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Article 2618 of comp.ai.philosophy:
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>From: andreas@etl.go.jp (Andreas Knobel)
Newsgroups: sci.philosophy.tech,sci.logic,comp.ai.philosophy,sci.math
Subject: Re: Aleph-1 - What?
Message-ID: <1992Jan10.045819.27538@etl.go.jp>
Date: 10 Jan 92 04:58:19 GMT
References: <1992Jan7.214019.6969@neptune.inf.ethz.ch> <1992Jan8.170943.15772@aio.jsc.nasa.gov> <1992Jan9.141029.13881@ulrik.uio.no>
Organization: Electrotechnical Lab.
Lines: 27

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 converse, 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? 

Sure it makes sense. For any cardinal aleph, you define its successor aleph^+
as the least ordinal greater than aleph; since the ordinals are well ordered,
this ordinal exists and is now called a cardinal.

>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?

Nothing stops you, as long as you do not violate the rules of set theory.
For instance if aleph < aleph' then necessarily 2^aleph <= 2^aleph'. You
can find models of ZF through forcing where your statement holds and hence
demonstrate its consistency with the rest of ZF.

>Solan

Andreas

-- 

Andreas Knobel                      e-mail andreas@etlcom.etl.go.jp
ETL, Tsukuba, Japan.


