Worked solution: Independence of Suslin's Hypothesis from ZFC (Jensen, Solovay–Tennenbaum, 1971)

Step 5 of 8: Martin's Axiom: guaranteed generic-like filters without leaving

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In plain words

Recall from forcing that a generic filter has to meet every dense set that a ground model can name; ordinarily this forces you to step outside that model into a genuinely new universe $V[G]$ . Martin's Axiom is the bold hypothesis that, for ccc posets, the universe you already live in is generous enough to contain such filters itself, as long as you only need to meet a limited number ( $\kappa$ -many) of dense sets at once.

Because a modest version of this ( $\kappa = \aleph_0$ ) is provable in ordinary $\mathrm{ZFC}$ , Martin's Axiom only becomes an interesting extra assumption once $2^{\aleph_0} > \aleph_1$ , so it is almost always stated together with $\neg\mathrm{CH}$ .

\mathrm{MA}_\kappa: \forall \text{ ccc } \mathbb{P},\ \forall \mathcal{D} \ (|\mathcal{D}| \le \kappa \text{ dense sets}),\ \exists \text{ filter meeting all of } \mathcal{D}

Detailed analysis

Martin's Axiom at cardinal $\kappa$ , written $\mathrm{MA}_\kappa$ , states: for every ccc poset $\mathbb{P}$ and every family $\mathcal{D}$ of at most $\kappa$ dense subsets of $\mathbb{P}$ , there is a filter $G \subseteq \mathbb{P}$ meeting every member of $\mathcal{D}$ . " $\mathrm{MA}$ " alone means $\mathrm{MA}_\kappa$ holds for every $\kappa < 2^{\aleph_0}$ .

$\mathrm{MA}_{\aleph_0}$ is a theorem of $\mathrm{ZFC}$ (a Rasiowa–Sikorski style genericity argument, the same one behind Cohen's own construction), so $\mathrm{MA}$ only says something new once $2^{\aleph_0} > \aleph_1$ , and under $\mathrm{CH}$ itself $\mathrm{MA}_{\aleph_1}$ is provably false. Donald Martin isolated the principle while studying analytic sets and measure-theoretic problems in the late 1960s; combined with $\neg\mathrm{CH}$ it has an enormous range of combinatorial consequences, making $\aleph_1$ behave, for ccc-related purposes, almost like $\aleph_0$ .

Crucially, since any candidate Suslin tree, ordered by extension (reversed), is itself a ccc poset, $\mathrm{MA} + \neg\mathrm{CH}$ guarantees enough generic-like filters to defeat every one of them at once — precisely the mechanism the next step turns into an actual model.

Terms in this step

Martin's Axiom (MA): The statement that, for every ccc poset and every collection of fewer than $2^{\aleph_0}$ dense subsets of it, a filter meeting all of them exists inside the universe itself, without needing to force to a bigger one.

Knowledge used in this step

The ZFC axioms