We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using this method we prove that the forcing axiom for the class of all finitely proper posets of size ℵ 1 is compatible with 2 ℵ 0 > ℵ 2 . In particular, this answers a question of Moore by showing that does not follow from this arithmetical assumption.

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in $\mathsf {ZF}$ , and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in $\mathsf {ZF}+\mathsf {DC}$ and $\mathsf {ZFC}$ . Our results confirm $\mathsf {ZF} + \mathsf {DC}$ as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

By forcing over a model of ZFC + GCH (above ?) with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H (?) (? = ? a regular cardinal) is a well-order of H (?) definable over the structure by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in addition to forcing a locally defined well-order of the universe, preserve many of the n-huge cardinals from the ground model (for all n). © 2008 Elsevier B.V. All rights reserved

Abstract. We define the ℵ 1.5 -chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom; in fact, MA 1.5 <κ implies MA <κ . Also, MA 1.5 <κ implies certain uniform failures of club-guessing on ω 1 that do not seem to have been considered in the literature before. We show, assuming CH and given any regular cardinal κ ≥ ω 2 such that µ ℵ0 < κ for all µ < κ and such that ♦({α < κ : cf(α) ≥ ω 1 }) holds, that there is a proper ℵ 2 -c.c. partial order of size κ forcing 2 ℵ0 = κ together with MA 1.5 <κ .1. A generalization of Martin's Axiom. And some of its applications.Martin's Axiom, often denoted by MA, is the following very wellknown and very classical forcing axiom: If P is a partial order (poset, for short) with the countable chain condition 1 and D is a collection of size less than 2 ℵ 0 consisting of dense subsets of P, then there is a filterMartin's Axiom is obviously a weakening of the Continuum Hypothesis. Given a cardinal λ, MA λ is obtained from considering, in the above formulation of MA, collections D of size at most λ rather than of size less than 2 ℵ 0 . Martin's Axiom becomes interesting when 2 ℵ 0 > ℵ 1 . MA ℵ 1 was the first forcing axiom ever considered ([9]). As observed by D. Martin, the consistency of MA together with 2 ℵ 0 > ℵ 1 follows from generalizing the Solovay-Tennenbaum construction of a model of Suslin's Hypothesis by iterated forcing using finite supports ([14]). Since then, a plethora of applications of MA (+ 2 ℵ 0 > ℵ 1 ) have been