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Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

Posted: Sat Jul 15, 2006 2:52 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

David R Tribble wrote:

 Quote: Hatto von Aquitanien wrote: "There is an algorithm that "generates" the members of S. That means that its output is simply a list of the members of S: s1, s2, s3, ... If necessary it runs forever." And for the sake of talking about generating .3, .33, .333,..., it is quite sufficient. David R Tribble wrote: Are you saying that this eventually generates 0.333... (1/3)? Why would you think such a thing? Is it because you can set up the pattern and then apply strong induction? Is there some way of communicating or arriving at that idea which doesn't involve some kind of generalization from an iterative process? Yes. 1/3 = sum{i=1 to oo} 3x10^-i, which is a sum of infinite terms, not an iterative process.

What do you mean by sum of infinite terms?

 Quote: Hatto von Aquitanien wrote: The only addition I need to make in order to get my original point of it producing a bijection with a subset of N is that I need to keep track of the loop count. David R Tribble wrote: And what is that loop count after you've generated 0.333... ? Hatto von Aquitanien wrote: I don't have a fast enough CPU, nor enough RAM to answer that. What we do when we set up an induction hypothesis and then show it hold for some start value as well as for n+1 for arbitrary n is to conclude that _if_ we could do this infinitely, then.... Such open ended conclusions are useful when evaluating certain kinds of questions such as whether a series converges. Yes, the series 0.3, 0.33, 0.333, etc.., does indeed converge to 1/3. But the (countable) sequence of these values does not actually contain the limit value 0.333... (1/3).

The limit is called a supremum - IIRC from 10 years ago. But the only way I
can know that 0.333...=1/3 is by induction, which implies a repeated
process. It's really nice that 1/3 happens to have a simple repeating
pattern, but that is not ture of all such infinite decimal point
representations of numbers.

 Quote: Hatto von Aquitanien wrote: We have a tendency to accept .333... as being as good a specification of a number as is 3, for example. That is to say "for all intents and purposes, .333... is as good a specification of a number as is 3." But is it? Are there some attributes which we should not attribute to .333... which we attribute to 3 by virtue of its being a number? 1/3 = sum{1 to oo} 3 x 10^-1 = 0.333... 3 = 3 + sum{1 to oo} 0 x 10^i. I don't see any significant difference in the two as "numbers", that is, I don't see how "3" is a "better" specification of a number than "0.333...". We only have a "tendency to accept" 0.333... as a good "specification" of 1/3 because we can prove that they are exactly one and the same number. Why would you think otherwise?

The form I was given for Cantor's diagonal proof by contradiction used
infinite decimal point notation[*]. It therefore assumes that it is
meaningful to represent the entire ordered set of real numbers in such a
form. I am first told to assume that there is a bijection between the
elements of this set and the natural numbers.

I have to agree with Wittgenstein on this one. I really don't know what
that means. But I'll play along anyway.

I am then presented with an iterative method for generating a number not in
the existing set that was supposed to be in 1-1 correspondence with N. I'm
not permitted to say "whoops!, looks like you missed one! Better add it to
the count." I'm told: "Nope, we're clean out of natural numbers. We
already used em all up on ... well you saw what we did with the rational
numbers right?" "Well, I saw what you *started* doing with the rational
numbers. I had the impression that was going to take a bit of time to
finish. Are you now trying to tell me you finished?" To that I am
told: "No, no, the demonstration that the rational numbers are countable
was a completely different matter."

So here I have an infinite sequence of infinite sequences of discretely
valued atoms. When I point out the atomicity of this structure and
say "wait a minute. Let's examine the implications of this", the result is
that it is condescended unto me that contemplating the atomicity shows my
lack of understanding of the very principles of mathematics.

I can pretty well convince myself that it makes some sense to arrange a
representation of the real numbers in such a way. It's probably best that
I find some notational technique for implying the infinite interpolation
rather than attempting to write it out explicitly. AAMOF, I don't really
believe I *could* write it all out explicitly. It seems to me the best I
can do is say _IF_ I could write out one infinite sequence, and then _IF_ I
could write out an infinite sequence of infinite sequences, and then _IF_ I
could put them in 1-1 correspondence with the infinite well ordered set of
natural numbers, then I could show that I can't really do that.

But wait! Why do I believe I can represent every real number using infinite
decimal point notation? Well, I can imagine that every number is some kind
of cuttable thing of a specific size. Any number smaller than the one I'm
dealing with will be represented by a cuttable thing proportionally
smaller. Similarly for lager numbers. I chose a base, such as base ten,
get out Occam's razor, Occam's meter stick and some Occam's glue and start
cutting and gluing.

