Critical Cardinals
Logic
2020-05-07 v2
Abstract
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.
Keywords
Cite
@article{arxiv.1805.02533,
title = {Critical Cardinals},
author = {Yair Hayut and Asaf Karagila},
journal= {arXiv preprint arXiv:1805.02533},
year = {2020}
}
Comments
16 pages; revised version