>....00000123.
>
>Just change the implied 0, to an implied 1, and there you have it. The
>integer pair for the real 1/9.
************************
David C. Ullrich
Anonymous
integers plus all numbers that have an infinite progression of digits to the
left of the decimal. This set, though, is larger than the set of integers
since there is a 1 to 1 correspondence with the reals ... this set is not
used much in standard mathematics either.
When dealing with mathematics you have to be careful ... you see, you wrote
'why can't I just write 111111111...' ... and you can, but you no longer
have an integer, you have a member of my set of infinite integers.
"Curt Welch" <cu...@kcwc.com> wrote in message
news:20041114122706.926$P...@newsreader.com...
> José_Carlos_Santos <jcsa...@fc.up.pt> wrote:
>
>> Could you please tell me then which integer corresponds to the real
>> number 1/9 (or 0.11111111111111111111... if you prefer)?
>
> ....1111111
>
> Why is it ok to write 0.111... but not ...11111 ?
>
> My point is that 1/9 is in fact an algorithm for generating a real value.
> It is not the real value itself. It's just a name we use to talk about
> the
> real value which is 0.1111 repeating forever. And I can just as easily
> define the integer of 1 repeating forver. The only reason we do not do
> that is a matter of convention. It's not (so I claim) in violation of
> what
> integers are.
>
> If you start with 0, and continue to apply the +1 function to it, and
> ignore all the values you come up with which does not have all 1's, you
> find you have the exact same type of defintion that gives you 1/9 when you
> generate a string of one's running to the right, instead of running to the
> left.
>
> Also, all integers have an implied infinite string of 0's running to left
> (just like reals have am implied infinite string of 0's running to the
> left and right). So when you write 123, you are really writing
> ....00000123.
>
> Just change the implied 0, to an implied 1, and there you have it. The
> integer pair for the real 1/9.
>
Anonymous
How do you, year after year, reply to people who do not have the basic tools
of mathematics again and again, just to hear "you're wrong, you just blindly
follow the rules of mathematics," over and over? For god's sake, you're a
professor ... you can do this in class all day - do you just have an inner
drive to 'help people see'?
"David C. Ullrich" <ull...@math.okstate.edu> wrote in message
news:eg7hp0h7sp82kdubn...@4ax.com...
David Kastrup
> On 14 Nov 2004 17:27:06 GMT, cu...@kcwc.com (Curt Welch) wrote:
>
>>José_Carlos_Santos <jcsa...@fc.up.pt> wrote:
>>
>>> Could you please tell me then which integer corresponds to the real
>>> number 1/9 (or 0.11111111111111111111... if you prefer)?
>>
>> ....1111111
>>
>>Why is it ok to write 0.111... but not ...11111 ?
>
> It's ok to _write_ ...11111 . But ...11111 is not an _integer_.
The beautiful thing is that one can _prove_ it is not an integer, by
induction. You can prove that for every integer n there exists an
integer K(n) (well, you actually can generously choose K(n)=n) so that
for all k>K(n), the kth digit counted from the right is zero. And
that means that the "ultimate" left is zero as long as we are talking
about integers defined by the Peano axioms.
For other definitions (like p-adic numbers), the bets regarding the
ultraleftist digits are off.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
robert j. kolker
Curt Welch wrote:
> But now I'm starting to wonder if there may be value in looking at this
> differently. Might it be valid to think of division of 1 by 9 as an
> infinite processes (algorithm) that produces closer and closer
> approximations, yet is always unable to produce the actual "point" on the
> line?
Nonesense. One can always divide a line segment into N equal parts by a
well known geometric construction. So getting the point on the segment
corresponding to k/N for k = 0,1,...N is trivial. Or equivalently choose
a unit length and lay out multiples of this length on an infinite ray.
Again one easily constructs points corresponding to k*N. Why do you
complicate a very straightforward matter?
Bob Kolker
robert j. kolker
Curt Welch wrote:
> Virgil <ITSnetNOTcom#vir...@COMCAST.com> wrote:
>
>>To show that ONE table of integers is missing some integers does not
>>show that EVERY table of integers must be missing some integers, and it
>>is the "EVERY table" which is essential to the issue.
>
>
> What I thought I was showing, is that by using the diagonal argument, I
> could "prove" that a table which included every natural number, was
> actually missing some natural numbers.
This is a flat out contradiction. A table is essentially a function
whose domain is the natural integers. A function whose range includes
the integers does not miss a thing (by defintion). A table which as all
the integers clear is not missing any.
Question: Are you capable of comprehending simple definitions? Are you
capable of following a straightforward mathematical proof? If the answer
to either of these questions is no, then forget about trying to
understand mathematics and refrain from making judgements about matters
in which you are demonstrably incompetent.
