Consecutive Singular Cardinals and the Continuum Function

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
Keywords supercompact cardinal   supercompact Prikry forcing   GCH   symmetric inner model
Categories (categorize this paper)
DOI 10.1215/00294527-1960434
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 46,238
Through your library

References found in this work BETA

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
Blowing Up Power of a Singular Cardinal—Wider Gaps.Moti Gitik - 2002 - Annals of Pure and Applied Logic 116 (1-3):1-38.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Extender Based Forcings.Moti Gitik & Menachem Magidor - 1994 - Journal of Symbolic Logic 59 (2):445-460.
Some Structural Results Concerning Supercompact Cardinals.Arthur W. Apter - 2001 - Journal of Symbolic Logic 66 (4):1919-1927.
Ad and Patterns of Singular Cardinals Below Θ.Arthur W. Apter - 1996 - Journal of Symbolic Logic 61 (1):225-235.
Splitting Number at Uncountable Cardinals.Jindřich Zapletal - 1997 - Journal of Symbolic Logic 62 (1):35-42.
On Measurable Limits of Compact Cardinals.Arthur W. Apter - 1999 - Journal of Symbolic Logic 64 (4):1675-1688.
Leibnizian Models of Set Theory.Ali Enayat - 2004 - Journal of Symbolic Logic 69 (3):775-789.
The Stability Function of a Theory.H. Jerome Keisler - 1978 - Journal of Symbolic Logic 43 (3):481-486.
Unfoldable Cardinals and the GCH.Joel David Hamkins - 2001 - Journal of Symbolic Logic 66 (3):1186-1198.
A Gitik Iteration with Nearly Easton Factoring.William J. Mitchell - 2003 - Journal of Symbolic Logic 68 (2):481-502.

Analytics

Added to PP index
2013-03-01

Total views
9 ( #822,197 of 2,285,778 )

Recent downloads (6 months)
2 ( #573,699 of 2,285,778 )

How can I increase my downloads?

Downloads

My notes

Sign in to use this feature