## Journal of Symbolic Logic

### On Measurable Limits of Compact Cardinals

Arthur W. Apter

#### Abstract

We extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and also a limit of strongly compact cardinals having no non-trivial degree of supercompactness. We also show that it is consistent for the least supercompact cardinal $\kappa$ to be a limit of strongly compact cardinals and be so that every measurable limit of compact cardinals below $\kappa$ has a non-trivial degree of supercompactness. In this model, the only compact cardinals below $\kappa$ with a non-trivial degree of supercompactness are the measurable limits of compact cardinals.

#### Article information

Source
J. Symbolic Logic, Volume 64, Issue 4 (1999), 1675-1688.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745946

Mathematical Reviews number (MathSciNet)
MR1780078

Zentralblatt MATH identifier
0959.03041

JSTOR