Tame failures of the unique branch hypothesis and models of ADℝ + Θ is regular

2016 ◽  
Vol 16 (02) ◽  
pp. 1650007
Author(s):  
Grigor Sargsyan ◽  
Nam Trang
Keyword(s):  
Symbolic Logic ◽  
Generic Extension ◽  
Transitive Model ◽  
Woodin Cardinals ◽  
Iteration Trees

In this paper, we show that the failure of the unique branch hypothesis ([Formula: see text]) for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66(4) (2014) 903–923; Core models with more Woodin cardinals, J. Symbolic Logic 67(3) (2002) 1197–1226] for tame trees.

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

On generic extensions without the axiom of choice

1983 ◽  
Vol 48 (1) ◽  
pp. 39-52 ◽  
Author(s):  
G. P. Monro
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Generic Extension ◽  
Transitive Model ◽  
The Axiom Of Choice ◽  
Generic Extensions

AbstractLet ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let M be a countable transitive model of ZF. The method of forcing extends M to another model M[G] of ZF (a “generic extension”). If the axiom of choice holds in M it also holds in M[G], that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive an application to Boolean toposes.

Iterates of the core model

2006 ◽  
Vol 71 (1) ◽  
pp. 241-251 ◽  
Author(s):  
Ralf Schindler
Keyword(s):  
Core Model ◽  
Elementary Embedding ◽  
The Core ◽  
Transitive Model ◽  
Woodin Cardinals

AbstractLet N be a transitive model of ZFC such that “N ⊂ N and P(ℝ) ⊂ N. Assume that both V and N satisfy “the core model K exists.” Then KN is an iterate of K, i.e., there exists an iteration tree F on K such that F has successor length and . Moreover, if there exists an elementary embedding π: V → N then the iteration map associated to the main branch of F equals π յ K. (This answers a question of W. H. Woodin, M. Gitik, and others.) The hypothesis that P(ℝ) ⊂ N is not needed if there does not exist a transitive model of ZFC with infinitely many Woodin cardinals.

The self-iterability of L[E]

2009 ◽  
Vol 74 (3) ◽  
pp. 751-779 ◽  
Author(s):  
Ralf Schindler ◽  
John Steel
Keyword(s):  
The Self ◽  
Woodin Cardinals ◽  
Iteration Trees ◽  
Diamond Principle

AbstractLet L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that Jκ[E] is iterable above t with respect to iteration trees of length less than κ.As an application we show L[E] to be a model of the following two cardinals versions of the diamond principle. If λ > κ > ω1 are cardinals, then holds true, and if in addition λ is regular, then holds true.

scholarly journals Outer models and genericity

2003 ◽  
Vol 68 (2) ◽  
pp. 389-418 ◽  
Author(s):  
M. C. Stanley
Keyword(s):  
Set Theory ◽  
Generic Extension ◽  
Transitive Model ◽  
Class Forcing ◽  
The Universe ◽  
Models Of Set Theory

Why is forcing the only known method for constructing outer models of set theory?If V is a standard transitive model of ZFC, then a standard transitive model W of ZFC is an outer model of V if V ⊆ W and V ∩ OR = W ∩ OR.Is every outer model of a given model a generic extension? At one point Solovay conjectured that if 0# exists, then every real that does not construct 0# lies in L[G], for some G that is generic for some forcing ℙ ∈ L. Famously, this was refuted by Jensen's coding theorem. He produced a real that is generic for an L-definable class forcing property, but does not lie in any set forcing extension of L.Beller, Jensen, and Welch in Coding the universe [BJW] revived Solovay's conjecture by asking the following question: Let a ⊆ ω be such that L[a] ⊨ “0# does not exist”. Is there ab∈ L[a] such that a ∉ L[b] and a is set generic over L[b].

scholarly journals Non-tame Mice from Tame Failures of the Unique Branch Hypothesis

2014 ◽  
Vol 66 (4) ◽  
pp. 903-923 ◽  
Author(s):  
Grigor Sargsyan ◽  
Nam Trang
Keyword(s):  
Generic Extension ◽  
Transitive Model

AbstractIn this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees implies that in some homogenous generic extension of V there is a transitive model M containing Ord ∪ℝ such that M ⊧ AD+ + Θ > θ0. In particular, this implies the existence (in V) of a non-tame mouse. The results of this paper significantly extend J. R. Steel's earlier results for tame trees.

