Re: Incompleteness of Cantor's enumeration of the rational numbers

On 06.11.2024 15:59, Mikko wrote:

On 2024-11-05 11:26:58 +0000, WM said:
 

  Cantor's results are

conclusions of proofs and you have not shown any error in the proofs.

>
I have. This example for instance proves that he did not enumerate all rationals, because the rationals are dense, the intervals are not dense.

 You have not proven that. It is fairly easy to prove that there are
no positive rationals other than those enumerated by Cantor (if I
recall correctly he enumerated only positive rationals). To prove
that there are positive rationals that are not included in Cantor's
enumeration it suffices to show one but you have not shown any.

I have shown that without rational numbers outside of the intervals with irrational endpoints covering 3 of infinitely many units the real axis has measure 3. Completely independent of the form and configuration of the intervals between them no real number could exist if all rationals were included.

 

You are free to deny one of more of the assumptions that constitue
the foudations of the results but you havn't.

>
Cantor's bijections concern only potentially infinite sets, but are assumed and claimed to concern the complete sets.

 Everything Cantor said was about complete sets. He did neither deny the
possibility of potentially infinte sets nor said anything about them (as
far as I know and remember).

Georg Cantor did not get tired to explain the difference and the importance of the actual infinite over and over again.
"In spite of significant difference between the notions of the potential and actual infinite, where the former is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably often that the one is confused with the other." [Cantor, p. 374]

That is the grave mistake. His result says for all infinite "countable sets" that they are infinite, nothing more.

 He very clearly says and proves that all infinite sets are not equinumerous.

All countable infinite sets are equinumerous according to him, but not in reality.
Regards. WM