Bob Kolker
Richard Tobin
Curt Welch <cu...@kcwc.com> wrote:
>The obvious difference (as has been pointed out to me) is that 123 with
>leading zeros can be thought of as a series which converges on a single
>number. 123 with leading ones is a series which does not converge on a
>single number. But why is "convergence" a required part of what a natural
>number is?
You're looking at it the wrong way round. The usual way is to start
from a definition of the natural numbers in which they are 0 and its
successors. The representation in a finite number of decimal digits
is just a more convenient notation. Once you have that, you can start
considering extending the representation to an infinite number of
digits, and see what consistent interpretations you come up with.
-- Richard
robert j. kolker
Curt Welch wrote:
>
> What I have seen with a few seconds of looking at web pages on those
> subjects is that the basic axioms used to define math from always seem to
> include the belief that infinite sets are allowed to exist.
Infinite sets do not need permission. There is one infinite set we all
deal with, the set of natural integers 1,2,...... . The set can be
shown to be infinite in a number of ways. One way is to notice there is
no largest integer. If the set were finite there would be a largest.
Another to to show the infinite cardinality of the integers is to note
that the correspondence n <->2*n maps matches up the integers with a
proper subset of the integers. In fact, the existence of a mapping one
to one onto a proper subset can be taken as a definition of infinte.
So we know infinite sets exist. This is not cotroversial.
>
> My thought is that once you do that, you have created problems in the world
> of math that do not exist in this universe. My thought is that you should
> be able to use a set of axioms that does not include infinite sets, yet
> still, do all the math we do today.
There are plenty of finite structures (along with axioms). But they are
not the integers. How about the set of permuations on N elements
1,2,...N. There are N! of those. They form a group.
>
> That is, we define the natural numbers not as an infinite set, but as a
> counting algorithm which can never complete.
Peano has already done that. Gooogle Peano Axioms.
>
> I believe a lot of my ideas parallels work done with computability and
> Turing machines. But that's more that I don't know enough about yet.
Actually you don't know what you are talking about. Do not quit your day
job.
Bob Kolker
>
robert j. kolker
Curt Welch wrote:
>
> Could everything I believe be wrong? Of course. Time will tell if this
> belief turns out to be useful or just a silly waste of time.
About 90 percent of it. You spelled wrong right though. It is a silly
waste of time. Take up a useful occupation.
Bob Kolker
>
robert j. kolker
Curt Welch wrote:
>
> I use it in the general sense of being able to build a machine with all the
> powers of cognition and behavior that humans have. I like to call that
> "strong AI" but that term actually means something different already in the
> AI argument.
I will tell you how to make an intelligent machine and have lots of fun
doing it, too. Find a fertile female person, have sexual congress with
her, and make babies. There are your intelligent machines. You don't
have to be an engineer or a mathematician to do it, either.
Bob Kolker
robert j. kolker
Curt Welch wrote:
>
> Actually it's easy to refute and hard to defend. If I say, "we are not
> machines, we are conscious and machines are not, and never can be", how can
> you prove me wrong? All you can really do is say "no, that's not true".
>
> That AI debate is 50+ years old and no one has settled it.
And no one will. In point of fact you cannot prove than anyone but you
has a "mind". The only things you can observe about other people are
1. Their behaviour.
2. Their physical makeup.
3. Some of the chemical and electrical processes.
What do you notice about this list? Consciousness and mind is not on it.
You cannot establish by empricial means that anyone but you has a mind.
The rest of us can be very complicated zombies. Or maybe there is not
difference between being a complicated zombie and having a mind. Maybe
there is no such thing as a mind. After all, humans have been slicing
each other open for over 10,000 years and no one has ever found a mind.
> Yeah, that is how I think. And I take it a step further and say infinity
> is not even possible. But this is clearly not how everyone looks at this.
Is there a largest integer? What is it? Add one and be ashamed.
Bob Kolker
robert j. kolker
Chairman of the David Hilbert Appreciation Society wrote:
>>
> I think the current attitude among most mathematicians is that
> mathematics doesn't need to have anything to do with the physical
> world.
All mathematical concepts are abstractions on actual experience. The
fact that a mathematical idea needs to be expressed in a language tells
you the mathematics has some connection to the real world.
Bob Kolker
robert j. kolker
Curt Welch wrote:
>
>
> Thanks, but that's still too advanced for me. It's using what seems to be
> some very basic set theory nomenclature which I am forced to guess at it's
> meaning.
Anyone who cannot cope with mathematics is not fully human. At best he
is a tolerable subhuman who has learned to wear shoes, bathe, and not
make messes in the house. — Robert A. Heinlein
Bob Kolker
robert j. kolker
Curt Welch wrote:
>
> That's exactly why AI is not an easy problem. It is the solution to how
> the two worlds connect. If you can not explain the connection, you can not
> solve AI.
