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## Extender based forcings, fresh sets and Aronszajn trees (2011)

Citations: | 1 - 0 self |

### Citations

16 |
The singular cardinal hypothesis revisited
- Gitik, Magidor
- 1992
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Citation Context ...nd projecting to all permitted coordinates of p. Denote such q by p_〈η1, ..., ηn〉. For each α < κ+ and p ∈ P there are n < ω and pα ≥∗ p such that any n-extension of pα decides b∼(α), as was shown in =-=[5]-=-. Note that the branch b∼ α + 1 is decided as well, since T ∈ V and so the value at the level α determines uniquely the branch to it below. Denote by n(p, α) the least such n. 1 Lemma 1.2 For each p... |

16 | Extensions with the Approximation and Cover Properties have No New Large Cardinals”, Fundamenta Mathematicae 180
- Hamkins
- 2003
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Citation Context .... Now proceed as before and define qα,p ′ α, να and pα. This completes the construction. There are a stationary S ⊆ κ+ and ν∗ < κ such that for every α ∈ S we have (να)0 = ν∗. J. Hamkins defined in =-=[7]-=- the following two useful notions: Definition 1.3 (Hamkins) Let V ⊆ V1. δ-approximation property holds between V and V1 iff for every set A of ordinals in V1, if A∩ a ∈ V for all a ∈ V with V |= |a| <... |

13 |
Blowing up power of a singular cardinal—wider gaps
- Gitik
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Citation Context ...long or short extenders Prikry or extender based Prikry forcing over κ over V Cohen(ω). Then in V Cohen(ω)∗P∼ there is no new fresh subsets of ordinals of cofinality bigger than κ. Proof. We refer to =-=[3]-=-, sections 1,2 for definitions and basic properties of Long (and short) extenders forcing , a more detailed account may be found in [6]. Let us give here only a brief description. Conditions in this f... |

8 | Prikry-type forcings
- Gitik
- 2010
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Citation Context ...y skip the first section and go directly to the second. 1 No branches to κ+- Aronszajn trees. We deal here with Extender Based Prikry forcing, Long and short extenders Prikry forcing. Let us refer to =-=[2]-=- for definitions. Theorem 1.1 Extender based Prikry forcing over κ cannot add a cofinal branch to a κ+- Aronszajn tree. Proof. Let 〈T,≤T 〉 be a κ+-Aronszajn tree. Denote by P the extender based Prikry... |

2 |
The Tree Property at ℵω+2
- Friedman, Halilović
- 2011
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Citation Context ... are studied with respect of adding branches to Aronszajn trees. We construct a model with no Aronszajn tree over ℵω+2 from the optimal assumptions. This answers a question of Friedman and Halilović =-=[1]-=-. The reader interested only in Friedman and Halilović question may skip the first section and go directly to the second. 1 No branches to κ+- Aronszajn trees. We deal here with Extender Based Prikry... |

2 |
Intermediate models of Prikry generic extensions
- Gitik, Kanovei, et al.
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Citation Context ... a generic subset. Consider a set A := {α < κ+ | ∃p ∈ G, `(p) > 0, p = 〈pn | n < ω〉, p0(α) = 0}. Then for each β < κ+ the set Acapβ is in V , since a single one extension decides it completely. 2. By =-=[4]-=-, the Prikry forcing does not add new fresh subsets to κ+ (or to ordinals of cofinality ≥ κ+. 6 Theorem 1.7 Let Q be a forcing of cardinality ≤ κ and P be a forcing in V Q that preserves κ+ and does n... |