Martin's axiom

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, consider the following statement: MA(κ) For any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense subsets of P such that |D| ≤ κ, there is a filter F on P such that F ∩ d is non-empty for every d in D. 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. MA(ℵ0) is simply true — the Rasiowa–Sikorski lemma. MA(2ℵ0) is false: [0, 1] is a separable compact Hausdorff space, and so (P, the poset of open subsets under inclusion, is) ccc. But now consider the following two size-2ℵ0= families of dense sets in P: no x∈[0, 1] is isolated, and so each x defines the dense subset {S : x∉S}. And each r∈(0, 1], defines the dense subset {S : diam(S)