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

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*

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