Martin's axiom

In themathematicalfield ofset theory,Martin's axiom,introduced byDonald A. MartinandRobert M. Solovay,is a statement that is independent of the usual axioms ofZFC set theory.It is implied by thecontinuum hypothesis,but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than thecardinality of the continuum,𝔠, behave roughly like ℵ₀.The intuition behind this can be understood by studying the proof of theRasiowa–Sikorski lemma.It is a principle that is used to control certainforcingarguments.

Statement

For a cardinal numberκ,define the following statement:

MA(κ)

For anypartial orderPsatisfying thecountable chain condition(hereafter ccc) and any setD= {D_i}_i∈Iof dense subsets ofPsuch that|D|≤κ,there is afilterFonPsuch thatF∩D_iis non-emptyfor everyD_i∈D.

In this context, a setDis called dense if every element ofPhas a lower bound inD.For application of ccc, an antichain is a subsetAofPsuch that any two distinct members ofAare 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 oftrees.

MA(ℵ₀) is provable in ZFC and known as theRasiowa–Sikorski lemma.

MA(2^ℵ₀) is false: [0, 1] is aseparablecompactHausdorff 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 inP:nox∈ [0, 1] isisolated,and so eachxdefines the dense subset {S|x∉S}. And eachr∈ (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 filterFcontaining sets of arbitrarily small diameter must contain a point in ⋂Fby compactness. (See also§ Equivalent forms of MA(κ).)

Martin's axiom is then that MA(κ) holds for everyκfor which it could:

Martin's axiom (MA)

MA(κ) holds for everyκ< 𝔠.

Equivalent forms of MA(κ)

The following statements are equivalent to MA(κ):

• IfXis a compact Hausdorfftopological spacethat satisfies thecccthenXis not the union ofκor fewernowhere densesubsets.
• IfPis a non-empty upwards cccposetandYis a set of cofinal subsets ofPwith|Y|≤κthen there is an upwards-directed setAsuch thatAmeets every element ofY.
• LetAbe a non-zero cccBoolean algebraandFa set of subsets ofAwith|F|≤κ.Then there is a boolean homomorphism φ:A→Z/2Zsuch that for everyX∈F,there is either ana∈Xwith φ(a) = 1 or there is an upper boundb∈Xwith φ(b) = 0.

Consequences

Martin's axiom has a number of other interestingcombinatorial,analyticandtopologicalconsequences:

• The union ofκor fewernull setsin an atomless σ-finiteBorel measureon aPolish spaceis null. In particular, the union ofκor fewer subsets ofRofLebesgue measure0 also has Lebesgue measure 0.
• A compact Hausdorff spaceXwith|X|< 2^κissequentially compact,i.e., every sequence has a convergent subsequence.
• No non-principalultrafilteronNhas a base of cardinality less thanκ.
• Equivalently for anyx∈ βN\Nwe have 𝜒(x) ≥κ,where 𝜒 is thecharacterofx,and so 𝜒(βN) ≥κ.
• MA(ℵ₁) implies that a product of ccc topological spaces is ccc (this in turn implies there are noSuslin lines).
• MA + ¬CH implies that there exists a Whitehead group that is not free;Shelahused this to show that theWhitehead problemis independent of ZFC.

Further development

• Martin's axiom has generalizations called theproper forcing axiomandMartin's maximum.
• Sheldon W. Davis has suggested in his book that Martin's axiom is motivated by theBaire category theorem.

Further reading

• Fremlin, David H. (1984).Consequences of Martin's axiom.Cambridge tracts in mathematics, no. 84. Cambridge:Cambridge University Press.ISBN0-521-25091-9.
• Jech, Thomas,2003.Set Theory: The Third Millennium Edition, Revised and Expanded.Springer.ISBN3-540-44085-2.
• Kunen, Kenneth,1980.Set Theory: An Introduction to Independence Proofs.Elsevier.ISBN0-444-86839-9.