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_{ω1, ω}, 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 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 ℵ₁. But without assuming CH, it is easy to show that there is no such absolute criterion for model existence in ℵ₂. 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 ℵ_α is not absolute for GCH models, for any countable alpha different from ω. The ℵ_&omega 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.