Forcing and consistency properties

Juan M Santiago Suárez (U Paris Cité)

Jun 27. 2024, 09:15 — 10:00

The relation between forcing, Boolean valued models and consistency properties was implicit from the beginning of forcing in the works of Mansfield, Keisler, Solovay, Scott and Makkai. However, no concrete application was given. The aim of this talk is to argue that consistency properties and infinintary logics provide a natural setting for building forcing notions. First, we will present general logic results such as completeness, interpolation and omittying types. All can be proved for the most general case, the logic $\mathcal{L}_{\infty \infty}$. Nonetheless, wether or not these theorems can be proved with respect to Boolean valued models with the mixing property allows to strictly separate $\mathcal{L}_{\infty \omega}$ from $\mathcal{L}_{\infty \infty}$. Second, we will analyse the relation between consistency properties and forcing. Motivated by the fact that every generic filter can be seen as the model built by a consistency property, we will discuss for what formulas there is a consistency property forcing a model in a nice way. The results presented here build on the proof of MM$^{++}$ implies $(*)$ by Asperò and Schindler together with the AS condition recently isolated by Kasum and Velickovic.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Determinacy, Inner Models and Forcing Axioms (Workshop)
Organizer(s):
Sandra Müller (TU Vienna)
Grigor Sargsyan (Polish Academy of Science, Warsaw)
Ralf Schindler (WWU Münster)
John Steel (UC, Berkeley)