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 $κ$ 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.

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