Re: Replacement of Cardinality

Am 28.08.2024 um 15:05 schrieb Moebius:

Am 28.08.2024 um 06:23 schrieb joes:

Am Tue, 27 Aug 2024 19:26:25 +0000 schrieb WM:

Le 25/08/2024 à 23:18, Jim Burns a écrit :
>

Therefore, there is no ω-1,

 *sigh*
 Therefore there is no ordinal number o such that o + 1 = ω.
 

If the set of ordinal numbers is complete, then ω-1 precedes ω - by
definition.

 

How does it go again?

 WM defines "complete" comcerning (sets of) ordinal numbers the following way:
      A set of ordinals is /complete/ iff each and every ordinal
         in the set (except 0) has an immediate predecessor (which
         precedes it).
 Simple as that.
 In this sense, {0, 1, 2, 3, ... ω} is not (Mückenheim) complete.
 Though clearly no ordinal between 0 and ω is "missing" - LOL. :-)
 Hint: ~Eo e ORD: An e IN: n < o < ω.
 Hence the term "complete" as defined by Mückenheim is quite "misleading" (to say the least).

He's playing shell games, as usual.
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Actually, his argument is: If all moons consist of green cheese, than our moon consists of green cheese.
Yeah! Keep up the good work, Mückenheim!