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In set theory, an Aronszajn tree is an uncountable tree with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ such that all levels have size less than κ and all branches have height less than κ (so Aronszajn trees are the same as -Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by . A cardinal κ for which no κ-Aronszajn trees exist is said to have the tree property. (sometimes the condition that κ is regular and uncountable is included.) ==Existence of κ-Aronszajn trees== König's lemma states that -Aronszajn trees do not exist. The existence of Aronszajn trees (=-Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of König's lemma does not hold for uncountable trees. The existence of -Aronszajn trees is undecidable (assuming a certain large cardinal axiom): more precisely, the continuum hypothesis implies the existence of an -Aronszajn tree, and Mitchell and Silver showed that it is consistent (relative to the existence of a weakly compact cardinal) that no -Aronszajn trees exist. Jensen proved that V=L implies that there is a κ-Aronszajn tree (in fact a κ-Suslin tree) for every infinite successor cardinal κ. showed (using a large cardinal axiom) that it is consistent that no -Aronszajn trees exist for any finite ''n'' other than 1. If κ is weakly compact then no κ-Aronszajn trees exist. Conversely if κ is inaccessible and no κ-Aronszajn trees exist then κ is weakly compact. 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「Aronszajn tree」の詳細全文を読む スポンサード リンク
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