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The consistency strength of choiceless failures of SCH. (English) Zbl 1202.03056

The Singular Cardinal Hypothesis (\(\mathsf{SCH}\)) states that if \(\kappa\) is a strong limit singular cardinal then \(\kappa^{\mathrm{cof}(\kappa)}=\kappa^+\). It has been one of the central problems in the development of modern set theory. Through works of Silver, Shelah, Gitik and Woodin, the study of \(\mathsf{SCH}\) has produced many powerful tools in combinatorics, forcing and inner model theory. However, these works all involve the Axiom of Choice (\(\mathsf{AC}\)) in essential ways. In this article, the authors examine the consistency strengths of various failures of \(\mathsf{SCH}\) in the choiceless context.
In the non-\(\mathsf{AC}\) situation there are two versions of \(\neg\mathsf{SCH}\), one is witnessed by a surjective function \(f: [\kappa]^{\mathrm{cof}(\kappa)} \to\kappa^{++}\), the other by an injective function \(g: \kappa^{++} \to [\kappa]^{\mathrm{cof}(\kappa)}\). For the surjective version the authors show that the surjective failure of \(\mathsf{SCH}\) at a strong limit cardinal (below which \(\mathsf{GCH}\) holds) of countable cofinality (such as \(\aleph_\omega\)) in \(\mathsf{ZF}+\neg\mathsf{AC}\) is equiconsistent with the existence of one measurable cardinal in \(\mathsf{ZFC}\). It is still open how to obtain an analogue for strong limit cardinals of uncountable cofinality.
For the injective version, three theorems are presented in the paper, each measuring the consistency strength of the failure of \(\mathsf{SCH}\) at a strong limit cardinal \(\lambda\) of cofinality \(\omega\), \(\omega_1\) or \(\omega_2\), respectively, together with \(\mathsf{GCH}\) holding below \(\lambda\). For instance, the case of \(\lambda=\aleph_\omega\) is equiconsistent with \(\mathsf{ZFC}\) plus the existence of a limit cardinal of countable cofinality such that for any \(i<\omega\) there are unboundedly many \(\delta\) below \(\kappa\) such that \(o(\delta)\geq \delta^{+i}\), where \(o(\delta)\) is the Mitchell order of \(\delta\). The case of \(\lambda=\aleph_{\omega_2}\) is equiconsistent with \(\mathsf{ZFC}\) plus the existence of a cardinal \(\kappa\) of Mitchell order \(\kappa^{++} + \omega_2\). The case \(\lambda=\aleph_{\omega_1}\) is a bit subtle. Only upper and lower bounds are provided. Its consistency strength is bounded between the consistency strengths of \(\mathsf{ZFC}+\exists \kappa (o(\kappa)=\kappa^{++})\) and \(\mathsf{ZFC}+\exists \kappa (o(\kappa)=\kappa^{++}+\omega_1)\).
For all these results the upper bound is established by producing a choiceless, symmetric submodel \(N\) of a generic extension via a new notion of parallel Prikry forcing. The authors also give a general framework for symmetrically collapsing large cardinals at which \(\mathsf{SCH}\) fails down to small singular limit cardinals, such as \(\aleph_{\omega}\), \(\aleph_{\omega_1}\) or \(\aleph_{\omega_2}\). The lower bound on consistency strength of surjective failure of \(\mathsf{SCH}\) at \(\aleph_\omega\) is determined using the Dodd-Jensen core model for sequences of measures. The lower bound for the three injective failures of \(\mathsf{SCH}\) are obtained by using Gitik and Mitchell’s work.
Note that the above failures of \(\mathsf{SCH}\) at a cardinal of countable cofinality (both surjective and injective) in a choiceless context is beyond what we currently know in \(\mathsf{ZFC}\), and the injective failures at cardinals of uncountable cofinality are in sharp contrast to Silver’s \(\mathsf{ZFC}\) result that \(\mathsf{GCH}\) cannot first fail at a singular strong limit cardinal of uncountable cofinality.

MSC:

03E25 Axiom of choice and related propositions
03E35 Consistency and independence results
03E45 Inner models, including constructibility, ordinal definability, and core models
03E55 Large cardinals
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