New axioms in set theory

The paper under review provides a general introduction to the foundations of set theory and independence results and discusses two modern programs meant to overcome these independence results: forcing axioms and Woodin's =-. The article is aimed at readers with some mathematical background but does not assume extensive knowledge of set theory or logic; with a few exceptions that can be skipped without sacrificing understanding of the rest of the article. The exposition is fairly non-technical. The article opens with a brief historical overview focusing on the Continuum Hypothesis (), beginning with the work of Cantor and continuing through Cohen's development of forcing. The authors then introduce the axioms of , motivated by a discussion of Russell's paradox, followed by an overview of cardinals, ordinals, and the cumulative hierarchy. Section 3 uses the example of to introduce the primary set-theoretic tools for proving independence results: canonical inner models (particularly Gödel's constructible universe L) and forcing. The method of forcing is illustrated with an extended example: a presentation of random forcing via a Boolean-valued model approach. This is the most technically demanding section of the article but will provide the dedicated reader with a glimpse into the inner workings of forcing. Section 4 turns to more philosophical questions and discusses Gödel's program, which aims to overcome the phenomenon of independence over by supplementing with additional axioms that will be able to decide independent statements such as . The authors introduce the distinction of intrinsic vs. extrinsic justification for new axioms via quotes by Gödel. The last two sections of the article turn to modern developments in set theory. Section 5 discusses large cardinals, determinacy hypotheses, and generic absoluteness for second-order arithmetic, culminating in a discussion of Woodin's result stating that, assuming the existence of a proper class of Woodin cardinals, the truth values of statements in second-order arithmetic cannot be changed by set forcing. Since many problems throughout mathematics can be formulated in second-order arithmetic, this gives rise to the possibility that by accepting large cardinal axioms, forcing can be turned into a tool for proving theorems instead of just establishing independence results. In the final section, the authors introduce forcing axioms and Woodin's -, two programs meant to at least partially overcome the phenomenon of independence (and which decide in opposite ways). The discussion of forcing axioms focuses on Martin's maximum () and mentions its effect on Whitehead's problem and outer automorphisms of the Calkin algebra, as well as work of the second author indicating that strengthenings of provide certain instances of generic absoluteness for third-order arithmetic. The discussion of inner models begins with the observation that the assumption V=L decides , Whitehead's problem, and the existence of outer automorphisms of the Calkin algebra in the opposite way from but has the drawback that it is incompatible with the existence of very large cardinals. This leads to a broad overview of Woodin's - program.