Speaker

# Chris Lambie-Hanson

Czech Academy of Sciences

### Talks at this conference:

Thursday, 15:20, J330 |
## Guessing Models, Cardinal Arithmetic, and Ultrafilters |

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 This talk contains joint work with Šárka Stejskalová and with Assaf Rinot and Jing Zhang. |