Definition
Martin's axiom (MA) is a statement in set theory asserting that for any partially ordered set (poset) satisfying the countable chain condition (c.c.c.) and any collection of fewer than $2^{\aleph_{0}}$ dense subsets of that poset, there exists a filter intersecting every dense set in the collection. Formally, for a cardinal $\kappa < 2^{\aleph_{0}}$, MA$\kappa$ says: if $\mathbb{P}$ is a c.c.c. poset and ${D\alpha : \alpha < \kappa}$ is a family of dense subsets of $\mathbb{P}$, then there is a filter $G \subseteq \mathbb{P}$ such that $G \cap D_\alpha
eq \varnothing$ for all $\alpha < \kappa$. The full axiom, often simply called MA, is the conjunction of MA$\kappa$ for all $\kappa < 2^{\aleph{0}}$.
Overview
Martin's axiom was introduced in the 1970s as a combinatorial principle that lies strictly between the Zermelo–Fraenkel axioms with the Axiom of Choice (ZFC) and the Continuum Hypothesis (CH). It is consistent with ZFC, assuming ZFC itself is consistent, and is independent of both CH and its negation. MA has profound consequences for the structure of the real line, the behavior of topological spaces, and the theory of forcing. Under MA, many pathological constructions possible under CH are ruled out, while still allowing the continuum to be larger than $\aleph_{1}$. For example, MA implies that every compact Hausdorff space of weight less than the continuum is metrizable, and it yields strong partition properties for subsets of $\mathbb{R}$.
Etymology/Origin
The axiom is named after the American logician Donald A. Martin, who formulated it in collaboration with Robert M. Solovay. Their work appeared in the early 1970s, building on earlier results concerning the Baire category theorem and the behavior of c.c.c. forcing notions.
Characteristics
- Consistency Strength: MA is provably consistent relative to ZFC; it can be added to ZFC by a standard forcing construction that preserves ZFC.
- Relation to the Continuum Hypothesis: MA is compatible with both the negation of CH (e.g., assuming $2^{\aleph_{0}} = \aleph_{2}$) and with CH itself, though the full axiom (for all $\kappa < 2^{\aleph_{0}}$) contradicts CH when $2^{\aleph_{0}} > \aleph_{1}$.
- c.c.c. Posets: The axiom applies exclusively to posets with the countable chain condition, ensuring that antichains are at most countable. This restriction is crucial for its combinatorial power.
- Consequences in Topology: MA implies that every countable chain condition (c.c.c.) compact space is separable, that every Aronszajn tree is special, and that the product of $\aleph_{1}$ many separable metric spaces is again separable.
- Variants: Weaker forms such as MA${\aleph{1}}$ (Martin’s axiom for $\kappa = \aleph_{1}$) are often studied. Stronger principles, like the Proper Forcing Axiom (PFA), extend the ideas of MA to broader classes of forcing notions.
Related Topics
- Continuum Hypothesis (CH)
- Forcing and generic extensions
- Countable Chain Condition (c.c.c.)
- Proper Forcing Axiom (PFA)
- Suslin’s hypothesis
- Set-theoretic topology (e.g., separability, metrizability)
- Aronszajn trees and special trees
- Baire category theorem