## Notre Dame Journal of Formal Logic

### Consecutive Singular Cardinals and the Continuum Function

#### 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 (1) $\aleph_{1}$ and $\aleph_{2}$ are both singular and the continuum function at $\aleph_{1}$ can be precisely controlled, or (2) $\aleph_{\omega}$ 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.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 54, Number 2 (2013), 125-136.

Dates
First available in Project Euclid: 21 February 2013

https://projecteuclid.org/euclid.ndjfl/1361454970

Digital Object Identifier
doi:10.1215/00294527-1960434

Mathematical Reviews number (MathSciNet)
MR3028791

Zentralblatt MATH identifier
1284.03235

#### Citation

Apter, Arthur W.; Cody, Brent. Consecutive Singular Cardinals and the Continuum Function. Notre Dame J. Formal Logic 54 (2013), no. 2, 125--136. doi:10.1215/00294527-1960434. https://projecteuclid.org/euclid.ndjfl/1361454970

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