Critical embeddings

Hayut and first author isolated the notion of a critical cardinal in [1]. In this work we answer several questions raised in the original paper. We show that it is consistent for a critical cardinals to not have any ultrapower elementary embeddings, as well as that it is consistent that no target model is closed. We also prove that if $\kappa$ is a critical point by any ultrapower embedding, then it is the critical point by a normal ultrapower embedding. The paper contains several open questions of interest in the study of critical cardinals.Comment: 7 page

Iterated extended ultrapowers and supercompactness without choice

AbstractWorking in ZF + DC with no additional use of the axiom of choice, we show how to iterate the extended ultrapower construction of Spector (1988, 1991). This generalizes the technique of iterated ultrapowers to choiceless set theory. As an application, we prove the following theorem: Assume V = LU[Pκ(λ)] + “κ is λ-supercompact with normal ultrafilter U” + DC. Then for every sufficiently large regular cardinal ρ, there exists a set-generic extension V[G] of the universe in which there exists for some σ a set S ⊆ Pρ(σ) for which one can define an elementary embedding j mapping V to LD[S], where D is the filter in V[G] generated by the closed unbounded filter (according to V) on Pρ(σ). Moreover, we have j(κ) = ρ, j(λ) = σ, j(Pκ(λ)) = S (which is Pρ(σ) according to LD[S]), and j(itU) = D ∩ LD[S] i s a normal ultrafilter in LD[S] on Pρ(σ)

Critical Cardinals

We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the axiom of choice this is equivalent to measurability, but it is well-known that choice is necessary for the equivalence. Oddly enough, this central notion was never investigated on its own before. We prove a technical criterion for lifting elementary embeddings to symmetric extensions, and we use this to show that it is consistent relative to a supercompact cardinal that there is a critical cardinal whose successor is singular