From: utzoo!decvax!ucbvax!arens@UCBKIM
Newsgroups: net.math
Title: Re: Infinities
Article-I.D.: ucbvax.748
Posted: Fri Jan 28 23:24:54 1983
Received: Sat Jan 29 07:25:53 1983

From: arens@UCBKIM (Yigal Arens)
Received: from UCBKIM.BERKELEY.ARPA by UCBVAX.BERKELEY.ARPA (3.300 [1/17/83])
	id AA12922; 28 Jan 83 23:24:34 PST (Fri)
To: net-math@ucbvax


The cardinality of the set of integers is called aleph-0.  The cardinality
of the set of real numbers is called "two to the power of aleph-0", and is
the cardinality of the set of all subsets of the integers.

	[That is why it is called "two to the aleph-0".  The cardinality
	 of the set of all subsets of a set of N elements (including the
	 null set and the set itself) is 2^N]

2^aleph-0 is larger than aleph-0, meaning that there is no one-to-one
mapping of a set of cardinality 2^aleph-0 into a set of cardinality aleph-0,
but the reverse holds.  The proof is a cute and simple one, but I won't give
it here unless there's much public interest.

The cardinality of the set of all real functions is 2^(2^aleph-0) and is
larger still.

Aleph-1, on the other hand, is by definition the smallest cardinal larger
than aleph-0.  By the axioms of set theory such a cardinal exists, but it is
open to question whether aleph-1=2^aleph-0.  It would seem to be consistent
with set theory to believe that the above is either true, or false.

Yigal Arens
UC Berkeley