Guessing Models, Cardinal Arithmetic, and Ultrafilters
Chris Lambie-Hanson
Guessing models were introduced by Viale and Weiß in the course of their investigations into the consistency strength of the Proper Forcing Axiom (\(\mathsf{PFA}\)). It quickly became apparent that guessing models are powerful tools for obtaining instances of compactness, and over the last fifteen years they have seen numerous applications. For example, results Wei\ss \ and of Viale indicate that the existence of guessing models implies the failure of relatively weak square principles, and results of Viale and of Krueger combine to show that the existence of guessing models implies the Singular Cardinals Hypothesis (\(\mathsf{SCH}\)).
In this talk, we present two recent applications of guessing models. In the first, we pursue further investigations into the impact of guessing models on cardinal arithmetic. We show, for instance, that the existence of guessing models tightly correlates the values of \(2^{\aleph_0}\) and \(2^{\aleph_1}\), and also that the existence of guessing models implies Shelah’s Strong Hypothesis (\(\textsf{SSH}\)), a PCF-theoretic strengthening of \(\mathsf{SCH}\). We then use ideas from these arguments to show that the generalized narrow system property, a close relative of the strong tree property, at a cardinal \(\kappa\) implies \(\mathsf{SSH}\) above \(\kappa\).
The second application concerns indecomposable ultrafilters. A non-principal ultrafilter \(U\) over a cardinal \(\kappa > \aleph_1\) is indecomposable if, for every \(\lambda < \kappa\) and every \(f:\kappa \rightarrow \lambda\), there is a set \(A \in U\) such that \(f[A]\) is countable. Sheard, answering a question of Silver, proved in the 1980s that, relative to the consistency of a measurable cardinal, it is consistent that there is an inaccessible cardinal \(\kappa\) such that \(\kappa\) carries an indecomposable ultrafilter but \(\kappa\) is not weakly compact. Recently, Goldberg showed that this cannot happen above a strongly compact cardinal: If \(\mu\) is strongly compact and \(\kappa \geq \mu\) carries an indecomposable ultrafilter, then \(\kappa\) is either measurable or a limit of countably many measurable cardinals. Using guessing models, we show that the same conclusion follows from \(\mathsf{PFA}\). This provides another instance in the long list of compactness principles that were first shown to hold above a strongly compact or supercompact cardinal and later shown also to follow from \(\mathsf{PFA}\).
This talk contains joint work with Šárka Stejskalová and with Assaf Rinot and Jing Zhang.