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 formula_1-dense if every two points are separated by exactly formula_1 other points, where formula_1 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 formula_1-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.
Another axiom introduced by states that Martin's axiom for partially ordered sets MA"P"("κ") is true for all partially ordered sets "P" that are countable closed, well met and ℵ1-linked and all cardinals κ less than 2ℵ1.
Baumgartner's axiom A is an axiom for partially ordered sets introduced in . A partial order ("P", ≤) is said to satisfy axiom A if there is a family ≤"n" of partial orderings on "P" for "n" = 0, 1, 2, ... such that 
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