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 [math]\displaystyle{ \aleph_1 }[/math]-dense if every two points are separated by exactly [math]\displaystyle{ \aleph_1 }[/math] other points, where [math]\displaystyle{ \aleph_1 }[/math] 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 [math]\displaystyle{ \aleph_1 }[/math]-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 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). 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
- ≤0 is the same as ≤
- If p ≤n+1q then p ≤nq
- If there is a sequence pn with pn+1 ≤n pn then there is a q with q ≤n pn for all n.
- 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
- ↑ Baumgartner, James E. (1973), "All [math]\displaystyle{ \aleph_{1} }[/math]-dense sets of reals can be isomorphic", Fundamenta Mathematicae 79 (2): 101–106, doi:10.4064/fm-79-2-101-106
- ↑ Avraham, Uri; Shelah, Saharon (1981), "Martin's axiom does not imply that every two [math]\displaystyle{ \aleph_{1} }[/math]-dense sets of reals are isomorphic", Israel Journal of Mathematics 38 (1-2): 161–176, doi:10.1007/BF02761858
- Baumgartner, James E. (1975), Generalizing Martin's axiom, unpublished manuscript
- Baumgartner, James E. (1983), "Iterated forcing", in Mathias, A. R. D., Surveys in set theory, London Math. Soc. Lecture Note Ser., 87, Cambridge: Cambridge Univ. Press, pp. 1–59, ISBN 0-521-27733-7, https://books.google.com/books?id=6VSgYq-4kK4C&pg=PA34
- Kunen, Kenneth (2011), Set theory, Studies in Logic, 34, London: College Publications, ISBN 978-1-84890-050-9
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