Useful axioms

TL;DR

This paper surveys the relationship between forcing axioms and non-constructive principles like the axiom of choice, exploring their reformulations, model-theoretic perspectives, and potential for providing complete set-theoretic semantics.

Contribution

It offers a comprehensive overview of how forcing axioms relate to various principles and discusses their potential to serve as a complete semantics for certain set-theoretic models.

Findings

Forcing axioms can reformulate the axiom of choice and Baire's theorem.

Deep analogies exist between ultraproducts and Shoenfield's absoluteness.

Forcing axioms may provide a complete semantics for initial set-theoretic fragments.

Abstract

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 $\kappa>\omega_1$.