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Embeddability between orderings and GCH

Freire Rodrigo A.

Reports on Mathematical Logic

2021

Vol. 56

101--109

We provide some statements equivalent in ZF C to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete cardinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure theorem for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.