Chang's Conjecture and the Non-Stationary Ideal

2001 ◽  
Vol 66 (1) ◽  
pp. 144-170 ◽  
Author(s):  
Daniel Evan Seabold
Keyword(s):  
Partial Order ◽  
Order Type ◽  
Generic Extension ◽  
Forcing Axioms ◽  
Axiom Of Determinacy ◽  
Woodin Cardinals ◽  
Chang’S Conjecture ◽  
Nonstationary Ideal ◽  
Chang's Conjecture ◽  
Central Result

In The Axiom of Determinacy, Forcing Axioms and the Nonstationary Ideal [W1], Woodin constructs the partial order ℙmax, which in the presence of large cardinals yields a forcing extension of L(ℝ) where ZFC holds and the non-stationary ideal on ω1 (hereafter denoted NSω1) is ω2-saturated. The basic analysis of ℙmax forcing over L(ℝ) can be carried out assuming only the Axiom of Determinacy (AD). In the central result of this paper, we show that if one increases slightly the strength of the determinacy assumptions, then Chang's Conjecture—the assertion that every finitary algebra on ω2 has a subalgebra of order type ω1—holds in this extension as well. Specifically, we obtain:Corollary 4.6. Assume AD + V = L(ℝ, μ) + μ is a normal, fine measure on. Chang's Conjecture holds in any ℙmax-generic extension of L(ℝ).This technique for obtaining Chang's Conjecture is fairly general. We [Se] have adapted it to obtain Chang's Conjecture in the model presented by Steel and Van Wesep [SVW] and Woodin [Wl] has adapted it to his ℚmax forcing notion. In each of these models, as in the ℙmax extension, one forces over L(ℝ) assuming AD to obtain ZFC and NSω1 is ω2-saturated.By unpublished results of Woodin, the assumption for Corollary 4.6 is equiconsistent with the existence of ω2 many Woodin cardinals, and hence strictly stronger than ADL(ℝ). One would like to reduce this assumption to ADL(ℝ). Curiously, this reduction is not possible in the arguments for ℚmax or the Steel and Van Wesep model, and the following argument of Woodin suggests why it may not be possible for ℙmax either.

VARSOVIAN MODELS I

2018 ◽  
Vol 83 (2) ◽  
pp. 496-528 ◽  
Author(s):  
GRIGOR SARGSYAN ◽  
RALF SCHINDLER
Keyword(s):  
Core Model ◽  
Generic Extension ◽  
Strong Cardinal ◽  
Small Core ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Specific Fragment ◽  
Image Position ◽  
Woodin Cardinal

AbstractLet Msw denote the least iterable inner model with a strong cardinal above a Woodin cardinal. By [11], Msw has a fully iterable core model, ${K^{{M_{{\rm{sw}}}}}}$, and Msw is thus the least iterable extender model which has an iterable core model with a Woodin cardinal. In V, ${K^{{M_{{\rm{sw}}}}}}$ is an iterate of Msw via its iteration strategy Σ.We here show that Msw has a bedrock which arises from ${K^{{M_{{\rm{sw}}}}}}$ by telling ${K^{{M_{{\rm{sw}}}}}}$ a specific fragment ${\rm{\bar{\Sigma }}}$ of its own iteration strategy, which in turn is a tail of Σ. Hence Msw is a generic extension of $L[{K^{{M_{{\rm{sw}}}}}},{\rm{\bar{\Sigma }}}]$, but the latter model is not a generic extension of any inner model properly contained in it.These results generalize to models of the form Ms (x) for a cone of reals x, where Ms (x) denotes the least iterable inner model with a strong cardinal containing x. In particular, the least iterable inner model with a strong cardinal above two (or seven, or boundedly many) Woodin cardinals has a 2-small core model K with a Woodin cardinal and its bedrock is again of the form $L[K,{\rm{\bar{\Sigma }}}]$.

Stacking mice

2009 ◽  
Vol 74 (1) ◽  
pp. 315-335 ◽  
Author(s):  
Ronald Jensen ◽  
Ernest Schimmerling ◽  
Ralf Schindler ◽  
John Steel
Keyword(s):  
Special Interest ◽  
Generic Extension ◽  
Proper Class ◽  
Weakly Compact ◽  
Covering Theorem ◽  
Technical Result ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Strong Cardinals

AbstractWe show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ+). Of special interest is 1) with κ = ℵ3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ.

Core models with more Woodin cardinals

2002 ◽  
Vol 67 (3) ◽  
pp. 1197-1226 ◽  
Author(s):  
J. R. Steel
Keyword(s):  
Fine Structure ◽  
Model Theory ◽  
Core Model ◽  
Structure Theory ◽  
General Nature ◽  
Woodin Cardinals ◽  
The One ◽  
Iteration Trees

In this paper, we shall prove two theorems involving the construction of core models with infinitely many Woodin cardinals. We assume familiarity with [12], which develops core model theory the one Woodin level, and with [10] and [6], which extend the fine structure theory of [5] to mice having many Woodin cardinals. The most important new problem of a general nature which we must face here concerns the iterability of Kc with respect to uncountable iteration trees.Our first result is the following theorem, a slightly stronger version of which was proved independently and earlier by Woodin. The theorem settles positively a conjecture of Feng, Magidor, and Woodin [2].Theorem. Let Ω be measurable. Then the following are equivalent:(a) for all posets,(b) for every poset,(c) for every poset ℙ ∈ VΩ, Vℙ ⊨ there is no uncountable sequence of distinct reals in L(ℝ)(d) there is an Ω-iterable premouse of height Ω which satisfies “there are infinitely many Woodin cardinals”.It is an immediate corollary that if every set of reals in L(ℝ) is weakly homogeneous, then ADL(ℝ) holds. We shall also indicate some extensions of the theorem to pointclasses beyond L(ℝ), and mice with more than ω Woodin cardinals.

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