# 2011 seminar talk: Absoluteness and Model Existence for Infinitary Logic

Talk held by Sy-David Friedman (KGRC) at the KGRC seminar on 2011-03-03.

### Abstract

If a sentence of first-order logic has an infinite model then it has models of all infinite cardinalities. But this is not the case for infinitary logic, which for the purposes of this talk I take to be*L*, the extension of first-order logic which allows countably infinite conjunctions and disjunctions. In infinitary logic there are sentences which have only countably infinite models. Using Keisler's completeness theorem for the logic

_{ω1, ω}*L(Q)*(where

*Qx*means "there exist uncountably many

*x*") there is an absolute criterion for a sentence of infinitary logic to have a model of size

*ℵ*. But without assuming CH, it is easy to show that there is no such absolute criterion for model existence in

_{1}*ℵ*. In this talk I'll focus on model existence for infinitary logic under the assumption of GCH. Using Kurepa trees and Special Aronszajn trees, I'll show that model existence in

_{2}*ℵ*is not absolute for GCH models, for any countable alpha different from

_{α}*ω*. The

*ℵ*case remains open, as is the question of whether large cardinals can be eliminated from these results. This is joint work with Tapani Hyttinen and Martin Koerwien.

_{&omega}