## Bulletin of Symbolic Logic

### Square in Core Models

#### Abstract

We prove that in all Mitchell-Steel core models, $\square_\kappa$ holds for all $\kappa$. (See Theorem 2.). From this we obtain new consistency strength lower bounds for the failure of $\square_\kappa$ if $\kappa$ is either singular and countably closed, weakly compact, or measurable. (Corallaries 5, 8, and 9.) Jensen introduced a large cardinal property that we call subcompactness; it lies between superstrength and supercompactness in the large cardinal hierarchy. We prove that in all Jensen core models, $\square_\kappa$ holds iff $\kappa$ is not subcompact. (See Theorem 15; the only if direction is essentially due to Jensen.)

#### Article information

Source
Bull. Symbolic Logic Volume 7, Number 3 (2001), 305-314.

Dates
First available in Project Euclid: 20 June 2007

https://projecteuclid.org/euclid.bsl/1182353797

Mathematical Reviews number (MathSciNet)
MR1860606

Zentralblatt MATH identifier
0992.03062

JSTOR