In set theory, a Kurepa tree is a tree (T, <) of height ω1, each of whose levels is at most countable, and has at least ℵ2 many branches. This concept was introduced by Kurepa. The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe. More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, Silver showed that if a strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is in fact equiconsistent wit
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| - Kurepa tree (en)
- クレパ木 (ja)
- Hipoteza Kurepy (pl)
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| - 集合論において、クレパ木とは高さ の木(T, <)であって、全ての水準 (Level) の濃度が高々可算で、かつ少なくとも 本以上の枝をもつ木のこと。ユーゴスラビア人数学者ジューロ・クレパの名に因む。クレパ木の存在性(クレパの仮説(KH))は、ZFCと矛盾しないことが知られている。ソロヴェイは未発表の論文の中で、ゲーデルの構成的宇宙 L (構成可能集合全体からなるクラス)にクレパ木が存在することを示した（より正確には、ダイヤモンド原理からクレパ木の存在が従うことを示した）。一方、シルバーが1971年に示したように、強到達不能基数が へレヴィ崩壊しているとき、そのモデルではクレパ木が存在しない。実際には到達不能基数の存在とクレパの仮説の否定はであることが知られている。 (ja)
- Hipoteza Kurepy, KH (od ang. Kurepa hypothesis) – zdanie teorii mnogości postulujące istnienie obiektów nazywanych drzewami Kurepy. Jest ono niezależne od standardowych aksjomatów ZFC (nie można go udowodnić ani obalić na gruncie tych aksjomatów). (pl)
- In set theory, a Kurepa tree is a tree (T, <) of height ω1, each of whose levels is at most countable, and has at least ℵ2 many branches. This concept was introduced by Kurepa. The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe. More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, Silver showed that if a strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is in fact equiconsistent wit (en)
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| - In set theory, a Kurepa tree is a tree (T, <) of height ω1, each of whose levels is at most countable, and has at least ℵ2 many branches. This concept was introduced by Kurepa. The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay showed in unpublished work that there are Kurepa trees in Gödel's constructible universe. More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, Silver showed that if a strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is in fact equiconsistent with the failure of the Kurepa hypothesis, because if the Kurepa hypothesis is false then the cardinal ω2 is inaccessible in the constructible universe. A Kurepa tree with fewer than 2ℵ1 branches is known as a Jech–Kunen tree. More generally if κ is an infinite cardinal, then a κ-Kurepa tree is a tree of height κ with more than κ branches but at most |α| elements of each infinite level α<κ, and the Kurepa hypothesis for κ is the statement that there is a κ-Kurepa tree. Sometimes the tree is also assumed to be binary. The existence of a binary κ-Kurepa tree is equivalent to the existence of a Kurepa family: a set of more than κ subsets of κ such that their intersections with any infinite ordinal α<κ form a set of cardinality at most α. The Kurepa hypothesis is false if κ is an ineffable cardinal, and conversely Jensen showed that in the constructible universe for any uncountable regular cardinal κ there is a κ-Kurepa tree unless κ is ineffable. (en)
- 集合論において、クレパ木とは高さ の木(T, <)であって、全ての水準 (Level) の濃度が高々可算で、かつ少なくとも 本以上の枝をもつ木のこと。ユーゴスラビア人数学者ジューロ・クレパの名に因む。クレパ木の存在性(クレパの仮説(KH))は、ZFCと矛盾しないことが知られている。ソロヴェイは未発表の論文の中で、ゲーデルの構成的宇宙 L (構成可能集合全体からなるクラス)にクレパ木が存在することを示した（より正確には、ダイヤモンド原理からクレパ木の存在が従うことを示した）。一方、シルバーが1971年に示したように、強到達不能基数が へレヴィ崩壊しているとき、そのモデルではクレパ木が存在しない。実際には到達不能基数の存在とクレパの仮説の否定はであることが知られている。 (ja)
- Hipoteza Kurepy, KH (od ang. Kurepa hypothesis) – zdanie teorii mnogości postulujące istnienie obiektów nazywanych drzewami Kurepy. Jest ono niezależne od standardowych aksjomatów ZFC (nie można go udowodnić ani obalić na gruncie tych aksjomatów). (pl)
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