Are there ways to describe the Martin Axiom intuitively?

Question

I'm looking for some way of understanding the Martin Axiom for some $\kappa$ in an intuitive way. I know that the motivation is the Baire Category Theorem, however is there any other way of thinking about this axiom?

Thanks in advance.

Answer 1

Since you mentioned in your comment that you're not familiar with iterated forcing, I'll try to explain a different aspect of $MA$, which I still hope can shed some light on this subject.

One way of understanding $MA_{\aleph_1}$ is in the general context of forcing axioms. If $\mathbb P$ is a forcing notion such that each $p\in \mathbb P$ has two incompatible members above it, then for every generic set $G\subseteq \mathbb P$, $G\notin V$. However, we would still like to find "quite generic" sets in the universe. More formally, let $\Gamma$ be a class of forcing notions. We say that $MA(\Gamma)$ holds if for every $\mathbb{P}\in \Gamma$ and a collection $\{I_{\alpha} : \alpha<\omega_1\}$ of dense sets, there is a filter $G$ on $\mathbb{P}$ that intersects all the $I_{\alpha}$. It turns out that we can prove the consistency of $MA(\Gamma)$ for non pathological classses of forcing notions. In this case, $MA_{\aleph_1}$ is simply $MA(\Gamma)$ when $\Gamma$ is the collection of c.c.c. forcing notions. By enlarging $\Gamma$ we obtain stronger forcing axioms, such as $PFA$ for the class of proper forcing notions and $MM$ for the class of stationary set preserving forcing notions. In this view, $MA_{\aleph_1}$ is just one level in a hierarchy of axioms that imply the existence of sufficiently generic sets.