Extenders, embedding normal forms, and the Martin-Steel-theorem

1998 ◽  
Vol 63 (3) ◽  
pp. 1137-1176 ◽  
Author(s):  
Peter Koepke
Keyword(s):  
Set Theory ◽  
Normal Forms ◽  
Woodin Cardinals ◽  
Elementary Embeddings ◽  
Simple Notion ◽  
Models Of Set Theory

AbstractWe propose a simple notion of “extender” for coding large elementary embeddings of models of set theory. As an application we present a self-contained proof of the theorem by D. Martin and J. Steel that infinitely many Woodin cardinals imply the determinacy of every projective set.

scholarly journals Changing cofinalities and collapsing cardinals in models of set theory

2003 ◽  
Vol 120 (1-3) ◽  
pp. 225-236 ◽  
Author(s):  
Miloš S. Kurilić
Keyword(s):  
Set Theory ◽  
Models Of Set Theory

scholarly journals Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 910 ◽  
Author(s):  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Lower Limit ◽  
Set Theory ◽  
Generic Extension ◽  
Projective Class ◽  
Almost Disjoint ◽  
Projective Hierarchy ◽  
Models Of Set Theory

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .

Elementary extensions of models of set theory

2000 ◽  
Vol 39 (7) ◽  
pp. 509-514 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Set Theory ◽  
Models Of Set Theory

Subdifferentials in Boolean-valued models of set theory

1984 ◽  
Vol 24 (5) ◽  
pp. 735-746 ◽  
Author(s):  
A. G. Kusraev ◽  
S. S. Kutateladze
Keyword(s):  
Set Theory ◽  
Models Of Set Theory

scholarly journals On certain elementary extensions of models of set theory

1984 ◽  
Vol 283 (2) ◽  
pp. 705-705 ◽  
Author(s):  
Ali Enayat
Keyword(s):  
Set Theory ◽  
Models Of Set Theory

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 Chains of end elementary extensions of models of set theory

1998 ◽  
Vol 63 (3) ◽  
pp. 1116-1136 ◽  
Author(s):  
Andrés Villaveces
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Elementary Extension ◽  
Models Of Set Theory

AbstractLarge cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (‘unfoldable cardinals’) lie in the boundary of the propositions consistent with ‘V = L’ and the existence of 0#. We also provide an ‘embedding characterisation’ of the unfoldable cardinals and study their preservation and destruction by various forcing constructions.

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