Problem with Cantors Diagonalization Proof.

[talanum1]

Cantor supposes we may make an infinite list of real numbers between 0 and 1 and pair them with Natural Numbers. Then he supposes we make another real number by changing digits along a diagonal in the list. We take the first digit of the first number and change it, take the second digit from the second number and add one to it, and so on. Then he says that we constructed a number not on the list.

My objection is that in order to conclude this we must be able to state that the digit at infinity of the infinite numbered real number got taken and changed. But "digit at infinity" and "infinite numbered real number" is undefined.

One can also not say: "change the n'th digit by adding one and then let n tend to infinity." because the "tending to" operations operand must be specified and the process will never end: the operand at infinity must actually be evaluated.

We don't have access to the digit at infinity.

This is especially acute because we have wrong intuition about infinity.

 

[Brian_G]

 

[rustypup]

 

[EmmanuelGoldstein]

 

[konfab]

Cantor supposes we may make an infinite list of real numbers between 0 and 1 and pair them with Natural Numbers. Then he supposes we make another real number by changing digits along a diagonal in the list. We take the first digit of the first number and change it, take the second digit from the second number and add one to it, and so on. Then he says that we constructed a number not on the list.

My objection is that in order to conclude this we must be able to state that the digit at infinity of the infinite numbered real number got taken and changed. But "digit at infinity" and "infinite numbered real number" is undefined.

One can also not say: "change the n'th digit by adding one and then let n tend to infinity." because the "tending to" operations operand must be specified and the process will never end: the operand at infinity must actually be evaluated.

We don't have access to the digit at infinity.

This is especially acute because we have wrong intuition about infinity.

It is because you have the wrong definition of infinity. His innovation was to not care about the end digit of infinity, but to rather compare how you count it compared to natural numbers.


I am too retarded to explain it well, but this is a ridiculously good video on it:

 

[RedViking]

Thank you! Life is life changing.

 

[TheMightyQuin]

Cantor supposes we may make an infinite list of real numbers between 0 and 1 and pair them with Natural Numbers. Then he supposes we make another real number by changing digits along a diagonal in the list. We take the first digit of the first number and change it, take the second digit from the second number and add one to it, and so on. Then he says that we constructed a number not on the list.

My objection is that in order to conclude this we must be able to state that the digit at infinity of the infinite numbered real number got taken and changed. But "digit at infinity" and "infinite numbered real number" is undefined.

One can also not say: "change the n'th digit by adding one and then let n tend to infinity." because the "tending to" operations operand must be specified and the process will never end: the operand at infinity must actually be evaluated.

We don't have access to the digit at infinity.

This is especially acute because we have wrong intuition about infinity.

Ok calm down , Terence.

 

[Brian_G]

From the second thread;

I have got contact with Aliens. They say all physical processes should have definite in-between states. They take it as an Axiom.

 

[Brian_G]

 

[Brian_G]

It is because you have the wrong definition of infinity. His innovation was to not care about the end digit of infinity, but to rather compare how you count it compared to natural numbers.

Thanks, that part turns out to be within reason to understand.

It's what came after that nearly broke my head - that Real numbers are somehow a bigger infinity than other types of numbers
No sleep for me tonight :- (

 

[konfab]

Thanks, that part turns out to be within reason to understand.

It's what came after that nearly broke my head - that Real numbers are somehow a bigger infinity than other types of numbers
No sleep for me tonight :- (

I had to watch it a couple of times to get it.


This is basically how I feel when talking about math.

 

[ENIAC]

The diagonal argument never requires a 'digit at infinity'. It only defines each digit by position, not as a completed process. The resulting number is defined by a rule, not by actually performing an infinite procedure.

 

[killerbyte]

Cantor supposes we may make an infinite list of real numbers between 0 and 1 and pair them with Natural Numbers. Then he supposes we make another real number by changing digits along a diagonal in the list. We take the first digit of the first number and change it, take the second digit from the second number and add one to it, and so on. Then he says that we constructed a number not on the list.

My objection is that in order to conclude this we must be able to state that the digit at infinity of the infinite numbered real number got taken and changed. But "digit at infinity" and "infinite numbered real number" is undefined.

One can also not say: "change the n'th digit by adding one and then let n tend to infinity." because the "tending to" operations operand must be specified and the process will never end: the operand at infinity must actually be evaluated.

We don't have access to the digit at infinity.

This is especially acute because we have wrong intuition about infinity.

While I appreciate the earnest attempt to grapple with Cantor’s diagonal argument - a cornerstone of modern mathematics - I must confess that your objection reads less like a critique and more like a philosophical flail against the very concept of infinity, wielded with the precision of a butter knife in a fencing match.

Cantor’s construction does not, contrary to your assertion, require access to a “digit at infinity.” That phrase alone betrays a fundamental misapprehension of both the argument and the nature of infinite sequences. The diagonalization method operates entirely within the realm of finite indices: for each natural number n, we alter the n-th digit of the n-th number. This yields a new sequence whose n-th digit differs from the n-th digit of the n-th entry in the list. The elegance of the argument lies precisely in its avoidance of any need to evaluate a mythical “digit at infinity” - a concept as undefined in mathematics as it is unnecessary.

