## Determinacy and the tree property (2003)

### Citations

690 |
Set theory
- Jech
- 1978
(Show Context)
Citation Context ... height δ all of whose levels have size < δ has a cofinal branch. Hence König’s Lemma states that ω has the tree property, whereas Aronszajn himself discovered that there is an Aronszajn ω1-tree (cf. =-=[4]-=- on this and other background information). In his thesis (cf. [6]), Mitchell had shown that, e.g., any successor cardinal > ω1 may have the tree property, starting from a weakly compact cardinal, and... |

158 |
The fine structure of the constructible hierarchy
- Jensen
- 1972
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Citation Context ...e σ: KA → W being discontinuous at ω +KA 2 , contadicting a theorem of Steel. Claim 1. A ♯ exists for every A ⊂ ω2. Proof. First let A be a bounded subset of ω2, A ⊂ α < ω2, say. Jensen has shown (in =-=[5]-=-) that �κ holds in L[A] for any κ ≥ α. Also, the GCH holds in L[A] above α, so that in L[A], there is a special Aronszajn κ + -tree for any κ ≥ α (cf. [5] p. 283). Hence if ω2 were a successor cardina... |

57 |
Fine structure and iteration trees
- Mitchell, Steel
- 1994
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Citation Context ...remouse which is not n-small, i.e., which has an extender with critical point κ on its sequence such that J M κ |= there are n Woodin cardinals. An x-premouse N generalizes a premouse in the sense of =-=[8]-=- (which would then be a ∅-premouse) in that N is constructed over the real x instead of over ∅. However, to keep the induction going, we need a generalization of the M ♯ n-operator for arbitrary sets.... |

55 |
Aronszajn trees and the independence of the transfer property,
- MITCHELL
- 1972
(Show Context)
Citation Context ...Hence König’s Lemma states that ω has the tree property, whereas Aronszajn himself discovered that there is an Aronszajn ω1-tree (cf. [4] on this and other background information). In his thesis (cf. =-=[6]-=-), Mitchell had shown that, e.g., any successor cardinal > ω1 may have the tree property, starting from a weakly compact cardinal, and he had asked whether for example ω2 and ω3 may simultaneously adm... |

28 |
Marginalia to a theorem of Silver
- Devlin, Jensen
- 1975
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Citation Context ... ω +L[A] 2 < ω3. But this and the existence of Ā♯ for all bounded subsets Ā of ω2 implies the existence of A♯ as follows. Pick π: Lτ[ Ā] → Lβ[A] as in the proof of the Jensen Covering Lemma (cf. e.g. =-=[3]-=-). Then π can be extended in the usual fashion to ˜π: L[ Ā] → L[A], as otherwise β could be collapsed onto ω2 in L[A]. Then it is easily seen that there is a stationary class S of Ā-indiscernibles con... |

25 | Combinatorial principles in the core model for one Woodin cardinal
- Schimmerling
- 1995
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Citation Context ...itial segment of M and L[A] |= M is κ-strong with ρω M = κ. Clearly, On ∩ M ∗ ≤ ω3. Subclaim 2. On ∩ M ∗ < ω3. Proof. Suppose that On ∩ M ∗ = ω3. This means that κ is the largest cardinal in M ∗ . By =-=[10]-=-, � <ω κ holds in every K A , A ≥c B. However, as noted in [12], the proof of this fact is ”local,” so that the different � <ω κ -sequences so obtained can be neated together so as to give � <ω κ in M... |

18 |
Weak covering without countable closure
- Mitchell, Schimmerling
- 1995
(Show Context)
Citation Context ... may pick some A ≥c A¯ ξ for all ¯ ξ < ξ. If we then set Aξ = A then (1)ξ an (2)ξ are easily checked. 9sNow let A = Aω2+2, and set Q = J KA ω ++KA 2 � (Claim 2) . Clearly, M has size ω2 in V , and by =-=[7]-=-, cf V (ω +KA 2 ) = ω2. So the tree T from the proof of Lemma 2.1, searching for a countably complete π: KA → W being discontinuous at ω +KA 2 exists in some L[B], B ≥c A, and has a cofinal branch in ... |

18 |
Optimal proofs of determinacy
- Neeman
- 1995
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Citation Context ... going to prove 1.1 and 1.2. In order to establish Π 1 n+1 Determinacy it suffices to prove that for any x ∈ R, M ♯ n(x) exists. (Cf. [17] or below for a definition of M ♯ n(x).) To give a reference, =-=[9]-=- Theorem 2.5 states a slightly stronger, best possible, result which in particular says that M ♯ n(x) gives Π 1 n+1(x)-determinacy. In fact, we use (the boldface version of) [9] Corollary 2.3 to deduc... |

