Useful axioms

Matteo Viale

Published 2016-10-10Version 1

We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we outline how, using basic partial order theory, it is possible to reformulate the axiom of choice, Baire's category theorem, and many large cardinal axioms as specific instances of forcing axioms. We then address forcing axioms with a model-theoretic perspective and outline a deep analogy existing between the standard {\L}o\'s Theorem for ultraproducts of first order structures and Shoenfield's absoluteness for $\Sigma^1_2$-properties. Finally we address the question of whether and to what extent forcing axioms can provide a "complete" semantics for set theory. We argue that to a large extent this is possible for certain initial fragments of the universe of sets: The pioneering work of Woodin on generic absoluteness show that this is the case for the Chang model $L(\text{Ord}^\omega)$ in the presence of large cardinals, and recent works by the author show that this can also be the case for the Chang model $L(\text{Ord}^{\omega_1})$ in the presence of large cardinals and maximal strengthenings of Martin's maximum or of the proper forcing axiom. The major open question we leave open is whether this situation is peculiar to these Chang models or can be lifted up also to $L(\text{Ord}^\kappa)$ for cardinals $\kappa>\omega_1$.