STRUCTURE THEORY OFL(ℝ,μ) AND ITS APPLICATIONS

2015 ◽  
Vol 80 (1) ◽  
pp. 29-55 ◽  
Author(s):  
NAM TRANG
Keyword(s):  
Structure Theory ◽  
Consistency Strength ◽  
Woodin Cardinals

AbstractIn this paper, we explore the structure theory ofL(ℝ,μ) under the hypothesisL(ℝ,μ) ⊧ “AD +μis a normal fine measure on” and give some applications. First we show that “ ZFC + there existω2Woodin cardinals”1has the same consistency strength as “ AD +ω1is ℝ-supercompact”. During this process we show that ifL(ℝ,μ) ⊧ AD then in factL(ℝ,μ) ⊧ AD+. Next we prove important properties ofL(ℝ,μ) including Σ1-reflection and the uniqueness ofμinL(ℝ,μ). Then we give the computation of full HOD inL(ℝ,μ). Finally, we use Σ1-reflection and ℙmaxforcing to construct a certain ideal on(or equivalently onin this situation) that has the same consistency strength as “ZFC+ there existω2Woodin cardinals.”

scholarly journals THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

2019 ◽  
Vol 85 (1) ◽  
pp. 338-366 ◽  
Author(s):  
JUAN P. AGUILERA ◽  
SANDRA MÜLLER
Keyword(s):  
Set Theory ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Image Position ◽  
Woodin Cardinal

AbstractWe determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

scholarly journals The largest countable inductive set is a mouse set

1999 ◽  
Vol 64 (2) ◽  
pp. 443-459 ◽  
Author(s):  
Mitch Rudominer
Keyword(s):  
Consistency Strength ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Theoretic Proof

AbstractLet κℝ be the least ordinal κ such that Lκ (ℝ) is admissible. Let A = {x ϵ ℝ ∣ (∃α < κℝ) such that x is ordinal definable in Lα (ℝ)}. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the theory: ZFC − Replacement + “There exists ω Woodin cardinals which are cofinal in the ordinals.” T has consistency strength weaker than that of the theory ZFC + “There exists ω Woodin cardinals”, but stronger than that of the theory ZFC + “There exists n Woodin Cardinals”, for each n ϵ ω. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A = ℝ ∩ M. Since M is a mouse, we say that A is a mouse set. As an application, we use our characterization of A to give an inner-model-theoretic proof of a theorem of Martin which states that for all n, every real is in A.

Proper forcing and L(ℝ)

2001 ◽  
Vol 66 (2) ◽  
pp. 801-810 ◽  
Author(s):  
Itay Neeman ◽  
Jindřich Zapletal
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Consistency Strength ◽  
Woodin Cardinals ◽  
Proper Forcing

AbstractWe present two ways in which the model L(ℝ) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing: we show further that a set of ordinals in V cannot be added to L(ℝ) by small forcing. The large cardinal needed corresponds to the consistency strength of ADL(ℝ): roughly ω Woodin cardinals.

UNIVERSALLY BAIRE SETS AND GENERIC ABSOLUTENESS

2017 ◽  
Vol 82 (4) ◽  
pp. 1229-1251
Author(s):  
TREVOR M. WILSON
Keyword(s):  
Measurable Cardinal ◽  
Consistency Strength ◽  
Woodin Cardinals ◽  
Generic Absoluteness ◽  
Relative Consistency ◽  
Consistency Results ◽  
High Consistency ◽  
Image Position ◽  
General Method

AbstractWe prove several equivalences and relative consistency results regarding generic absoluteness beyond Woodin’s ${\left( {{\bf{\Sigma }}_1^2} \right)^{{\rm{u}}{{\rm{B}}_\lambda }}}$ generic absoluteness result for a limit of Woodin cardinals λ. In particular, we prove that two-step $\exists ^&#x211D; \left( {{\rm{\Pi }}_1^2 } \right)^{{\rm{uB}}_\lambda } $ generic absoluteness below a measurable limit of Woodin cardinals has high consistency strength and is equivalent, modulo small forcing, to the existence of trees for ${\left( {{\bf{\Pi }}_1^2} \right)^{{\rm{u}}{{\rm{B}}_\lambda }}}$ formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many “failures of covering” for the models $L\left( {T,{V_\alpha }} \right)$ for α below a measurable cardinal.

Fine structure for tame inner models

1996 ◽  
Vol 61 (2) ◽  
pp. 621-639 ◽  
Author(s):  
E. Schimmerling ◽  
J. R. Steel
Keyword(s):  
Fine Structure ◽  
Large Cardinal ◽  
Structure Theory ◽  
Uniqueness Problem ◽  
Covering Property ◽  
Strong Cardinal ◽  
Woodin Cardinals ◽  
Inner Models ◽  
Normal Measure ◽  
Consistency Results

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3].Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that(1) every level of is a sound, tame mouse, and(2) ⊨ “There is a strong cardinal that is a limit of Woodin cardinals”.Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10].Suppose that fi is a normal measure on Ω and that all premice are tame. Then Kc, the background certified core model, exists and is a premouse of height Ω. Moreover, for μ-almost every α < Ω.Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application.If there is a cardinal κ such that κ is κ+-strongly compact, then there is a premouse that is not tame.

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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