Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.2 9/18/84; site dartvax.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!ucbvax!decvax!dartvax!chuck
From: chuck@dartvax.UUCP (Chuck Simmons)
Newsgroups: net.math
Subject: Re: How Many Continuous Functions Are There
Message-ID: <3686@dartvax.UUCP>
Date: Thu, 10-Oct-85 18:27:56 EDT
Article-I.D.: dartvax.3686
Posted: Thu Oct 10 18:27:56 1985
Date-Received: Sun, 13-Oct-85 04:10:25 EDT
References: <310@ihnet.UUCP> <10556@ucbvax.ARPA> <3368@pur-ee.UUCP> <656@petsd.UUCP>
Organization: Dartmouth College, Hanover, NH
Lines: 16

> C ** aleph-0 = aleph-1 ** aleph-0
>              = ( 2 ** aleph-0) ** aleph-0
>              = 2 ** (aleph-0 ** aleph-0)
>              = 2 ** aleph-0
>              = aleph-1.

Now wait a minute.  (1) Since when is the exponentiation operator
associative (lines 2-3).  (2) aleph-0 ** aleph-0 is NOT aleph-0.
Proof:

   2 < aleph-0  ->  2 ** aleph-0  <= aleph-0 ** aleph-0
                ->  aleph-1 <= aleph-0 ** aleph-0

Since aleph-0 < aleph-1, it follows that aleph-0 < aleph-0 ** aleph-0.

chuck@dartvax