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From: dmr@dutoit.UUCP
Newsgroups: net.math
Subject: Re: How Many Continuous Functions Are There
Message-ID: <2063@dutoit.UUCP>
Date: Sat, 12-Oct-85 00:22:43 EDT
Article-I.D.: dutoit.2063
Posted: Sat Oct 12 00:22:43 1985
Date-Received: Mon, 14-Oct-85 03:23:29 EDT
Lines: 22

There seems to be a general belief in recent messages here that aleph-1
(second infinite cardinal) equals C (power of the continuum).
C is defined as the cardinality of the real numbers, and by simple
Cantorian arguments is equal to the cardinality of the subsets
of integers, which is what is meant by 2^aleph-0.

But the other belief is called the Continuum hypothesis, and Paul Cohen
showed about 20 years ago that it is independent of the usual
axioms of set theory (can't be proved; may consistently be asserted
or denied).  You are free to assume that C is much bigger
than practically any aleph you can name (well, not aleph-C).

In particular, the following fragment is lunacy:

>There are C polynomials ---> there are aleph-0 * C countable subsets.
>
>Incidentally, C = aleph-1.
>( C <= the number of finite subsets of reals = aleph-1
>and C >= the number of reals = aleph-1 --> C = aleph-1)
>and so aleph-0 * C = aleph-2.

	Dennis Ritchie