The Continuum Problem and Forcing

Abstract

1.1. Cantor introduced two fundamental ideas in the theory of infinite sets: he discovered (or invented?) the scale of cardinalities of infinite sets, and gave a proof that this scale is unbounded. We recall that two sets M and N are said to have the same cardinality (card M = card N) if there exists a one-to-one correspondence between them. We write card M ≤ card N if M has the same cardinality as a subset of N. We say that M and N are comparable if either card M ≤ card N or card N ≤ card M. We write card M > card N if card M ≤ card N but M and N do not have the same cardinality.

1.2. Theorem (Cantor, Schröder, Bernstein, Zermelo)

(a) Any two sets are comparable. If both card M ≤ card N and card N ≤ card M, then card M = card N. In other words, the cardinalities are linearly ordered.

(b) Let P(M) be the set of all subsets of M. Then card P(M) > card M. In particular, there does not exist a largest cardinality.

(c) In any class of cardinalities there is a least cardinality. In other words, the cardinalities are well-ordered.