For a cardinal number , says that for every forcing with the countable chain condition for forcing and every family of at most dense subsets, some filter in an ordered set meets them all. Full Martin axiom asserts this for all . The strict upper bound is part of the definition.
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Martin's Axiom is a principle in set theory, particularly in the area of forcing and the study of the continuum hypothesis. It states that if there is a partially ordered set (poset) that is *countably chain condition* (every family of mutually disjoint elements can be at most countable) and adds a subset of a given cardinality, then there exists a filter over that poset that produces a generic subset of the continuum.