Incomparable ω₁‐like models of set theory

Abstract

We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω 1 -like models of set theory. Specifically, under the hypothesis and suitable consistency assumptions, we show that there is a family of 2 ω 1 many ω 1 -like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω 1 -like model of ZFC that does not embed into its own constructible universe; and there can be an ω 1 -like model of PA whose structure of hereditarily finite sets is not universal for the ω 1 -like models of set theory.

Split Principles, large cardinals, splitting families and split ideals

“…A version of the split principle was first considered by Fuchs, Gitman, and Hamkins in the course of their work on [FGH17], the intended use being the construction of ultrafilters with certain properties. Later, the first author observed that the split principle is an "anti-large-cardinal axiom" which characterizes the failure of a regular cardinal to be weakly compact.…”