Notre Dame Journal of Formal Logic 54 (2):125-136 (2013)
Abstract |
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 $\mathrm {ZF}+\lnot\mathrm {AC}_{\omega}$ in which either $\aleph_{1}$ and $\aleph_{2}$ are both singular and the continuum function at $\aleph_{1}$ can be precisely controlled, or $\aleph_{\omega}$ and $\aleph_{\omega+1}$ are both singular and the continuum function at $\aleph_{\omega}$ can be precisely controlled. Additionally, we discuss a result in which we separate the lengths of sequences of distinct subsets of consecutive singular cardinals $\kappa$ and $\kappa^{+}$ in a model of $\mathrm {ZF}$. Some open questions concerning the continuum function in models of $\mathrm {ZF}$ with consecutive singular cardinals are posed
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Keywords | supercompact cardinal supercompact Prikry forcing GCH symmetric inner model |
Categories | (categorize this paper) |
DOI | 10.1215/00294527-1960434 |
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References found in this work BETA
Strong Axioms of Infinity and Elementary Embeddings.Robert M. Solovay - 1978 - Annals of Mathematical Logic 13 (1):73.
Squares, Scales and Stationary Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (01):35-98.
Scales, Squares and Reflection.James Cummings, Matthew Foreman & Menachem Magidor - 2001 - Journal of Mathematical Logic 1 (1):35-98.
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