## Notre Dame Journal of Formal Logic

### Inaccessible Cardinals, Failures of GCH, and Level-by-Level Equivalence

Arthur W. Apter

#### Abstract

We construct models for the level-by-level equivalence between strong compactness and supercompactness containing failures of the Generalized Continuum Hypothesis (GCH) at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal $\delta$, $2^{\delta }\gt \delta ^{++}$. In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inaccessible cardinals at which GCH holds are also measurable. These results extend and generalize earlier work of the author.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 55, Number 4 (2014), 431-444.

Dates
First available in Project Euclid: 7 November 2014

https://projecteuclid.org/euclid.ndjfl/1415382950

Digital Object Identifier
doi:10.1215/00294527-2798691

Mathematical Reviews number (MathSciNet)
MR3276406

Zentralblatt MATH identifier
1335.03044

#### Citation

Apter, Arthur W. Inaccessible Cardinals, Failures of GCH, and Level-by-Level Equivalence. Notre Dame J. Formal Logic 55 (2014), no. 4, 431--444. doi:10.1215/00294527-2798691. https://projecteuclid.org/euclid.ndjfl/1415382950

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