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About: Baumgartner's axiom     Goto   Sponge   NotDistinct   Permalink

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In mathematical set theory, Baumgartner's axiom (BA) can be one of three different axioms introduced by James Earl Baumgartner. A subset of the real line is said to be -dense if every two points are separated by exactly other points, where is the smallest uncountable cardinality. This would be true for the real line itself under the continuum hypothesis. An axiom introduced by states that all -dense subsets of the real line are order-isomorphic, providing a higher-cardinality analogue of Cantor's isomorphism theorem that countable dense subsets are isomorphic. Baumgartner's axiom is a consequence of the proper forcing axiom. It is consistent with a combination of ZFC, Martin's axiom, and the negation of the continuum hypothesis, but not implied by those hypotheses.