Before you figure out what Artificial Intelligence is, have you figured
out what the real thing is?
> It's unproven if AI is a problem that we (humans) have the power
> to solve. I do happen to belive we have the power to solve it, (or else it
> would be kinda stupid for me to spend time on it), but I also know it is
> just a leap of faith on my part to believe that.
Yoda says: Do not hold your breath until you find AI, Young Curtis or
purple will you turn and pass out you will.
Bob Kolker
Torkel Franzen
>
> Anyone who cannot cope with mathematics is not fully human. At best he
> is a tolerable subhuman who has learned to wear shoes, bathe, and not
> make messes in the house. Robert A. Heinlein
Heinlein said a lot of silly things.
robert j. kolker
Curt Welch wrote:
>
> However, I believe that the world of ideas, and concepts, and math, are not
> disconnected from the physical world. I believe they are one and the same.
> I believe we simply just don't fully understand the connection yet. To
> understand that connection requires we understand what the brain does, and
> how it works.
There is only one world.
>
>
> What I'm seeing, is that there seems to be a disconnect - especially in
> this use of ideas of infinity.
Infinity is just a fancy way of saying "and so on...". Infinity is just
indefinite repetition of some operation.
Bob Kolker
Luis A. Rodriguez
> Chairman of the David Hilbert Appreciation Society
> <mathgee...@hotmail.com> wrote:
> > Curt Welch wrote:
the
> physical world, we find a very different looking foundation? What if we
> built a new type of mathematics defined from the realities of the physical
> foundation that the brain is actually built from? Would it be any
> different from what we already have created in mathematics?
My answer is no.
Whether the origin of language is scientifically stablished, that
would not change the way poetry, novels,essays etc will be written.
The same for mathematics, the knowledge of brain operations will not
change the form we make mathematics, because is a cultural product
that is accepted as a game with its own laws. Think in the Theory of
Prime Numbers. Is a theory founded on an algorithm for producing a
special type of numbers and its developmnet will not be influenced by
discoveries in the natural sciences.
Luis A. Rodriguez
> number 1/9 (or 0.11111111111111111111... if you prefer)?
The array that Cantor devised for demonstrating the countability of
rationals permit us to assign an integer N to each fraction n/d.
Be S = n + d . If S is odd then: N = (S-1)(S-2)/2 + d
If S in even then: N = (S-1)(S-2)/2 + n.
For your fraction 1/9 we have: N = 9x8/2 + 1 = 37
The famous fraction 355/113 occupy the position N = 109166.
Curt Welch
> The natural numbers are usually the starting point of arithmetic, and
> the standard basis for them is usually taken as something like the Peano
> axioms, which make no reference to the numerals by which they are
> represented.
>
> See, for example, http://en.wikipedia.org/wiki/Peano_axioms
Thanks for the starting point. And that quickly takes me to all sorts of
intersting things which shows how much more I have to learn on the history
of all the work done in these related areas.
Curt Welch
> The point is that any connection between mathematics and the physical
> world is dependent on the assumptions one makes about the nature of the
> physical world, and those assumptions are, of necessity, outside of
> mathematics.
>
> So that before a mathematician will agree to your application of
> mathematics to the physical world, he will require you to state clearly
> and unambiguously what you are assuming about the physical World, at
> least insofar as is relevant to your application of mathematics to that
> world.
>
> This is remarkably difficult to do, even for scientists who must do it
> on a regular basis.
The nasty part is that it seems the brain that creates mathematics is part
of the physical world it is trying use mathematics to describe. This
creates a nasty little loop.
The ultimate goal, if in fact this is a closed loop, is to describe the
entire loop. But as you say, you must start somewhere by making some
assumptions. You can make assumptions about the physical world, and then
try to describe the physical world based on those starting assumptions.
Then, from there, you try to describe how that physical world creates the
mental world. If you are able to do that, then you describe in full, the
mental world. When you are done, the description of the mental world
should connect back, and justify, the original assumptions you made to
describe the physical world. It should form a closed, and consistent loop.
Or, you can start in the mental world and make assumptions about the nature
of "thought", and attempt to then describe the full nature of the mental
world, and how in that world, the physical world is defined, and in turn,
describe the full nature of the physical world, and then close the loop
again by explaining how the mental world arises out of that physical world
model.
It's much like a Kline bottle in that it is not a simple loop. It's a loop
that turns the nature of things inside out as you loop around it.
Understanding the nature of this mind/body problem has been a central
problem in philosophy since the beginning.
