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In mathematical set theory, Baumgartner's axiom (BA) can be one of three different axioms introduced by James Earl Baumgartner. An axiom introduced by states that any two ℵ1-dense subsets of the real line are isomorphic. Another axiom introduced by states that Martin's axiom for partially ordered sets MA''P''(''κ'') is true for all posets ''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 posets 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 # ≤0 is the same as ≤ #If ''p'' ≤''n''+1''q'' then ''p'' ≤''n''''q'' #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''. #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== * * * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Baumgartner's axiom」の詳細全文を読む スポンサード リンク
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