Re: Replacement of Cardinality

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).
We'd better define:
A set of ordinals is /Mückenheim complete/ iff each and every
         ordinal in the set (except 0) has an immediate predecessor
         (which precedes it).
Theorem: {0, 1, 2, 3, ... ω} is not Mückenheim complete. :-)
Though it's not clear what's missing here. :-)