CHANG’S CONJECTURE, GENERIC ELEMENTARY EMBEDDINGS AND INNER MODELS FOR HUGE CARDINALS

2015 ◽  
Vol 21 (3) ◽  
pp. 251-269 ◽  
Author(s):  
MATTHEW FOREMAN
Keyword(s):  
Consistency Strength ◽  
Supercompact Cardinals ◽  
Inner Models ◽  
Chang’S Conjecture ◽  
Inner Model ◽  
Elementary Embeddings ◽  
Chang's Conjecture

AbstractWe introduce a natural principleStrong Chang Reflectionstrengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show thatdecisiveideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found in [10].1,2

Co-critical points of elementary embeddings

1985 ◽  
Vol 50 (1) ◽  
pp. 220-226
Author(s):  
Michael Sheard
Keyword(s):  
Critical Point ◽  
Measurable Cardinal ◽  
Elementary Embedding ◽  
Proper Class ◽  
Ground Model ◽  
Target Model ◽  
Inner Models ◽  
Inner Model ◽  
Elementary Embeddings

Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universeLinto itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model” (and in the latter case they are equal). It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: ifis a generic ultrafilter arising from forcing with a precipitous ideal on a successor cardinalκ, then the ultraproduct of the ground model viacollapsesκ. Such considerations suggest a classification of how close the target model comes to “fitting inside” the ground model.Definition 1.1. LetMandNbe inner models (transitive, proper class models) of ZFC, and letj:M→Nbe an elementary embedding. Theco-critical pointofjis the least ordinalλ, if any exist, such that there isX⊆λ, X∈NbutX∉M. Such anXis called anew subsetofλ.It is easy to see that the co-critical point ofj:M→Nis a cardinal inN.

Supercompact cardinals, elementary embeddings and fixed points

1982 ◽  
Vol 47 (1) ◽  
pp. 84-88
Author(s):  
Julius B. Barbanel
Keyword(s):  
Fixed Points ◽  
Supercompact Cardinals ◽  
Inner Models ◽  
Elementary Embeddings ◽  
Universe Of Sets ◽  
The Universe

Supercompactness is usually defined in terms of the existence of certain ultrafilters. By the well-known procedure of taking ultrapowers of V (the universe of sets) and transitive collapses, one obtains transitive inner models of V and corresponding elementary embeddings from V into these inner models. These embeddings have been studied extensively (see, e.g. [3] or [4]). We investigate the action of these embeddings on cardinals. In particular, we establish a characterization, based upon cofinality, of which cardinals are fixed by these embeddings.

scholarly journals On the consistency strength of the inner model hypothesis

2008 ◽  
Vol 73 (2) ◽  
pp. 391-400 ◽  
Author(s):  
Sy-David Friedman ◽  
Philip Welch ◽  
W. Hugh Woodin
Keyword(s):  
Set Theory ◽  
Lower Bounds ◽  
Internal Consistency ◽  
Countable Model ◽  
Upper And Lower Bounds ◽  
Consistency Strength ◽  
Class Theory ◽  
Inner Models ◽  
Inner Model ◽  
The Universe

The Inner Model Hypothesis (IMH) and the Strong Inner Model Hypothesis (SIMH) were introduced in [4]. In this article we establish some upper and lower bounds for their consistency strength.We repeat the statement of the IMH, as presented in [4]. A sentence in the language of set theory is internally consistent iff it holds in some (not necessarily proper) inner model. The meaning of internal consistency depends on what inner models exist: If we enlarge the universe, it is possible that more statements become internally consistent. The Inner Model Hypothesis asserts that the universe has been maximised with respect to internal consistency:The Inner Model Hypothesis (IMH): If a statement φ without parameters holds in an inner model of some outer model of V (i.e., in some model compatible with V), then it already holds in some inner model of V.Equivalently: If φ is internally consistent in some outer model of V then it is already internally consistent in V. This is formalised as follows. Regard V as a countable model of Gödel-Bernays class theory, endowed with countably many sets and classes. Suppose that V* is another such model, with the same ordinals as V. Then V* is an outer model of V (V is an inner model of V*) iff the sets of V* include the sets of V and the classes of V* include the classes of V. V* is compatible with V iff V and V* have a common outer model.

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.

INNER MODEL THEORETIC GEOLOGY

2016 ◽  
Vol 81 (3) ◽  
pp. 972-996 ◽  
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER
Keyword(s):  
Set Theory ◽  
Main Theme ◽  
Solid Core ◽  
Strong Cardinal ◽  
The Third ◽  
Inner Models ◽  
Inner Model ◽  
Basic Concepts ◽  
Woodin Cardinal ◽  
The Universe

AbstractOne of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P-constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ω (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model.

scholarly journals Ramsey-like cardinals II

2011 ◽  
Vol 76 (2) ◽  
pp. 541-560 ◽  
Author(s):  
Victoria Gitman ◽  
P. D. Welch
Keyword(s):  
Upper Bound ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Core Model ◽  
Consistency Strength ◽  
The Core ◽  
Countable Ordinal ◽  
Elementary Embeddings ◽  
Remarkable Cardinals

AbstractThis paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

scholarly journals Some two-cardinal results for O-minimal theories

1998 ◽  
Vol 63 (2) ◽  
pp. 543-548 ◽  
Author(s):  
Timothy Bays
Keyword(s):  
Minimal Theory ◽  
Chang’S Conjecture ◽  
Chang's Conjecture

AbstractWe examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some (κ, λ) must admit every (κ′, λ′). We also prove that every “reasonable” variant of Chang's Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the δ-cardinal case for arbitrary ordinals δ.

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