Baumgartner's axiom

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 ${\displaystyle \aleph _{1}}$-dense if every two points are separated by exactly ${\displaystyle \aleph _{1}}$ other points, where ${\displaystyle \aleph _{1}}$ is the smallest uncountable cardinality. This would be true for the real line itself under the continuum hypothesis. An axiom introduced by Baumgartner (1973) states that all ${\displaystyle \aleph _{1}}$-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,[1] but not implied by those hypotheses.[2]

Another axiom introduced by Baumgartner (1975) 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 2^ℵ₁.

Baumgartner's axiom A is an axiom for partially ordered sets introduced in (Baumgartner 1983, section 7). 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

1. ≤₀ is the same as ≤
2. If p ≤_n+1q then p ≤_nq
3. If there is a sequence p_n with p_n+1 ≤_n p_n then there is a q with q ≤_n p_n for all n.
4. If I is a pairwise incompatible subset of P then for all p and for all natural numbers n there is a q such that q ≤_n p and the number of elements of I compatible with q is countable.