In the mathematical field of set theory, Martin's axiom, introduced by, 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 forcingarguments.
Statement of Martin's axiom
For any cardinal k, we define a statement, denoted by MA:
For any partial orderP satisfying the countable chain condition and any family D of dense sets in P such that |D| ≤ k, there is a filterF on P such that F ∩ d is non-empty for every d in D.
Since it is a theorem of ZFC that MA fails, Martin's axiom is stated as:
Martin's axiom : For every k <, MA holds.
In this case, an antichain is a subsetA of P such that any two distinct members of A are incompatible. This differs from, for example, the notion of antichain in the context of trees. MA is simply true. This is known as the Rasiowa–Sikorski lemma. MA is false: is a compactHausdorff 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.
If X is a compact Hausdorff topological space that satisfies the ccc then X is not the union of k or fewer nowhere dense subsets.
If P is a non-empty upwards ccc poset and Y is a family of cofinal subsets of P with |Y| ≤ k then there is an upwards-directed set A such that A meets every element ofY.
Let A be a non-zero ccc Boolean algebra and F a family of subsets of A with |F| ≤ k. Then there is a boolean homomorphism φ: A → Z/2Z such that for every X in F either there is an a in X with φ = 1 or there is an upper boundb for X with φ = 0.
The union of k or fewer null sets in an atomless σ-finite Borel measure on a Polish space is null. In particular, the union of k or fewer subsets of R of Lebesgue measure 0 also has Lebesgue measure 0.
A compact Hausdorff space X with |X| < 2k is sequentially compact, i.e., every sequence has a convergent subsequence.
No non-principal ultrafilter on N has a base of cardinality < k.
Equivalently for any x in βN\N we have χ ≥ k, where χ is the character of x, and so χ ≥ k.
