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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sat, 11 Jan 2025 05:17:01 +0100
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Am 11.01.2025 um 04:51 schrieb Moebius:
> Am 10.01.2025 um 23:18 schrieb Chris M. Thomasson:
>> On 1/10/2025 1:52 AM, WM wrote:

>>> Then for every [ordinal number] it is determined whether [it] is a 
>>> natural number [or not].
> 
> Indeed!
> 
>>> Although we cannot determine it because <bla>
> 
> Well, there's a difference between mathematical facts and what we KNOW 
> about these facts (or even if they a r e facts).
> 
> There are many mathematical facts we cannot "determine" (at least not yet).

Wenn Du auch nur ein GRUNDVERSTÄNDNIS der Mathematik hättest, würdest Du 
das verstehen. Hier ein Auszug aus dem Buch "Nonstandard Analysis" von 
Alain M. Robert:

"[...] the new term 'standard' used in internal NSA is undefined. Only 
the use of this 'predicate' is codified by the axioms. Nevertheless, a 
suitable interpretation of this term is useful. The main point is that 
it can be applied to any object of set theory (or simply to any set, 
since all objects of ZF are sets). Any set E is either standard or 
nonstandard, ... any function is either standard or nonstandard, etc. 
This is the principle of the excluded middle. But of course, it may be 
difficult to discover which is true in specific examples."

Man könnte den letzten Satz auch so formulieren: "But of course, it may 
be difficult -if possible at all- to discover which is true in specific 
examples."

*sigh*

> .
> .
> .
>