13 |
Aronszajn Trees on ℵ2 and ℵ3
- Abraham
- 1983
(Show Context)
Citation Context ...e property, starting from a weakly compact cardinal, and he had asked whether for example ω2 and ω3 may simultaneously admit the tree property. This was in fact shown to be consistent by Abraham (cf. =-=[1]-=-) later on, starting from a supercompact cardinal and a weakly compact above. (We also remark that in Abraham’s model 2 ℵ0 = ℵ2.) More recently, Cummings and Foreman generalized Abraham’s construction... |

9 |
Projectively well-ordered inner models
- Steel
- 1995
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Citation Context ....14 (3). 4 Getting Determinacy. � (Theorem 3.1) We are now finally going to prove 1.1 and 1.2. In order to establish Π 1 n+1 Determinacy it suffices to prove that for any x ∈ R, M ♯ n(x) exists. (Cf. =-=[17]-=- or below for a definition of M ♯ n(x).) To give a reference, [9] Theorem 2.5 states a slightly stronger, best possible, result which in particular says that M ♯ n(x) gives Π 1 n+1(x)-determinacy. In ... |

4 |
The core model up to one strong cardinal
- Schindler
- 1997
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Citation Context ...δ in V , and every X ⊂ κ ++M of cardinality < δ can be covered by some Y ∈ M of cardinality < δ. Then there is a non-trivial M-ultrafilter on κ +M being δ-complete in V . 5sIt is essentially shown in =-=[14]-=- Lemma 5.10 that if 0 ¶ does not exist and if we let M be the core model and κ ++M < δ +V be smaller than the least M-measurable above δ, M and κ satisfy the assumptions of 2.2. However, we shall only... |

2 |
The tree property, preprint
- Cummings, Foreman
(Show Context)
Citation Context ...ngsgemeinschaft (DFG). He is indebted to John Steel for his guidiance. 1shas the tree property (and the continuum function below ℵω has the least possible values, i.e., 2 ℵn = ℵn+2 for 0 ≤ n < ω; cf. =-=[2]-=-). Cardinals much larger than weakly compacts are indeed necessary for these patterns of cardinals with the tree property to exist. Magidor saw that if there is a single consecutive pair of cardinals ... |

2 |
HOD as a core model
- Steel, Woodin
(Show Context)
Citation Context ...arly, On ∩ M ∗ ≤ ω3. Subclaim 2. On ∩ M ∗ < ω3. Proof. Suppose that On ∩ M ∗ = ω3. This means that κ is the largest cardinal in M ∗ . By [10], � <ω κ holds in every K A , A ≥c B. However, as noted in =-=[12]-=-, the proof of this fact is ”local,” so that the different � <ω κ -sequences so obtained can be neated together so as to give � <ω κ in M ∗ . But then, similar as in the proof of Claim 1, there is an ... |

1 |
The Jensen covering property
- Woodin
- 2001
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Citation Context ...n fact be carried out and does not break down below θ. This is one of the things referred to as a ”technality” in the Introduction. The resulting model V is what is called a lower part Y -premouse in =-=[13]-=-. It resembles the core model of Dodd and Jensen in that it has no total extenders on its sequence. This is exploited in [13] to get a ”Jensen like” covering property for lower part models. We shall u... |

1 |
singular or weakly compact cardinals
- Successive
(Show Context)
Citation Context ...s 1.1 and 1.2 are shown in ZF C, if one could as well get rid of the use of the Axiom of Choice in the proof of 1.2 then after all this would in fact (almost?) yield the best possible result (compare =-=[15]-=-). The third author has shown in unpublished work that if κ is an inaccessible limit of cardinals δ < κ such that both δ and δ + have the tree property then the Axiom of Determinacy holds in L(R ∗ ), ... |

1 |
covering and the tree property
- Weak
(Show Context)
Citation Context ... and σ is ℵ1-complete then ˜ H is easily seen to be well-founded, and we thus may and will identify ˜ H with its transitive collapse. The proof of the following lemma is a variation of an argument in =-=[16]-=-. Lemma 2.1 Assume that δ has the tree property. Let M be a transitive model of a sufficiently large fragment of ZF C as well as of GCH such that δ is inaccessible in M. Suppose that HM δ ++M , the se... |