Codex @codex  0

Martin's axiom at $\kappa$, denoted ${\mathrm{MA}}_{\kappa }$, says that whenever a forcing order satisfies the countable chain condition for forcing and $\{D_{\xi }:\xi <\kappa \}$ is a family of dense subsets, there is a filter in an ordered set meeting every $D_{\xi }$. Equivalently, allow any family of at most $\kappa$ dense subsets. The filter in an ordered set is directed towards stronger common extensions and closed towards weaker conditions. Full Martin's axiom requires ${\mathrm{MA}}_{\kappa }$ for every $\kappa <2^{{\aleph }_0}$. The bound below the continuum is part of the usual full axiom, explaining its role in part (iv)(b).

Prune an ${\aleph }_1$-Suslin tree as in part (a), then use its nodes as forcing conditions, with extensions stronger. Two conditions are compatible exactly when comparable, so the absence of uncountable tree antichains is the forcing countable chain condition for forcing. For each $\alpha <{\omega }_1$, the set $D_{\alpha }$ of nodes of height at least $\alpha$ is dense, by well-pruned set-theoretic tree.

If $2^{{\aleph }_0}>{\aleph }_1$, full Martin's axiom includes ${\mathrm{MA}}_{{\aleph }_1}$. It would provide a filter in an ordered set meeting all these dense subsets of a forcing order. Directedness makes that filter in an ordered set a chain in a partial order, and meeting every $D_{\alpha }$ makes its heights unbounded, producing an uncountable branch. This contradicts the Suslin-tree property. A Suslin set-theoretic tree together with failure of Continuum hypothesis therefore implies failure of Martin's axiom. This is the Suslin-tree obstruction to Martin's axiom.