Forcing Axioms and the Continuum Problem

In this note, we give a survey on recent developments pertaining to forcing axioms with emphasis on their characterizations and consequences connected to the Continuum Problem. Nonspecialists in set theory and/or students in mathematics are supposed to be the readers, and thus we tried to make this as self-contained as possible. Due to the limitation on the extent of the article, however, many proofs are simply omitted. This is in particular the case for the results cited in the later sections. Those interested in the technical details may consult with papers and textbooks cited in the references at the end of the article. In Section 1, we introduce Martin’s axiom formulated in terms of general topology and show how this axiom is used to establish the independence of certain mathematical assertions from the usual axiom system of set theory. After reviewing some basic facts about forcing in Section 2, we give in Section 3 a characterization of Martin’s axiom in terms of notions connected to the method of forcing. In most of the modern textbooks on set theory, the characterization given in this section is the official definition of Martin’s axiom. In Section 4, we give two further characterizations of Martin’s axiom in terms of forcing: a characterization due to the author and another characterization by J. Bagaria. Both of these characterizations assert a certain absoluteness of generic extension over the universe of set theory, and they suggest that Martin’s Axiom is a very