Baumgartner's axiom

In mathematical set theory, Baumgartner's axiom (BA) can be one of three different axioms introduced by James Earl Baumgartner.

An axiom introduced by Baumgartner (1973) states that any two 1-dense subsets of the real line are isomorphic.

Another axiom introduced by Baumgartner (1975) states that Martin's axiom for partially ordered sets MAP(κ) is true for all posets P that are countable closed, well met and ℵ1-linked and all cardinals κ less than 21.

Baumgartner's axiom A is an axiom for posets 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 pn with pn+1 n pn then there is a q with q n pn 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.

References

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