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In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay,[1] 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, 𝔠, behave roughly like ℵ0. 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 a cardinal number κ, define the following statement:
In this context, a set D is called dense if every element of P has a lower bound in D. 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 provable in ZFC and known as 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 𝔠-size sets 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) < r }. The two sets combined are also of size 𝔠, and a filter meeting both must simultaneously avoid all points of [0, 1] while containing sets of arbitrarily small diameter. But a filter F containing sets of arbitrarily small diameter must contain a point in ⋂F by compactness. (See also § Equivalent forms of MA(κ).)
Martin's axiom is then that MA(κ) holds for every κ for which it could:
The following statements are equivalent to MA(κ):
Martin's axiom has a number of other interesting combinatorial, analytic and topological consequences:
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