Maxwell Levine: Forcing Square Sequences

KGRC research seminar on 2017-11-30 at 4pm.

Speaker: Maxwell Levine (KGRC)

Abstract: In the 1970’s, Jensen proved that Gödel’s constructible universe $L$ satisfies a combinatorial principle called $\square_\kappa$ for every uncountable cardinal $\kappa$. Its significance is partially in that it clashes with the reflection properties of large cardinals – for example, if $\mu$ is supercompact and $\kappa \ge \mu$ then $\square_\kappa$ fails – and so it characterizes the minimality of $L$ in an indirect way. Schimmerling devised an intermediate hierarchy of principles $\square_{\kappa,\lambda}$ for $\lambda \le \kappa$ as a means of comparing a given model of set theory to $L$, the idea being that a smaller value of $\lambda$ yields a model that is more similar to $L$ at $\kappa$.

Cummings, Foreman, and Magidor proved that for any $\lambda<\kappa$, $\square_{\kappa,\lambda}$ implies the existence of a PCF-theoretic object called a very good scale for $\kappa$, but that $\square_{\kappa,\kappa}$ (usually denoted $\square_\kappa^\ast$) does not. They asked whether $\square_{\kappa,<\kappa}$ implies the existence of a very good scale for $\kappa$, and we resolve this question in the negative.

We will discuss the technical background of the problem, provide a complete solution, and discuss further avenues of research.

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