scholarly journals IN SEARCH OF ULTIMATE-LTHE 19TH MIDRASHA MATHEMATICAE LECTURES

2017 ◽  
Vol 23 (1) ◽  
pp. 1-109 ◽  
Author(s):  
W. HUGH WOODIN
Keyword(s):  
Model Theory ◽  
Model Problem ◽  
Supercompact Cardinal ◽  
Inner Model Theory ◽  
Inner Model ◽  
Complete Account

AbstractWe give a fairly complete account which first shows that the solution to the inner model problem for one supercompact cardinal will yield an ultimate version ofLand then shows that the various current approaches to inner model theory must be fundamentally altered to provide that solution.

scholarly journals Descriptive inner model theory

2013 ◽  
Vol 19 (1) ◽  
pp. 1-55 ◽  
Author(s):  
Grigor Sargsyan
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Recent Progress ◽  
Model Problem ◽  
Inner Model Theory ◽  
Inner Model ◽  
General Program ◽  
Descriptive Set

AbstractThe purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture (MSC). One particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large.

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Los Angeles ◽  
Set Theory ◽  
Model Theory ◽  
Large Cardinals ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Definable Sets ◽  
Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

scholarly journals Diagonal Prikry extensions

2010 ◽  
Vol 75 (4) ◽  
pp. 1383-1402 ◽  
Author(s):  
James Cummings ◽  
Matthew Foreman
Keyword(s):  
Model Theory ◽  
Measurable Cardinal ◽  
Core Model ◽  
Singular Cardinal ◽  
Inner Model Theory ◽  
Regular Cardinals ◽  
Inner Model ◽  
Singular Cardinals ◽  
Free Abelian Groups ◽  
Woodin Cardinal

§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.Strong covering implies weak covering.In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include1. X is “0# exists”, Kx is L, Y is “strongly”.2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

LARGE CARDINALS BEYOND CHOICE

2019 ◽  
Vol 25 (03) ◽  
pp. 283-318 ◽  
Author(s):  
JOAN BAGARIA ◽  
PETER KOELLNER ◽  
W. HUGH WOODIN
Keyword(s):  
Model Theory ◽  
Systematic Investigation ◽  
Large Cardinals ◽  
Inner Model Theory ◽  
The Third ◽  
Dichotomy Theorem ◽  
Regular Cardinals ◽  
Inner Model ◽  
Singular Cardinals ◽  
The Future

AbstractThe HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails.

A boundedness lemma for iterations

2001 ◽  
Vol 66 (3) ◽  
pp. 1058-1072
Author(s):  
Greg Hjorth
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Large Cardinal ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Inner Model ◽  
Direct Limits ◽  
The One ◽  
Image Position ◽  
Descriptive Set

The purpose of this paper is to present a kind of boundedness lemma for direct limits of coarse structural mice, and to indicate some applications to descriptive set theory. For instance, this allows us to show that under large cardinal or determinacy assumptions there is no prewellorder ≤ of length such that for some formula ψ and parameter zif and only ifIt is a peculiar experience to write up a result in this area. Following the work of Martin, Steel, Woodin, and other inner model theory experts, there is an enormous overhang of theorems and ideas, and it only takes one wandering pebble to restart the avalanche. For this reason I have chosen to center the exposition around the one pebble at 1.7 which I believe to be new. The applications discussed in section 2 involve routine modifications of known methods.A detailed introduction to many of the techniques related to using the Martin-Steel inner model theory and Woodin's free extender algebra is given in the course of [1]. Certainly a familiarity with the Martin-Steel papers, [5] and [6], is a prerequisite, as is some knowledge of the free extender algebra. Probably anyone interested in this paper will already know the necessary descriptive set theory, most of which can be found in [4]. Discussion of earlier results in this direction can be found in [3] or [2].

Deconstructing inner model theory

2002 ◽  
Vol 67 (2) ◽  
pp. 721-736 ◽  
Author(s):  
Ralf-Dieter Schindler ◽  
John Steel ◽  
Martin Zeman
Keyword(s):  
Model Theory ◽  
Initial Segment ◽  
Large Cardinal ◽  
Inner Model Theory ◽  
Inner Models ◽  
Inner Model

In this paper we shall repair some errors and fill some gaps in the inner model theory of [2]. The problems we shall address affect some quite basic definitions and proofs.We shall be concerned with condensation properties of canonical inner models constructed from coherent sequences of extenders as in [2]. Condensation results have the general form: if x is definable in a certain way over a level , then either x ∈ , or else from x we can reconstruct in a simple way.The first condensation property considered in [2] is the initial segment condition, or ISC. In section 1 we show that the version of this condition described in [2] is too strong, in that no coherent in which the extenders are indexed in the manner of [2], and which is such that L[] satisfies the mild large cardinal hypothesis that there is a cardinal which is strong past a measurable, can satisfy the full ISC of [2]. It follows that the coherent sequences constructed in [2] do not satisfy the ISC of [2]. We shall describe the weaker ISC which these sequences do satisfy, and indicate the small changes in the arguments of [2] this new condition requires.

Distinct iterable branches

2005 ◽  
Vol 70 (4) ◽  
pp. 1127-1136
Author(s):  
John R. Steel
Keyword(s):  
Model Theory ◽  
Basic Problem ◽  
Important Thing ◽  
Inner Model Theory ◽  
Elementary Submodels ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Open Question

§1. Introduction. The basic problem of inner model theory is how to construct mice satisfying hypotheses appreciably stronger than “there is a Woodin limit of Woodin cardinals”. We have a family of constructions, the Kc-constructions, which ought to produce such mice under the appropriate hypotheses on V. Perhaps the most important thing we lack is a proof that the countable elementary submodels of premice produced by a Kc-construction are ω1 + 1-iterable. The best partial results in this direction are those of Neeman ([4]) for Kc-constructions making use of full background extenders over V, and those of Andretta, Neeman, and Steel ([1]) for arbitrary Kc-constructions.Let be a countable premouse embedded by π into a level of the Kc-construction ℂ. If ℂ uses only full extenders over V as its background extenders, then π and ℂ enable one to lift an evolving iteration tree on to an iteration tree * on V. (See [3, §12].) The good behavior of * guarantees that of . The natural conjecture here is that V is ω1 + 1-iterable with respect to such trees* by the strategy of choosing the unique wellfounded branch. The open question here is uniqueness, since by [2] the uniqueness of the wellfounded branch chosen by * at limit stages strictly less than λ implies the existence of a wellfounded branch to be chosen at λ.

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