Martin's axiom

In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay (1970), is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH. Indeed, it is only really of interest when the continuum hypothesis fails (otherwise it adds nothing to ZFC). It can informally be considered to say that all cardinals less than the cardinality of the continuum, ${\mathfrak c}$, behave roughly like $\aleph_0$. The intuition behind this can be understood by studying the proof of the Rasiowa-Sikorski lemma. More formally it is a principle that is used to control certain forcing arguments.

Statement of Martin's axiom

The various statements of Martin's axiom typically take two parts. MA(k) says that for any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P, whose cardinality | D | is at most k, there is a filter F on P such that F ∩ d is non-empty for every $d \in D$. MA, then, says that MA(k) holds for every k less than the continuum. (It is a theorem of ZFC that MA(${\mathfrak c}$) fails.) Note that, 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($\aleph_0$) is simply true. This is known as the Rasiowa-Sikorski lemma.

MA($2^{\aleph_0}$) 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 $2^{\aleph_0}$ many points.

Equivalent forms of MA(k)

The following statements are equivalent to Martin's axiom:

• If X is a compact Hausdorff topological space which 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 $|Y| \leq k$ then there is an upwards directed set A such that A meets every element of Y.

• Let A be a non-zero ccc Boolean algebra and F a family of subsets of A with $|F| \leq k$. Then there is a boolean homomorphism $\phi : A \to \Bbb{Z}_2$ such that for every $X \in F$ either there is an $a \in X$ with ϕ(a) = 1 or there is an upper bound b for X with ϕ(b) = 0.

Consequences of MA(k)

Martin's axiom has a number of other interesting combinatorial, analytic and topological consequences:

• 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 $\Bbb R$ of Lebesgue measure 0 also has Lebesgue measure 0.

• A compact Hausdorff space X with | X | < 2^k is sequentially compact, i.e., every sequence has a convergent subsequence.

• No non-principal ultrafilter on $\Bbb{N}$ has a base of cardinality < k.

• Equivalently for any $x \in \beta \Bbb{N} \setminus \Bbb{N}$: $\chi(x) \geq k$, where χ is the character of x, and so $\chi(\beta \Bbb{N}) \geq k$.

MA($\aleph_1$) is particularly interesting. Some consequences include:

• A product of ccc topological spaces is ccc (this in turn implies there are no Suslin lines).

MA together with the negation of the continuum hypothesis implies:

• There exists a Whitehead group that is not free; Shelah used this to show that the Whitehead problem is independent of ZFC.

See also

• Martin's axiom has generalizations called the proper forcing axiom and Martin's maximum.
• Sheldon W.Davis has suggested that Martin's axiom is motivated by Baire category theorem in his book.

References

• Fremlin, David H. (1984). Consequences of Martin's axiom. Cambridge tracts in mathematics, no. 84. Cambridge: Cambridge University Press. ISBN 0521250919. 
• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
  • Martin, D. A.; Solovay, R. M. (1970), "Internal Cohen extensions.", Ann. Math. Logic 2 (2): 143–178, doi:10.1016/0003-4843(70)90009-4, MR0270904