Title: The Choice of New Axioms in Set Theory

The development of axiomatic set theory originated from the need for a rigorous investigation of the foundations of mathematics. The classical theory of sets ZFC offers a rich framework for mathematics, nevertheless many crucial problems cannot be solved within this theory. For this reason, set theorists have been exploring new principles that would allow one to answer some fundamental questions that are independent from ZFC. Research in this area has led to consider several candidates for a new axiomatisation such as Large Cardinal Axioms, Forcing Axioms, Projective Determinacy and others. The legitimacy of such new axioms is, however, heavily debated and gave rise to extensive discussions around an intriguing philosophical problem: what is an axiom in mathematics? What criteria would lead us to welcome or reject a new axiom for set theory? Self-evidence, intuitive appeal, fruitfulness are some of the many criteria that have been proposed. The future of set theory very much depends on how we answer such questions. The main objective of this tutorial is to illustrate the main candidates for new axioms in contemporary set theory and discuss the basic motivations and objections to their legitimacy. The tutorial will consist in 4 lectures:

– Lecture 1: we will briefly survey the evolution of the concept of axiom in mathematics and we will discuss to what extent the axiom of the theory ZFC can be supported by self-evidence, intuitive appeal or practical reasons

– Lecture 2: we discuss the axiom of constructibility and large cardinals axioms

– Lecture 3: we discuss Determinacy hypotheses

– Lecture 4: we discuss Forcing Axioms

Advertisements