Martin’s Axiom

Abstract

In this chapter, we shall introduce a set-theoretic axiom, known as Martin’s Axiom, which is independent of ZFC. In the previous chapter we have compared forcing extensions with group extensions. Similarly, we could also compare forcing extensions with field extensions. Now, if we start, for example, with the field of rational numbers $\mathbb{Q}$ and extend $\mathbb{Q}$ step by step with algebraic extensions, we finally obtain an algebraic closure $\mathbb{F}$ of $\mathbb{Q}$. Since $\mathbb{F}$ is algebraically closed, we cannot extend $\mathbb{F}$ with an algebraic extension. With respect to forcing extensions, we have a somewhat similar situation: If we start, for example, with Gödel’s model $\boldsymbol{\mathop{\mathrm{L}}\nolimits }$, which is a model of ZFC + CH, and extend $\boldsymbol{\mathop{\mathrm{L}}\nolimits }$ step by step with forcing notions of a certain type, we finally obtain a model of ZFC which cannot be extended by a forcing notion of that type. The model we obtain in this way is a model in which Martin’s Axiom holds. In other words, models in which Martin’s Axiom holds are closed under certain forcing extensions, like algebraically closed fields are closed under algebraic extensions.

As a matter of fact we would like to mention that in the presence of the Continuum Hypothesis, Martin’s Axiom is vacuously true. However, if the Continuum Hypothesis fails, then Martin’s Axiom becomes an interesting combinatorial statement as well as an important tool in Combinatorics which has many applications in Topology, but also in areas like Analysis and Algebra.