If you pick the wrong staring assumptions in either case, you can find that
by the time you try to close the loop, nothing fits together. Or worse,
that any starting assumption you pick, with the help of the loop, becomes a
self fulfilling prophecy.
Which leaves us to wonder which of the many ideas about the nature of the
mental world, or the nature of the physical world, will be the right set of
ideas to allows us to close the loop. We will not know if we have the
correct set of assumptions until we are able to duplicate the power of the
brain, and build a machine which duplicates our own mental powers. When we
do that, we will have proved that the assumptions we were using to build
the machine were sufficient to close the loop. Until we do that, we won't
know for sure if it even is a closed loop. The belief that it is a closed
loop us just one of my starting assumptions.
Brian Chandler
> Virgil <ITSnetNOTcom#vir...@COMCAST.com> wrote:
>
> > There is no such thing in the physical world as even a natural number.
> > "One", "two", "three", etc. are all entirely conceptual, not physical.
> > If you insist on physicality, give up mathematics.
>
> I am exploring things that you believe do not exist. And your outlook is
> not uncommon in the world. It's by far most common view all of mankind
> seems to like to share.
I don't think this is true at all. In my experience, if you ask a
roomful of non-mathematicians whether i, the square root of -1,
"exists" they will mostly claim it doesn't. Ask "Does 3 exist?" and
overwhelmingly they respond that yes, it does. Then you can play an
amusing little game, trying to establish where their particular
boundaries between "existing" and "not actually existing" numbers lie.
Virgil, of course, or any other mathematician, will just look a bit
blank, and say "What do you mean 'Exist'?", but I think this is a
minority view. Of course, mathematicians will hedge their response,
because in some cases, your "Exist?" will obviously apply to
mathematical existence, or not, as in four-sided triangles, even
primes greater than 63, and so on.
> It's that very fact that makes me at times, believe I've found something
> that has been missed for 100's of years. Matematics, by design, limits
> it's focus to a scope which does not include the things I'm investigating.
Ah! Two points.
a) You have misspelled "it's", which is a pretty serious blunder.
b) You've also missed quite a lot that was found 100's of years ago.
Several people answered your "proof that the integers cannot be put in
1-1 correspondence with themselves" (or something like that), but I
think the answers were not terribly good (a bit Micro$oft-like, if you
know the helicopter joke). You have a somewhat ill-defined set of
objects, including the integers, 1, 2, 3, 57, 264, etc., and some
things like ...11111; you haven't really said quite what else. Which
of these is one of your cwintegers (as I'll call them, in honour of
yourself!):
...2121212121 * recurring decimals backwards
...5356295141 * fractional part of pi backwards
-...1111111111 * cwintegers with a minus sign
While you are free to finish off the definition, so we know exactly
what is and isn't a cwinteger, then you have to start doing some grunt
work, proving results that suggest that your cwintegers are useful.
The integers used by mathematicians have at least the property that
they possess the qualities children learn about informally at a _very_
young age. In particular:
You can always add, subtract, or multiply two numbers. You can't
always divide, but if you can't, it's because there is no answer at
all, not because there are lots of answers. You can compare two
numbers, and it works: if a>b, b<a; if a<b and b<c, then a<c. Numbers
are odd and even, and even+even=even.
And lots and lots more. Here are some simple questions about
cwintegers you should ask yourself:
1 + 2 = ?
3 - 2 = ?
...999 + 2 = ?
1 - 2 = ?
Is ...2222 > 5 ?
Is ...2222 < 5 ?
Rather obviously, the subset of cwintegers (without minus signs) that
have either an infinite succession of 0s or 9s on the left, form a
representation of the integers in 2's complement (idealised to
infinite registers). So algebra 'works' within that subset, where
anything starting ...999 is negative. But if you try to include
...222, how are you going to decide if it's positive or negative? And
so on (I should really read about p-adics some time, as perhaps so
might you).
As for all this stuff about AI, I recommend Daniel Dennett's book
"Darwin's Dangerous Idea", particularly the section beginning with
Chapter 15, The Emperor's New Mind, and Other Fables. (He has a nice
start to the second paragraph: "What Goedel's Theorem promise the
romantically inclined...")
Sort of review here: http://imaginatorium.org/books/dennett.htm - of
course this is philosophy, and not to everyone's taste, but personally
I find Dennett rather readable. Good at demolishing other
philosophers, if that's anything special...
Anyway, I think a bit of reading would be a good idea, before you
announce your Great Discovery.
> You do not believe the "conceptual" world and the "physical" world are one
> in the same. I do. And once you believe that, everything starts to get
> very interesting, and everything starts to look very different.
But does it make any sense? Do you mean that numbers (and things)
Really Exist in the physical universe, for you? Including i?
Brian Chandler