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The generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S. That is, for any infinite cardinal \lambda\, there is no cardinal \kappa\, such that \lambda <\kappa <2^{\lambda}.\, GCH is equivalent to: \aleph_{\alpha+1}=2^{\aleph_\alpha} for every ordinal \alpha.\, The beth numbers provide an alternate notation for this condition: \aleph_\alpha=\beth_\alpha for every ordinal \alpha.\, This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers. Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the axiom of choice (AC), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than some Aleph number, and thus can be ordered. This is done by showing that n is smaller than 2^{\aleph_0+n}\, which is smaller than its own Hartogs number (this uses the equality 2^{\aleph_0+n}\, = \,2\cdot\,2^{\aleph_0+n} ; for the full proof, see Gillman (2002). Kurt Gödel showed that GCH is a consequence of ZF + V=L (the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen's model in which CH fails is a model in which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed by Cohen to prove Easton's theorem, which shows it is consistent with ZFC for arbitrarily large cardinals \aleph_\alpha to fail to satisfy 2^{\aleph_\alpha} = \aleph_{\alpha + 1}. Much later, Foreman and Woodin proved that (assuming the consistency of very large cardinals) it is consistent that 2^\kappa>\kappa^+\, holds for every infinite cardinal \kappa.\, Later Woodin extended this by showing the consistency of 2^\kappa=\kappa^{++}\, for every \kappa\,. A recent result of Carmi Merimovich shows that, for each n≥1, it is consistent with ZFC that for each κ, 2κ is the nth successor of κ. On the other hand, László Patai (1930) proved, that if γ is an ordinal and for each infinite cardinal κ, 2κ is the γth successor of κ, then γ is finite. For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsets of B. Thus for any infinite cardinals A and B, A < B \to 2^A \le 2^B. If A and B are finite, the stronger inequality A < B \to 2^A < 2^B \! holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals. Implications of GCH for cardinal exponentiation Although the Generalized Continuum Hypothesis refers directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation in all cases. It implies that \aleph_{\alpha}^{\aleph_{\beta}} is (see: Hayden & Kennison (1968), page 147, exercise 76): |
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