Embeddability Between Orderings and GCH

Rodrigo A. Freire


We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete car- dinals, 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.

AMS subject classification: 03E05.

Słowa kluczowe: continuum hypothesis, structure of linear orderings, partition relations.

[1] P. Erdös and R. Rado, A Problem on Ordered Sets, Journal of the London Mathematical Society 28:4 (1953), 426-438.

[2] R. Fraisse, Theory of Relations, Elsevier, 1986.

[3] A. Levy, Basic Set Theory, Dover, 2002.

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