Martin's axiom

In the mathematical field of set theory, Martin's axiom, named after Donald A. Martin, 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 + not 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.

The various statements of Martin's axiom typically take two parts. MA(k) is the assertion that for any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P, with |D| at most k, there is a filter F on P such that F ∩ d is non-empty for every d ε D. MA is then the statement 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.

David Fremlin also states this in terms of the cardinal invariant ${\mathfrak {m}}$, defined to be the least k such that MA(k) is false. MA(k) is then the statement that ${\mathfrak {m}}$ > k, and MA that ${\mathfrak {m}}$ = ${\mathfrak {c}}$.

Various equivalent forms of MA(k) follow:

• 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 P.

• 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 $\phi :A\to {\mathbb {Z}}_{2}$ such that for every $X\in F$ either there is an $a\in X$ with $\phi (a)=1$ or there is an upper bound b for X with $\phi (b)=0$.

MA($\aleph _{0}$) is simply true. This is known as the Rasiowa-Sikorski lemma.

Note that 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.

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

Consequences of MA(k):

• The union of k or fewer null sets in an atomless $\sigma$-finite Borel measure on a Polish space is null. In particular, the union of k or fewer subsets of ${\mathbb {R}}$ 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 ${\mathbb {N}}$ has a base of cardinality < k.

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

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 undecidable.

References

• 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.