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

Am 02.12.2024 um 08:41 schrieb Moebius:

Am 02.12.2024 um 00:11 schrieb Chris M. Thomasson:

On 11/30/2024 3:12 AM, WM wrote:

 FISON(s)
 

Finite initial segment[s]: F(n) = {1, 2, 3, ..., n}   (n e IN)

 You see it's a DEFINITION:
          F(n) = {1, 2, 3, ..., n}   (n e IN) .

 From this definition we may get an explicit definition for the term /FISON/ (short for "finite initial segment"):
            X is a /FISON/ iff there is an n e IN such that X = F(n).

This "means" (implies): F(1) (i.e. {1}) is a FISON, F(2) (i.e. {1, 2}) is a FISON, F(3) (i.e. {1, 2, 3}) is a FISION, and so on (ad infinitum). F(1), F(2), F(3), ... are FISONs.
 

Finite?

 Yeah, finite. For each and eveer n e IN F(n) (i.e. {1, 2, 3, ..., n}) is finite (i.e. a finite set).
 Huh? The natural numbers don't stop at n! WTF!!!
 No one (except possibly Mückenheim) said they did.
 Hint: There are _infinitely many_ finite initial segments (one for each and every natural number n). :-)