An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice

Abstract

We show that in the theory ZF + DC + for every cardinal {\lambda}, the set of infinite subsets of {\lambda} is well-ordered (i.e., Shelah's AX4), the {\theta}-function measuring the surjective size of the powersets P({\kappa}) can take almost arbitrary values on any set of uncountable cardinals. This complements our results from [FK16], where we prove that in ZF (without DC), any possible behavior of the {\theta}-function can be realized; and answers a question of Shelah in [She16], where he emphasizes that ZF + DC + AX4 is a reasonable theory, where much of set theory and combinatorics is possible.