[*] I believe Cantor's original paper can be shown to be logically
equivalent to this approach.
--
Nil conscire sibi
mike3

Joined: 27 May 2005
Posts: 52

Posted: Sat Jul 15, 2006 3:15 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

Hatto von Aquitanien wrote:
 Quote: I'm interested to know what attempts have been made to refute Cantor's proof that the real numbers are not denumerable? Quite honestly, I find the second diagonal method unconvincing. There are a few directions from which one might attempt to refute his argument. But before I spend a lot to time trying to formulate my own argument, it seems reasonable to seek prior art. Can anybody suggest a source which examines this topic? -- Nil conscire sibi

Why is it not convincing? I don't know of any attempts offhand, but
AFAIK it has
withstood all scrutiny.

In case you're curious, another (informal) argument, I don't know how
good, could
also be made for the real numbers' uncountability. Consider the
interval [-1, 1]. With
just the endpoints, we have 2. Add the middle, 0, and we have 3. We can
always
find a smaller number to stick in, and thus our total keeps going up.
We can never
"fill the gap" with naturals, because no matter how many points we
denumerate,
we can always make more in the same interval, so it evades all our
attempts to try
and assign a natural to each element. Since [-1, 1] is countless, and
it can be
mapped to all the real numbers via the one-to-one function f(x) =
tan(pix/2),
we conclude it's impossible to count the reals. If there's any way to
give the above
argument rigor, I'd be curious to hear about it.
imaginatorium@despammed.c
science forum Guru

Joined: 14 Sep 2005
Posts: 387

Posted: Sat Jul 15, 2006 3:30 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

mike4ty4@yahoo.com wrote:

<snip>

 Quote: In case you're curious, another (informal) argument, I don't know how good, could also be made for the real numbers' uncountability. Consider the interval [-1, 1]. With just the endpoints, we have 2. Add the middle, 0, and we have 3. We can always find a smaller number to stick in, and thus our total keeps going up. We can never "fill the gap" with naturals, because no matter how many points we denumerate, we can always make more in the same interval, so it evades all our attempts to try and assign a natural to each element. Since [-1, 1] is countless, and it can be mapped to all the real numbers via the one-to-one function f(x) = tan(pix/2), we conclude it's impossible to count the reals. If there's any way to give the above argument rigor, I'd be curious to hear about it.

I doubt it. Your argument above also appears to apply to the rationals
in [-1, 1]. There are an unlimited number of rationals, and any attempt
to count them (in the meaning of normal language, in which "counting"
is a terminating process) will fail. You will never get to the end,
even though you get to every individual rational eventually. (Last
sentence may induce QD in the feeble-brained.)

Brian Chandler
http://imaginatorium.org
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Sat Jul 15, 2006 4:13 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

In article <96mdnc8vzvxIyyXZnZ2dnUVZ_vmdnZ2d@speakeasy.net>,

 Quote: Virgil wrote: In article , Hatto von Aquitanien wrote: "finite extensions being of circumscribable effect". Can't say I really know what that means. "Circumscribable" means I can draw a circle around something. What are you drawing a circle around? Epsilon is the radius and the potential limit value is the center. For each epsilon greater than zero, only finitely many partial sums are allowed outside this circle if the series is to converge to that limit. that you are confining? You can give me some expression for delta which satisfies an arbitrary epsilon, but that seems to assume that I can prove something about every possible term in a potentially infinite sum. Whether I am to understand that the sum has actually been calculated or merely contemplated is irrelevant to the question of whether the idea of repeating a process ad infinitum is involved. As convergence depends on the finiteness of a set depending on epsilon, no appeal to an endless sequence of calulations is required or implied. Finiteness of a set? How can I show that all possible values of a potentially infinite series of partial sums will lie within some confinement without implicitly contemplating all members of that series via some form of extrapolation?

Given the sequence f(n) = 1/n, and any epsilon greater than zero, one
can show that |1/n - 0| > epsilon only when n < 1/epsilon, so that it
only occurs for finitely many values for n.

What happens for n >= 1/epsilon is irrelevant to the proof.
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Sat Jul 15, 2006 4:20 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

mike4ty4@yahoo.com wrote:

 Quote: Since [-1, 1] is countless, and it can be mapped to all the real numbers via the one-to-one function f(x) = tan(pix/2), we conclude it's impossible to count the reals.

The mapping as described does not work since the function is not defined
on the stated domain. f(-1) and f(1) are not defined.

While it is possible to map the closed interval [-1,1] bijectively to
the set of reals, it is not possible to do it with a continuous
function, nor with a function no defined on all of [-1,1].

