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

On 12/29/2024 10:15 AM, Ross Finlayson wrote:

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.
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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.
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>

That there is "infinite union" in sets of course
is necessary to not have inductive accounts that
induction never completes, among reasons why the
illative/univalent are considered axioms by some.
You like induction that something's so? There's
another that it's not. That's why thorough reasoners
necessarily include accounts of point and space,
void and universal, atoms and wholes, about
deductive inference, when exhaustion has somewhere
to go, else it just goes "away" (ad absurdam).
So, some have that your inductive accounts are
ultimately.untrue, and, you don't know the
difference though of course there are inductive
accounts that inductive accounts are ultimately.untrue
that are as well not.first.false and never.first.false.
It's like saying you're closed-minded,
yet nobody's ever invented a perfect mouse-trap.