Martin's axiom

Martin's axiom states that if a partially ordered set satisfies the countable chain condition, then any family of fewer than continuum many dense subsets has a common filter meeting every member.

Forcing: A method for extending models of set theory by adjoining generic objects through partially ordered sets. Martin's axiom abstracts the dense-set argument used to construct generic filters.
Baire category theorem: A theorem stating that complete metric spaces cannot be countable unions of nowhere-dense sets. Martin's axiom extends related category arguments to unions of fewer than continuum many sets.
Donald A. Martin: An American mathematician known for work in set theory, descriptive set theory, and mathematical logic. Martin's name is attached to the axiom developed from forcing arguments.
Martin's axiom for sigma-centered posets: The restriction of Martin's axiom to partially ordered sets that are countable unions of centered subsets. This weaker form isolates how much follows from a narrower class of forcing notions.
Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Conditions in the axiom's forcing formulation are ordered by extension.
Suslin's hypothesis: The assertion that every dense, complete, densely ordered linear order without endpoints is isomorphic to the real line. Martin's axiom rules out Suslin lines, a central consequence in order theory.
Robert M. Solovay: An American mathematician whose work helped shape modern set theory and the theory of forcing. Solovay and Martin introduced the principle in their study of forcing.
Proper forcing axiom: A forcing axiom asserting filter existence for proper partial orders and families of fewer than continuum many dense sets. It strengthens Martin's axiom by replacing the ccc class with proper forcing.
Dense subset: A subset of a partially ordered set in which every condition has a stronger condition. The axiom asks for a filter meeting every set in the chosen dense family.
Cardinality of the continuum: The cardinal number of the set of real numbers. Martin's axiom controls the number of dense sets relative to this cardinal.

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