Forcing Axioms, Finite Conditions and Some More

Abstract

Abstract. We survey some classical and some recent results in the theory of forcing axioms, aiming to present recent breakthroughs and interest the reader in further developing the theory. The article is written for an audience of logicians and mathematicians not necessarily familiar with set theory. IntroductionWe shall work within the axioms of the Zermelo-Fraenkel set theory with Choice (ZFC). These axioms were introduced basically starting from 1908 and improving to a final version in the 1920s as an attempt to axiomatize the foundations of mathematics. There have been other such attempts at about the same time and later, but it is fair to say that for the purposes of much of modern mathematics the axioms of ZFC represent the accepted foundation (see [13] for a detailed discussion of foundational issues in set theory). Gödel's Incompleteness theorems [16] prove that for any consistent theory T which implies the Peano Axioms and whose axioms are presentable as a recursively enumerable set of sentences, so for any reasonable theory one would say, there is a sentence ϕ in the language of T such that T does not prove or disprove ϕ. In some sense the discussion of which axioms to use is made less interesting by these theorems, which can be interpreted as saying that a perfect choice of axioms does not exists. We therefore do like the most, we concentrate on the axioms that correctly model most of mathematics, and for the rest, we try to understand the limits and how we can improve them. For us ZFC is a basis for a foundation which in some circumstances can be extended to a larger set of axioms which provide an insight into various parts of mathematics. In here we concentrate on the forcing axioms (and their negations).