If one restricts the function to domain (-1,1), on the other hand, ...
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

Posted: Sat Jul 15, 2006 4:54 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

Virgil wrote:

 Quote: Given the sequence f(n) = 1/n, and any epsilon greater than zero, one can show that |1/n - 0| > epsilon only when n < 1/epsilon, so that it only occurs for finitely many values for n. What happens for n >= 1/epsilon is irrelevant to the proof.

It may, in this case be trivially dispensed with, but you have implicitly
considered every value of n >= 1/epsilon. Somewhere in your thought
process you have applied a rule that says for n >= 1/epsilon the truth
of |1/n| > epsilon doesn't change, and the statement is false under these
conditions. It may seem so blatantly obvious that you don't see a need to
write out the induction rule, but that really is the only formal way to say
that such values are "irrelevant".
--
Nil conscire sibi
Dik T. Winter
science forum Guru

Joined: 25 Mar 2005
Posts: 1359

Posted: Sat Jul 15, 2006 11:36 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

In article <1152917292.168488.112220@i42g2000cwa.googlegroups.com> "David R Tribble" <david@tribble.com> writes:
 Quote: Hatto von Aquitanien wrote: .... We have a tendency to accept .333... as being as good a specification of a number as is 3, for example. That is to say "for all intents and purposes, .333... is as good a specification of a number as is 3." But is it? Are there some attributes which we should not attribute to .333... which we attribute to 3 by virtue of its being a number? 1/3 = sum{1 to oo} 3 x 10^-1 = 0.333...

You should stress that sum{1 to oo} is *not* an iterative process. As it
stands, it appears that you are taking the sum of infinitely many terms,
and that is not the case. sum{1 to oo} is (in my opinion) abuse of
notation, as it is defined as lim{n -> oo} sum{1 to n}. Which (in turn)
also is not an iterative process (think epsilon and delta).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Sun Jul 16, 2006 8:00 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

In article <iJOdnfATh6UuuCXZnZ2dnUVZ_qGdnZ2d@speakeasy.net>,

 Quote: Nathan wrote: Hatto von Aquitanien wrote: Nathan wrote: You're supposed to learn to understand the meaning of the limit of a sequence, particularly the idea of the sum of an infinite series as the limit of the sequence of partial sums. Not everybody does learn to understand this, which explains the .999...=1 threads. Is that what happens to all the smart people in college? They have their common sense psychologically pounded out of them. They either give lip service to the orthodox canon, accept it as truth, or transfer to computer science. No, there's another option you left out; the one I gave. They can also learn to understand the concept of a limit. As I said, this is what is supposed to happen. This doesn't require that they "accept it as truth", as some sort of dogma. There may indeed be those who "give lip service to the canon", because they really don't grasp it. There certainly are those who "accept it as truth"; it can be very hard to help certain students to learn to understand concepts and not just to do mechanical calculations. Some in that category do transfer to computer science, where they still tend to have trouble understanding concepts. And no, this doesn't necessarily happen in college. I said "around the second semester of calculus", which for a lot of "the smart people" means in high school. Perhaps you can explain to me how one arrives at 1/3 = .333... without appeal to an iterative process extended ad infinitum.

The process of multiplying the numerator by 10 then finding the integer
quotient and remainder on division by 3 is immediately seen to be
cyclic. Once something is cyclic like this, there is no need to view it
as an iterative process that needs extension ad infinitum.
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

Posted: Sun Jul 16, 2006 12:36 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

Virgil wrote:

 Quote: In article , Hatto von Aquitanien wrote: Perhaps you can explain to me how one arrives at 1/3 = .333... without appeal to an iterative process extended ad infinitum. The process of multiplying the numerator by 10 then finding the integer quotient and remainder on division by 3 is immediately seen to be cyclic. Once something is cyclic like this, there is no need to view it as an iterative process that needs extension ad infinitum.

That's what cyclic means.

--
Nil conscire sibi
Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Sun Jul 16, 2006 5:39 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

In article <XfSdnQDiqbiIrifZnZ2dnUVZ_oWdnZ2d@speakeasy.net>,

 Quote: Virgil wrote: In article , Hatto von Aquitanien wrote: Perhaps you can explain to me how one arrives at 1/3 = .333... without appeal to an iterative process extended ad infinitum. The process of multiplying the numerator by 10 then finding the integer quotient and remainder on division by 3 is immediately seen to be cyclic. Once something is cyclic like this, there is no need to view it as an iterative process that needs extension ad infinitum. That's what cyclic means.

Cyclic means that further repetition does not provide any different
result.
Lasse

Joined: 29 Apr 2005
Posts: 71

Posted: Mon Jul 17, 2006 12:56 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

 Quote: Yes, the series 0.3, 0.33, 0.333, etc.., does indeed converge to 1/3. But the (countable) sequence of these values does not actually contain the limit value 0.333... (1/3). The limit is called a supremum - IIRC from 10 years ago. But the only way I can know that 0.333...=1/3 is by induction, which implies a repeated process. It's really nice that 1/3 happens to have a simple repeating pattern, but that is not ture of all such infinite decimal point representations of numbers.

Nonsense. Induction is a method that can be used to prove a statement
for all natural numbers. What is that statement here? Also, do you have
the conception that induction is the *only* way to prove a statement
for all natural numbers? This is decidedly *not* the case.

Anyway, for the number x = 0.333... , it is easy to see from the
definition, and basic rules for limits, that 10*x = 3 + x. (In fact,
you can make this the definition of 0.333... . It's not a bad
definition for numbers with periodic decimal extensions, since it
avoids the use of limits altogether.)

Hence we have 9*x = 3, and thus x = 1/3.

Do you see induction anywhere in the above proof?

 Quote: The form I was given for Cantor's diagonal proof by contradiction used infinite decimal point notation[*]. It therefore assumes that it is meaningful to represent the entire ordered set of real numbers in such a form. I am first told to assume that there is a bijection between the elements of this set and the natural numbers.

No, you do not have to assume any such thing. Given *any* function from
the naturals into the reals, you can show that there is some real
number which is not in the image of this function. I posted a proof of
this (or rather, the underlying fact that no set is equipollent to its
power set) a while ago. You should try to understand this proof; it is
not that long.

The rest of your post seems rather confused and is difficult to
understand, but it seems to me that you have a problem grasping the
following: given a list of real numbers (i.e., a function from N to R),
we find an element which is not on this list. Could we not then just
add that element to our list?

The answer is: of course we can. However, by the same argument as
before, this list will still be missing some real number, and so on.
After all, we have proved the theorem not for a *particular* list of
real numbers, but for *all possible* lists!

Consider the following statements:

(i) For every real number x, there is a function f:N -> R whose image
contains x.

(ii) For every function f:N -> R, there is some real number x which is
*not* in the image of f.

Both of these statements are true, and there is no contradiction (look
at the order of the quantifiers)!
David R Tribble
science forum Guru

Joined: 21 Jul 2005
Posts: 1005

Posted: Mon Jul 17, 2006 8:52 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

Hatto von Aquitanien wrote:
 Quote: We have a tendency to accept .333... as being as good a specification of a number as is 3, for example. That is to say "for all intents and purposes, .333... is as good a specification of a number as is 3." But is it? Are there some attributes which we should not attribute to .333... which we attribute to 3 by virtue of its being a number?

David R Tribble writes:
 Quote: 1/3 = sum{1 to oo} 3 x 10^-1 = 0.333...

Dik T. Winter wrote:
 Quote: You should stress that sum{1 to oo} is *not* an iterative process.

I thought I did (in the next sentence).

 Quote: As it stands, it appears that you are taking the sum of infinitely many terms, and that is not the case. sum{1 to oo} is (in my opinion) abuse of notation, as it is defined as lim{n -> oo} sum{1 to n}. Which (in turn) also is not an iterative process (think epsilon and delta).

It's the limitations of the notations that appear to the be crux of
Hatto's problem:
1/3 = sum{i=1 to oo} 3x10^-i
1/3 = 3x10^-1 + 3x10^-2 + 3x10^-3 + ...
1/3 = 0.333...

Of course these are not "iterative processes", but the notations
"to" and "..." seem to imply some kind of iteration. It's a question
of comprehension and not reading too much into the notation.
Dik T. Winter
science forum Guru

Joined: 25 Mar 2005
Posts: 1359

Posted: Tue Jul 18, 2006 1:25 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

In article <1153169578.808273.40000@i42g2000cwa.googlegroups.com> "David R Tribble" <david@tribble.com> writes:
 Quote: David R Tribble writes: 1/3 = sum{1 to oo} 3 x 10^-1 = 0.333... Dik T. Winter wrote: You should stress that sum{1 to oo} is *not* an iterative process. .... It's the limitations of the notations that appear to the be crux of Hatto's problem: 1/3 = sum{i=1 to oo} 3x10^-i 1/3 = 3x10^-1 + 3x10^-2 + 3x10^-3 + ... 1/3 = 0.333... Of course these are not "iterative processes", but the notations "to" and "..." seem to imply some kind of iteration. It's a question of comprehension and not reading too much into the notation.

Yes, it is the problem with anybody who thinks too much about notation, and
thinks that that will provide him with all information.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

Posted: Tue Jul 18, 2006 10:22 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

Dik T. Winter wrote:

 Quote: In article <1153169578.808273.40000@i42g2000cwa.googlegroups.com> "David R Tribble" writes: David R Tribble writes: 1/3 = sum{1 to oo} 3 x 10^-1 = 0.333... Dik T. Winter wrote: You should stress that sum{1 to oo} is *not* an iterative process. ... It's the limitations of the notations that appear to the be crux of Hatto's problem: 1/3 = sum{i=1 to oo} 3x10^-i 1/3 = 3x10^-1 + 3x10^-2 + 3x10^-3 + ... 1/3 = 0.333... Of course these are not "iterative processes", but the notations "to" and "..." seem to imply some kind of iteration. It's a question of comprehension and not reading too much into the notation. Yes, it is the problem with anybody who thinks too much about notation, and thinks that that will provide him with all information.

It's not so much that I'm thinking about notation. In the circumstance,
notation is what is used to present the proof. The proof itself appears to
rely on the notational structure. I know that Cantor's argument can be
expressed in ways which do not use decimal (base 10) notation. Cantor's
paper did not even use decimal notation of any base. Nonetheless, I
believe his argument is basically lexicographical.

If I do address the proof given in decimal point notation, then what I see
is an argument which says: if I run down the diagonal of a decimal point
representation of an ordered set of real numbers far enough
(0.a_11a_22a_33...a_ii...), I will encounter a number not yet in the list.
these are all finite concepts as far as I can see. At some diagonal a_nn,
I will generate a new number which has not yet been put in 1-1
correspondence with N. Is this not the argument?

Is the base 10 form of the proof incorrect?

--
Nil conscire sibi
Hatto von Aquitanien
science forum Guru

Joined: 19 Nov 2005
Posts: 410

Posted: Tue Jul 18, 2006 11:17 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof?

Lasse wrote:

 Quote: Yes, the series 0.3, 0.33, 0.333, etc.., does indeed converge to 1/3. But the (countable) sequence of these values does not actually contain the limit value 0.333... (1/3). The limit is called a supremum - IIRC from 10 years ago. But the only way I can know that 0.333...=1/3 is by induction, which implies a repeated process. It's really nice that 1/3 happens to have a simple repeating pattern, but that is not ture of all such infinite decimal point representations of numbers. Nonsense. Induction is a method that can be used to prove a statement for all natural numbers. What is that statement here?

That for every decimal place n places to the right in the decimal point
representation of 1/3 the digit will be 3.

 Quote: Also, do you have the conception that induction is the *only* way to prove a statement for all natural numbers? This is decidedly *not* the case.

Starting with Peano's axioms, I'm not sure how much can be said about the
natural numbers as a whole without appeal, directly or implicitly to
induction.

 Quote: Anyway, for the number x = 0.333... , it is easy to see from the definition, and basic rules for limits, that 10*x = 3 + x. (In fact, you can make this the definition of 0.333... . It's not a bad definition for numbers with periodic decimal extensions, since it avoids the use of limits altogether.) Hence we have 9*x = 3, and thus x = 1/3. Do you see induction anywhere in the above proof?

I don't see how you are getting 0.333... from the basic rules of limits if
not through induction. More to the point, the concept of decimal point
notation is based on the idea of summing of negative powers of 10
multiplied by single digit integers.

 Quote: The form I was given for Cantor's diagonal proof by contradiction used infinite decimal point notation[*]. It therefore assumes that it is meaningful to represent the entire ordered set of real numbers in such a form. I am first told to assume that there is a bijection between the elements of this set and the natural numbers. No, you do not have to assume any such thing. Given *any* function from the naturals into the reals, you can show that there is some real number which is not in the image of this function. I posted a proof of this (or rather, the underlying fact that no set is equipollent to its power set) a while ago. You should try to understand this proof; it is not that long.

I asked what the axioms are for your set theory. The different
presentations of set theory I am drawing on are not mutually consistent in
the details of their axioms. I'll have to look again to see if you replied
to that request.

 Quote: The rest of your post seems rather confused and is difficult to understand, but it seems to me that you have a problem grasping the following: given a list of real numbers (i.e., a function from N to R), we find an element which is not on this list. Could we not then just add that element to our list? The answer is: of course we can. However, by the same argument as before, this list will still be missing some real number, and so on. After all, we have proved the theorem not for a *particular* list of real numbers, but for *all possible* lists! Consider the following statements: (i) For every real number x, there is a function f:N -> R whose image contains x.

I'm not sure exactly what is meant by R, or even N in this case.

--
Nil conscire sibi

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