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 ℵ1-dense subsets of the real line are order-isomorphic. Todorcevic showed that this Baumgartner's Axiom is a consequence of the Proper Forcing Axiom[1].

Another axiom introduced by Baumgartner (1975) states that Martin's axiom for partially ordered sets MAP(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 (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
  1. Todorcevic's proof of Baumgartner Axiom by Garrett Ervin
  • Baumgartner, James E. (1973), "All ℵ1-dense sets of reals can be isomorphic" (PDF), Fundamenta Mathematicae, 79 (2): 101–106, doi:10.4064/fm-79-2-101-106, MR 0317934
  • Baumgartner, James E. (1975), Generalizing Martin's axiom, unpublished manuscript
  • Baumgartner, James E. (1983), "Iterated forcing", in Mathias, A. R. D. (ed.), Surveys in set theory, London Math. Soc. Lecture Note Ser., 87, Cambridge: Cambridge Univ. Press, pp. 1–59, ISBN 0-521-27733-7, MR 0823775
  • Kunen, Kenneth (2011), Set theory, Studies in Logic, 34, London: College Publications, ISBN 978-1-84890-050-9, MR 2905394, Zbl 1262.03001
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.