This chapter describes Zermelo–Fraenkel (ZF) set theory method with some modifications and applies it to a proof that the continuum hypothesis is independent of ZF. The language of ZF is the first order language with identity and with one binary predicate ε. The chapter denotes this language by L. The logical symbols, which are used, are defined. The chapter defines the logical symbols and eight axioms of ZF, including axiom of extensionality, existence of pairs, existence of unions, existence of power sets, existence of infinite sets, axiom of foundation, axiom scheme of comprehension, and axiom scheme of replacement. The meta-theory in which the chapter studies models of ZF will be the set theory ZFC enriched by one additional axiom SM due to Cohen. © 1979, PWN-Polish Scientific Publisheres.
Mostowski, A. (1979). An exposition of forcing. Studies in Logic and the Foundations of Mathematics, 93, 416–478. https://doi.org/10.1016/S0049-237X(09)70464-1