faustus (01/29/83)
Now, if I remember correctly the next higher infinity after the number of points in a line (or in n-space) is the number of curves in any n-space. This makes sense: it is infinity to the infinity, in a way. This is I think aleph two. Does anybody know what aleph[3, 4, ...] are?? Wayne Christopher faustus@berkeley ucbvax!faustus
arens@UCBKIM (01/29/83)
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