Martin's axiom

The following statements are equivalent to Martin's axiom:

(Freiling used a similar argument to claim that Martin's axiom is false.)

Partial orders and spaces satisfying the ccc are used in the statement of Martin's Axiom.

In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin's axiom.

Martin's Axiom is consistent with .

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

This is why MM is called the maximal extension of Martin's axiom.

If Martin's axiom and the negation of the continuum hypothesis both hold, then there is a non-free Whitehead group.

Martin's axiom has generalizations called the proper forcing axiom and Martin's maximum.

PFA directly implies its version for ccc forcings, Martin's axiom.

The Rasiowa-Sikorski lemma can be viewed as a weaker form of an equivalent to Martin's axiom.

Martin's axiom, a weakening of CH, implies that all cardinals in the diagram (except perhaps ) are equal to .

Martin's axiom (which is not a ZFC axiom)

Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom.

If D is uncountable, but of cardinality strictly smaller than and the poset has the countable chain condition, we can instead use Martin's axiom.

Martin's axiom plus the negation of the continuum hypothesis, Whitehead's problem cannot be resolved in ZFC.

Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.

Sheldon W.Davis has suggested that Martin's axiom is motivated by Baire category theorem in his book.

On the other hand, if one assumes Martin's Axiom (MA) all common invariants are "big", that is equal to , the cardinality of the continuum.

Assuming Martin's Axiom and the negation of the continuum hypothesis, there are no Luzin spaces (or Luzin sets).

Some of these conjectures are provable with the addition of axioms such as Martin's axiom, large cardinal axioms to ZFC.

In set theory, Martin's maximum, introduced by , is a generalization of the proper forcing axiom, which is in turn a generalization of Martin's axiom.

It starts from basic notions, including the ZFC axioms, and quickly develops combinatorial notions such as trees, Suslin's problem, , and Martin's axiom.

The constructible universe satisfies the Generalized Continuum Hypothesis, the Diamond Principle, Martin's Axiom and the Kurepa Hypothesis.

In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory.