Taylor KO Factor

2013-01-27T03:42:07Z

50.30.192.115: /* Alternative Approaches */ Corrected a cartridge name reference

In [[set theory]], the '''singular cardinals hypothesis (SCH)''' arose from the question of whether the least [[cardinal number]] for which the [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]] (GCH) might fail could be a [[singular cardinal]].

According to Mitchell (1992), the singular cardinals hypothesis is:
:If κ is any singular [[limit cardinal|strong limit cardinal]], then 2κ = κ+.
Here, κ+ denotes the [[successor cardinal]] of κ.

Since SCH is a consequence of GCH which is known to be [[consistent]] with [[Zermelo–Fraenkel set theory|ZFC]], SCH is consistent with ZFC. The negation of SCH has also been shown to be consistent with ZFC, if one assumes the existence of a sufficiently large cardinal number. In fact, by results of [[Moti Gitik]], ZFC + the negation of SCH is equiconsistent with ZFC + the existence of a measurable cardinal κ of [[Mitchell order]] κ++.

Another form of the SCH is the following statement:
:2cf(κ) < κ implies κcf(κ) = κ+,
where cf denotes the [[cofinality]] function. Note that κcf(κ)= 2κ for all singular strong limit cardinals κ. The second formulation of SCH is strictly stronger than the first version, since the first one only mentions strong limits; from a model in which the first version of SCH fails at &alefsym;ω and GCH holds above &alefsym;ω+2, we can construct a model in which the first version of SCH holds but the second version of SCH fails, by adding &alefsym;ω Cohen subsets to &alefsym;n for some n.

[[Jack Silver|Silver]] proved that if κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Silver's original proof used [[generic ultrapowers]]. The following important fact follows from Silver's theorem: if the singular cardinals hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals. In particular, then, if <math> \kappa </math> is the least counterexample to the singular cardinals hypothesis, then <math> cf(\kappa) = \omega </math>.

The negation of the singular cardinals hypothesis is intimately related to violating the GCH at a measurable cardinal. A well-known result of [[Dana Scott]] is that if the GCH holds below a measurable cardinal <math> \kappa </math> on a set of measure one—i.e., there is normal <math> \kappa </math> -complete ultrafilter D on <math> \mathcal{P}(\kappa) </math> such that <math> \{\alpha < \kappa: 2^{\alpha} = \alpha^+\}\in D </math>, then <math> 2^\kappa = \kappa^+ </math>. Starting with <math> \kappa </math> a [[supercompact cardinal]], Silver was able to produce a model of set theory in which <math> \kappa </math> is measurable and in which <math> 2^\kappa > \kappa^+ </math>. Then, by applying [[Prikry forcing]] to the measurable <math> \kappa </math>, one gets a model of set theory in which <math> \kappa </math> is a strong limit cardinal of countable cofinality and in which <math> 2^\kappa > \kappa^+ </math>—a violation of the SCH. [[Moti Gitik|Gitik]], building on work of [[W. Hugh Woodin|Woodin]], was able to replace the supercompact in Silver's proof with a measurable of Mitchell order <math> \kappa^{++} </math>. That established an upper bound for the consistency strength of the failure of the SCH. Gitik again, using results of [[Inner model theory]], was able to show that a measurable of Mitchell order <math> \kappa^{++} </math> is also the lower bound for the consistency strength of the failure of SCH.

A wide variety of propositions imply SCH. As was noted above, GCH implies SCH. On the other hand, the [[proper forcing axiom]] which implies <math> 2^{\aleph_0} = \aleph_2 </math> and hence is incompatible with GCH also implies SCH. [[Robert M. Solovay|Solovay]] showed that large cardinals almost imply SCH—in particular, if <math> \kappa </math> is [[strongly compact cardinal]], then the SCH holds above <math> \kappa </math>. On the other hand, the non-existence of (inner models for) various large cardinals (below a measurable of Mitchell order <math> \kappa^{++} </math>) also imply SCH.

==References==
* [[Thomas Jech|T. Jech]]: [http://matwbn.icm.edu.pl/ksiazki/fm/fm81/fm8116.pdf Properties of the gimel function and a classification of singular cardinals], ''[[Fundamenta Mathematicae]]'', '''81'''(1974), 57-64.
* William J. Mitchell, "On the singular cardinal hypothesis," ''Trans. Amer. Math. Soc.'', volume 329, number 2, pages 507–530, 1992.
* Jason Aubrey, ''The Singular Cardinals Problem'' ([http://www.math.lsa.umich.edu/vigre/Expositions/Aubrey.pdf PDF]), VIGRE expository report, Department of Mathematics, University of Michigan.

[[Category:Cardinal numbers]]

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