Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 12/29/2024 06:43 AM, joes wrote:

Am Sun, 29 Dec 2024 12:05:26 +0100 schrieb WM:

On 28.12.2024 20:35, joes wrote:

Am Sat, 28 Dec 2024 14:58:50 +0100 schrieb WM:

>

Therefore the union of all of them contains less than half of all
natural numbers.

Worng.

My theorem: If all definable numbers (FISONs) stay below a certain
threshold, then every union of those FISONs stays below that threshold.

The infinite union doesn’t.
>

There is no "infinite union" in ZF, only "pair-wise union",
according to the axiom of union. The "Axiom of Infinity"
in ZF is actually an introduced constant, that there
exists an inductive set what would be an infinite union.
If there was "infinite union" via comprehension itself
then it's easy to arrive at contradictions about the
complete ordered field with usual sorts of things.
Of course making "infinite union" is the same thing
as either "the illative" or "univalency", neither
of which are things in ZF/ZFC or the standard accounts.
Because without fixing function theory about the
continua, thusly contradictions are demonstrable.