In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, , behave roughly like . The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments.
For any cardinal 𝛋, we define a statement, denoted by MA(𝛋):
For any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P such that |D| ≤ 𝛋, there is a filter F on P such that F ∩ d is non-empty for every d in D.
Since it is a theorem of ZFC that fails, Martin's axiom is stated as:
Martin's axiom (MA): For every 𝛋 < , MA(𝛋) holds.
In this case (for application of ccc), an antichain is a subset A of P such that any two distinct members of A are incompatible (two elements are said to be compatible if there exists a common element below both of them in the partial order). This differs from, for example, the notion of antichain in the context of trees.
is simply true. This is known as the Rasiowa–Sikorski lemma.
is false: [0, 1] is a compact Hausdorff space, which is separable and so ccc. It has no isolated points, so points in it are nowhere dense, but it is the union of many points. (See the condition equivalent to below.)
The following statements are equivalent to MA(𝛋):