There's been a lot of diversion in this stream of Posts, partly because of Mark Adkins admirable attempt to explain his reasoning in multiple different ways. Let's get back to basics. As I read these posts, there are two claims at issue:
1) Georg Cantor's reasoning which he used to justify his proofs is flawed. Therefore his theorems are unproven. Not necessarily false.
2) The set of all Real numbers CAN be listed. (If I misinterpret this, Mark, please correct me.)

If Claim 2 is true, then Claim one must be true and Cantor's theorems must be false. Similarly, if Claim 1 is false, Claim 2 must be false. But we can also have Claim 1 being true and Claim 2 being false, in the absence of a valid proof of either Claim. I believe I can demonstrate that Claim 1 is true; it is left to the reader to see if they can find any demonstration, true OR false, of Claim 2. Although I happen to believe Claim 2 is true, I know of no proof or disproof of this statement which does not rely on Cantor's flawed logic. I wish to only focus my attention on Claim number 1. If Claim 1 is true, and enough people can understand the reason WHY it is true, then Claim 2 can be more substantively addressed, and perhaps even resolved.

Some of the diversion efforts have focused on the meaning of “list” and axiom versus hypothesis versus Lemma. These efforts merely divert our attention from the basic issue.

For the purposes of this discussion of basic logic, I interpret “list” to mean a string of individual elements whose position within the string has meaning (as opposed to a set in which the individual elements can be in any order). I interpret “infinity” to mean without end / boundless / continuing forever. Cantor did not actually agree with this interpretation, which led to statements such as
∞ < ∞ + 1

In each of Cantor's most lauded proofs, he (implicitly, never explicitly) relied on being able to “complete” infinity, i.e. being able to process to the “end” of an infinite list. This is uniformly true for his nested interval proof, his proofs concerning whether or not the Reals could be listed, his proof concerning whether or not the Transcendentals could be listed, and, by use of his Anti-Diagonalization method, whether or not the Power Set of an infinite list could be listed.

Specifically, he felt that if he could perform a process for all integers n, then his process must be true AT n “=” ∞.

If you do NOT accept that ∞ < ∞ + 1, then his error is easy to understand. Otherwise, in a little bit I will show that accepting his view is self-contradictory.

If you accept that ∞ < ∞ + 1, and there is no proof of this, then you must accept it as an hypothesis (or better, an Axiom, since it embodies something which cannot be proven but which you rely upon to generate derivative theorems). If we first accept it as an hypothesis, then we immediately arrive at a contradiction. In my view, when you start with an hypothesis which logically leads to a contradiction, you must either discard the hypothesis or show specifically the error in logic which was used to arrive at the contradiction. Note that Cantor starts in each of his proofs with an hypothesis which he claims leads by a logical chain to a contradiction. I will show that his chain of logic is where the error lies.

The Principle Mathematical Induction is limited to finite “n”. This is obvious in a simple way:
Let f be a function on the natural integers such that f(n) = n.
Let P(n) be the proposition that the value f(n + 1) > f(n).
Clearly f(1) = 1 and f(2) > f(1) so P(1) is true.
Suppose P(n) is true. Then f(n+1) > f(n). Substitute “n+1” for n and you get:
f((n+1)+1) = f(n+2) = n +2, and f(n+1) = n +1.
Obviously f(n+2) = n + 2 > n + 1 = f(n=1).
Therefore P(n) is true for all (finite) n.

But what is P(n) if we let n reach (“=”) ∞? If we assume such arithmetic even has meaning, we arrive at P(∞+1) = ∞+1 >? ∞ = P(∞). Clearly false for many of us. IF WE DO not ACCEPT that ∞ < ∞ + 1, P(∞) = P(∞+1). Thus we conclude that P(n) is true for any finite value of n but is unproven for n = ∞.

In order for Cantor's proofs to be correct, in lay terms he must be able to extend his proof's process TO ∞; i.e. he must be able to process an entire infinite list to reach the last logical step of each proof. So, in my view, his singular error is in his belief of “completable” infinite induction.

Now, let us take as an hypothesis that completable infinite induction is a valid logically. Using it, because we can create an Anti-Diagonal which is not in our original “list” for any value of n, the Anti-Diagonal is valid not just for any arbitrarily large n, but specifically for n “=” ∞.

For my contradiction of this hypothesis, I will rely on Cantor's actual proof-by-contradiction in which he worked with an infinite “list” of elements, each infinitely long and each consisting of only of permutations of the symbols “m” and “w”. Note that if Cantor's proof is valid for these symbols, it is also valid for symbols such as 0, 1, 2, …, 9. (For the picky, it simplifies to require that if the digit being changed happens to be 8 or 9, choose 1 as the Anti-Diagonal digit) I create successively longer lists of these permutations, each list containing all possible permutations of “m” and “w” for the n-digit length of the list at each step.

Step 1: All possible 1-digit strings of permutations of “m” and w”.
m w

Step 2: All possible 2-digit strings of …
mm mw wm ww

Step 3: All possible 3-digit strings
mmm mmw mwm mww wmm wmw wwm www

Step n: All possible n-digit strings
m...m … w...w

If I could extend n to actually “=” ∞, then I would have an infinitely long list of all infinitely long permutations of the symbols “m” and “w”. IF you assume Cantor can extend his logic to n “=” ∞, then you must also permit me to extend my process to n “=” ∞. Since the Anti-Diagonal which Cantor create must be one of these infinitely long permutations, then it must be in my list.

Therefore the assumption that induction can be completed at infinity is self-contradictory. Therefore none of Cantor's proofs are logically valid.

A Bogart formulated on Thursday :
> There's been a lot of diversion in this stream of Posts, partly because of
> Mark Adkins admirable attempt to explain his reasoning in multiple different
> ways. Let's get back to basics. As I read these posts, there are two claims
> at issue: 1) Georg Cantor's reasoning which he used to justify his proofs is
> flawed. Therefore his theorems are unproven. Not necessarily false. 2) The
> set of all Real numbers CAN be listed. (If I misinterpret this, Mark, please
> correct me.)

I'll wait.

On 1/26/2024 4:21 AM, FromTheRafters wrote:
> A Bogart formulated on Thursday :
>> There's been a lot of diversion in this stream of Posts, partly
>> because of Mark Adkins admirable attempt to explain his reasoning in
>> multiple different ways.  Let's get back to basics.  As I read these
>> posts, there are two claims at issue: 1) Georg Cantor's reasoning
>> which he used to justify his proofs is flawed.  Therefore his theorems
>> are unproven.  Not necessarily false. 2) The set of all Real numbers
>> CAN be listed.  (If I misinterpret this, Mark, please correct me.)
>
> I'll wait.

No shit! :^D

On 1/25/2024 7:12 PM, A Bogart wrote:
> There's been a lot of diversion in this stream of Posts, partly because of Mark Adkins admirable attempt to explain his reasoning in multiple different ways. Let's get back to basics. As I read these posts, there are two claims at issue:
> 1) Georg Cantor's reasoning which he used to justify his proofs is flawed. Therefore his theorems are unproven. Not necessarily false.
> 2) The set of all Real numbers CAN be listed. (If I misinterpret this, Mark, please correct me.)
[...]
Here is a good start... Remember to buy his books:

https://youtu.be/ja6QSyct6SE

;^)