To suggest that “the process will never end” is to miss the point entirely. Of course it doesn’t end; it's a definition of an infinite object. The diagonal number is not constructed by completing an infinite process in time, but by specifying its digits via a rule. This is standard fare in mathematics: we define infinite sequences by rules, not by waiting for infinity to arrive like a delayed train.
Moreover, your invocation of “wrong intuition about infinity” is ironically apt, though not in the way you intend. It is precisely this intuition; this insistence on treating infinity as a destination rather than a domain, that leads to the kind of muddled reasoning on display here. Infinity is not a place we go to pluck digits from; it is a conceptual framework within which we define and compare sizes of sets, sequences, and cardinalities.

Cantor’s diagonal argument has withstood over a century of scrutiny not because it relies on sleight of hand, but because it is built on the bedrock of formal logic. To dismiss it on the basis of a mischaracterized “digit at infinity” is akin to rejecting calculus because one cannot physically touch a limit.
In short, the objection fails not because it challenges Cantor, but because it misunderstands him. And while philosophical discomfort with infinity is understandable, it is not a substitute for mathematical rigor.

 

[talanum1]

Be honest with yourself: we need to access the digit at infinity in order to change it. It's easy to specify a process that never ends, but then to suggest you build a number based on the process ending is unsound logic.

 

[Brian_G]

Be honest with yourself: we need to access the digit at infinity in order to change it. It's easy to specify a process that never ends, but then to suggest you build a number based on the process ending is unsound logic.

Presume you mean the real (irrational) number being built in the later parts of the video; there is no linear end-process basis in the theory.
Firstly, an irrational decimal number can't have an ending as we perceive endings, which of course any mathematician knows.
Secondly the concept is additionally within an infinite (continuum) consideration.

Infinity as a fixed result can't ever be accessed, you can't apply such linear concepts to it. You're struggling with accepting that infinity is nevertheless logical, as it doesn't make sense to your related more-narrow acceptance of logic.

Do I understand it? Do any of us properly?
I suspect not, infinity has strange rules. Best I'm able to do is get a ("solid") feeling for it.

 

[killerbyte]

Be honest with yourself: we need to access the digit at infinity in order to change it. It's easy to specify a process that never ends, but then to suggest you build a number based on the process ending is unsound logic.

Let us dispense with pleasantries and address the assertion directly: the claim that Cantor’s diagonal argument requires “access to the digit at infinity” is not merely mistaken—it is a category error of such magnitude that it threatens to collapse the very distinction between process and definition.

To say that we must “access the digit at infinity” is to commit a grievous conflation between the construction of an infinite object and the evaluation of a terminal step in a finite process. Infinity, I regret to inform the objector, is not a place one arrives at, nor a digit one retrieves from some metaphysical shelf. It is a conceptual framework—a limit, a horizon, a mode of definition. The diagonal number is not built by completing an infinite process; it is defined by a rule that specifies its n-th digit for every natural number n. That is the entirety of the construction. No digit at infinity is summoned, consulted, or altered, because no such digit exists.

To insist otherwise is akin to demanding the final verse of an eternal poem before accepting its authorship. It is to mistake the map for the journey, the definition for the execution, and the infinite for the unreachable. Mathematics, unlike intuition, does not require that a process “end” in order to define its output. We do not “wait” for infinity to arrive; we define what happens at every finite stage and thereby specify the whole.

Indeed, the diagonal argument’s genius lies in its immunity to such objections. It does not rely on the completion of an infinite list, but on the impossibility of listing all real numbers between 0 and 1 in a countable sequence. The diagonal number differs from each entry at a finite position. That suffices. The contradiction is immediate, elegant, and devastating.

So let us be honest with ourselves, as you so nobly request. The logic is sound. The process is well-defined. And the objection, though earnestly offered, is a valiant misfire—a lance aimed at a phantom, while the real dragon flies overhead, untouched and triumphant.

I remain, as ever, your humble servant in the pursuit of mathematical clarity.

 

[talanum1]

we define what happens at every finite stage and thereby specify the whole.

That requires evaluating what happens "at" infinity.

 

[paul5186]

tldr

LSM=−41FμνFμν+iψˉγμDμψ+∣Dμh∣2−V(h)+ψˉYhψ+h.c

 

[Brian_G]

Reading your post #1 again helped. (And I see it is regarding what I thought it was.)

My objection is that in order to conclude this we must be able to state that the digit at infinity of the infinite numbered real number got taken and changed. But "digit at infinity" and "infinite numbered real number" is undefined.

Seems like you're saying that there's suspicion in your mind re some hidden trick involved which would show itself if the infinite digit could be revealed, and using that to justify having to have that?


What I'm seeing is that the infinite nature of infinity can be stretched as a more linear concept, and then still results in the same... infinity.
So not actual stretched, just looks that way from our viewpoint. Maybe we could compare it to how space-time interacts.

 

[rh1]

Oh duelling over numbers. No idea what is being discussed in this thread, bjt reminded me of this: