Editing Continuum hypothesis (section)

==Generalized continuum hypothesis==<!-- This section is linked from [[Forcing (mathematics)]] -->
The ''generalized continuum hypothesis'' (GCH) states that if an infinite set's cardinality lies between that of an infinite set {{mvar|S}} and that of the [[power set]] <math>\mathcal{P}(S)</math> of {{mvar|S}}, then it has the same cardinality as either {{mvar|S}} or <math>\mathcal{P}(S)</math>. That is, for any [[infinite set|infinite]] cardinal <math>\lambda</math> there is no cardinal <math>\kappa</math> such that <math>\lambda <\kappa <2^{\lambda}</math>. GCH is equivalent to:

{{block indent|<math>\aleph_{\alpha+1}=2^{\aleph_\alpha}</math> for every [[ordinal number|ordinal]] <math>\alpha</math>{{r|Goldrei1996}}}}

(occasionally called ''Cantor's aleph hypothesis'').

The [[beth number]]s provide an alternative notation for this condition: <math>\aleph_\alpha=\beth_\alpha</math> for every ordinal <math>\alpha</math>. The continuum hypothesis is the special case for the ordinal <math>\alpha=1</math>. GCH was first suggested by [[Philip Jourdain]].{{r|Jourdain1905}} For the early history of GCH, see Moore.{{r|Moore2011}}

Like CH, GCH is also independent of ZFC, but [[Wacław Sierpiński|Sierpiński]] proved that ZF + GCH implies the [[axiom of choice]] (AC) (and therefore the negation of the [[axiom of determinacy]], AD), 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 <math>2^{\aleph_0+n}</math> which is smaller than its own [[Hartogs number]]—this uses the equality <math>2^{\aleph_0+n}\, = \,2\cdot\,2^{\aleph_0+n} </math>; for the full proof, see Gillman.{{r|Gillman2002}}

[[Kurt Gödel]] showed that GCH is a consequence of ZF + [[Axiom of constructibility|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.&nbsp;B.&nbsp;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 <math>\aleph_\alpha</math> to fail to satisfy <math>2^{\aleph_\alpha} = \aleph_{\alpha + 1}</math>. Much later, [[Matthew Foreman|Foreman]] and [[W. Hugh Woodin|Woodin]] proved that (assuming the consistency of very large cardinals) it is consistent that <math>2^\kappa>\kappa^+</math> holds for every infinite cardinal <math>\kappa</math>. Later Woodin extended this by showing the consistency of <math>2^\kappa=\kappa^{++}</math> for every {{nowrap|<math>\kappa</math>.}} Carmi Merimovich{{r|Merimovich2007}} showed that, for each {{math|''n''&nbsp;≥&nbsp;1}}, it is consistent  with ZFC that for each infinite cardinal {{mvar|κ}}, {{math|2<sup>''κ''</sup>}} is the {{mvar|n}}th successor of {{mvar|κ}} (assuming the consistency of some large cardinal axioms). On the other hand, László Patai{{r|Patai1930}} proved that if {{mvar|γ}} is an ordinal and for each infinite cardinal {{mvar|κ}}, {{math|2<sup>''κ''</sup>}} is the {{mvar|γ}}th successor of {{mvar|κ}}, then {{mvar|γ}} is finite.

For any infinite sets {{mvar|A}} and {{mvar|B}}, if there is an injection from {{mvar|A}} to {{mvar|B}} then there is an injection from subsets of {{mvar|A}} to subsets of {{mvar|B}}. Thus for any infinite cardinals {{mvar|A}} and {{mvar|B}}, <math>A < B \to 2^A \le 2^B</math>. If {{mvar|A}} and {{mvar|B}} are finite, the stronger inequality <math>A < B \to 2^A < 2^B </math> 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 <math>\aleph_{\alpha}^{\aleph_{\beta}}</math> in all cases. GCH implies that for ordinals {{mvar|α}} and {{mvar|β}}:{{r|HaydenKennison1968}}

*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\beta+1}</math> when {{math|''α'' ≤ ''β''+1}};
*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\alpha}</math> when {{math|''β''+1 < ''α''}} and <math>\aleph_{\beta} < \operatorname{cf} (\aleph_{\alpha})</math>, where '''cf''' is the [[cofinality]] operation; and
*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\alpha+1}</math> when {{math|''β''+1 < ''α''}} and {{nowrap|<math>\aleph_{\beta} \ge \operatorname{cf} (\aleph_{\alpha})</math>.}}

The first equality (when {{mvar|''α'' ≤ ''β''+1}}) follows from:

<math display="block">\aleph_{\alpha}^{\aleph_{\beta}} \le \aleph_{\beta+1}^{\aleph_{\beta}} =(2^{\aleph_{\beta}})^{\aleph_{\beta}} = 2^{\aleph_{\beta}\cdot\aleph_{\beta}} = 2^{\aleph_{\beta}} = \aleph_{\beta+1} </math>
while:

<math display="block">\aleph_{\beta+1} = 2^{\aleph_{\beta}} \le \aleph_{\alpha}^{\aleph_{\beta}} .</math>

The third equality (when {{mvar|''β''+1 < ''α''}} and <math>\aleph_{\beta} \ge \operatorname{cf}(\aleph_{\alpha})</math>) follows from:

<math display="block">\aleph_{\alpha}^{\aleph_{\beta}}  \ge \aleph_{\alpha}^{\operatorname{cf}(\aleph_{\alpha})} > \aleph_{\alpha} </math>

by [[Kőnig's theorem (set theory)#Kőnig's_theorem_and_cofinality|Kőnig's theorem]], while:

<math display="block">\aleph_{\alpha}^{\aleph_{\beta}} \le \aleph_{\alpha}^{\aleph_{\alpha}} \le (2^{\aleph_{\alpha}})^{\aleph_{\alpha}} = 2^{\aleph_{\alpha}\cdot\aleph_{\alpha}} = 2^{\aleph_{\alpha}} = \aleph_{\alpha+1}</math>