The Continuum Problem and Forcing

Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 53)

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.

Keywords

Dition Verse Summing 

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Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Max-Planck Institut für MathematikBonnGermany

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