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Inaccessible cardinals without the axiom of choice. (English) Zbl 1116.03045

In ZFC, a strongly inaccessible cardinal \(\kappa\) is a regular limit cardinal for which \(2^{\lambda}<\kappa\) whenever \(\lambda<\kappa\). The authors consider appropriate definitions for strong inaccessibility in the absence of AC. They produce four possible definitions that are equivalent in ZFC, but that are not equivalent in ZF. They provide a complete implication diagram (in ZF) for the different concepts: providing proofs for the positive directions and independence results for the directions that cannot be reversed.

MSC:

03E55 Large cardinals
03E25 Axiom of choice and related propositions
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