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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: Replacement of Cardinality
Date: Wed, 28 Aug 2024 15:05:04 +0200
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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. :-)