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Article 2612 of comp.ai.philosophy:
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>From: cpm5479@rigel.tamu.edu (Chris Menzel)
Newsgroups: sci.philosophy.tech,sci.logic,comp.ai.philosophy,sci.math
Subject: Re: Aleph-1 - What?
Message-ID: <9JAN199218364659@rigel.tamu.edu>
Date: 9 Jan 92 23:36:00 GMT
References: <1992Jan2.131048.18412@news.stolaf.edu> <1992Jan6.092440.25451@etl.go.jp> <1992Jan6.130008.16471@news.stolaf.edu> <1992Jan9.141029.13881@ulrik.uio.no>
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In article <1992Jan9.141029.13881@ulrik.uio.no>, solan@math.uio.no (Svein Olav Nyberg) writes...
>I just wondered: Given the continuum hypothesis, Aleph-1 = 2^Aleph-0 (=c).
>Given its [denial], that there exists (at least) a cardinal number between
>Aleph-0  and  2^Aleph-0, does it make sense to speak of Aleph-1, the
>least cardinal larger than Aleph-0? 
>With the denial of the continuum hypothesis, what is to stop us from
>saying that given any two cardinal numbers n,m, n<m, there might be some third
>cardinal number p such that n<p<m?

The axiom of choice, for one thing.  Given choice, < well-orders the 
cardinals and so in particular rules out denseness and makes sense 
of there always being a next largest cardinal.


