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Switch to: References
Citations of:
Blowing up power of a singular cardinal—wider gaps
Annals of Pure and Applied Logic 116 (1-3):1-38 (2002)
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We show that from a supercompact cardinal $\kappa$, there is a forcing extension $V[G]$ that has a symmetric inner model $N$ in which $\mathrm {ZF}+\lnot\mathrm {AC}$ holds, $\kappa$ and $\kappa^{+}$ are both singular, and the continuum function at $\kappa$ can be precisely controlled, in the sense that the final model contains a sequence of distinct subsets of $\kappa$ of length equal to any predetermined ordinal. We also show that the above situation can be collapsed to obtain a model of $\mathrm (...) |
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We determine exact consistency strengths for various failures of the Singular Cardinals Hypothesis (SCH) in the setting of the Zermelo-Fraenkel axiom system ZF without the Axiom of Choice (AC). By the new notion of parallel Prikry forcing that we introduce, we obtain surjective failures of SCH using only one measurable cardinal, including a surjective failure of Shelah's pcf theorem about the size of the power set of $\aleph _{\omega}$ . Using symmetric collapses to $\aleph _{\omega}$ , $\aleph _{\omega _{1}}$ , (...) |
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We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11]. |
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We show how to construct Gitik’s short extenders gap-3 forcing using a morass, and that the forcing notion is of Prikry type. |
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Woodin has shown that if there is a measurable Woodin cardinal then there is, in an appropriate sense, a sharp for the Chang model. We produce, in a weaker sense, a sharp for the Chang model using only the existence of a cardinal \ having an extender of length \. |
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The purpose of the present paper is to present new methods of blowing up the power of a singular cardinal κ of cofinality ω. New PCF configurations are obtained. The techniques developed here will be used in a subsequent paper to construct a model with a countable set which pcf has cardinality ℵ1. |
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We show that Gitik’s short extender gap-2 forcing is of Prikry type. |
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We introduce certain morass type structures and apply them to blowing up powers of singular cardinals. As a bonus, a forcing for adding clubs with finite conditions to higher cardinals is obtained. |
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We prove equiconsistency results concerning gaps between a singular strong limit cardinal κ of cofinality 0 and its power under assumptions that 2κ=κ+δ+1 for δ<κ and some weak form of the Singular Cardinal Hypothesis below κ. Together with the previous results this basically completes the study of consistency strength of the various gaps between such κ and its power under GCH type assumptions below. |
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