Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site sbcs.UUCP
Path: utzoo!linus!philabs!sbcs!stark
From: stark@sbcs.UUCP (Eugene Stark)
Newsgroups: net.math
Subject: Re: How Many Continuous Functions Are There
Message-ID: <483@sbcs.UUCP>
Date: Fri, 11-Oct-85 10:04:31 EDT
Article-I.D.: sbcs.483
Posted: Fri Oct 11 10:04:31 1985
Date-Received: Tue, 15-Oct-85 05:36:41 EDT
References: <310@ihnet.UUCP> <10556@ucbvax.ARPA> <3368@pur-ee.UUCP> <656@petsd.UUCP>
Organization: Computer Science Dept, SUNY@Stony Brook
Lines: 44

> 
> In fact the number of countable subsets is C ** aleph-0.
> >
> >Incidentally, C = aleph-1.
> ...
> >and so aleph-0 * C = aleph-2.
>
> Actually aleph-0 * aleph-1 = aleph-1.  But also,
> C ** aleph-0 = aleph-1.  Proof:
> 
> C ** aleph-0 = aleph-1 ** aleph-0
>              = ( 2 ** aleph-0) ** aleph-0
>              = 2 ** (aleph-0 ** aleph-0)

The previous line should read:  2 ** (aleph-0 * aleph-0)
	(exponentiation is not associative)

>              = 2 ** aleph-0
>              = aleph-1.
> 
> 
> References: College-level books on set theory.  One by Halmos,
> _Naive_Set_Theory_, is quite good.

Not to split hairs, but

	aleph-0		=	the cardinality of the natural numbers
			=	the least infinite cardinal

	aleph-1		=	the least cardinal strictly greater than
					aleph-0

	2**aleph-0	=	the cardinality of the continuum
			=	the cardinality of the powerset of a
					countably infinite set

	The equality of aleph-1 and 2**aleph-0 is the "continuum hypothesis",
	which is independent of the other axioms of set theory.

	(See, for example Shoenfield, "Mathematical Logic" Chapter 9,
		or any book on axiomatic set theory.)


						Gene Stark