A Course in Mathematical Logic for Mathematicians pp 105-150 | Cite as

# 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*.

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