Inner Models for Large Cardinals

Mitchell, William J.

doi:10.1016/b978-0-444-51621-3.50005-0

Cited by 3 publications

(3 citation statements)

References 71 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We refer the reader to [6], [18], [19], [43], and [56] for the details of the ongoing debate. Readers who are interested in the history of the inner model problem can consult one of the following excellent sources [12], the introduction of [21], [22], [38] and [51].…”

Section: The Inner Model Problemmentioning

confidence: 99%

Descriptive inner model theory

Sargsyan¹

2013

Bull. symb. log

View full text Add to dashboard Cite

The 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.The program of constructing canonical inner models for large cardinals has been a source of increasingly sophisticated ideas leading to a beautiful theory of canonical models of fragments * 2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. † of set theory known as mice. It reads as follows:The inner model program. Construct canonical inner models with large cardinals.It is preferred that the constructions producing such inner models are applicable in many different situations and are independent of the structure of the universe. Such universal constructions are expected to produce models that are "canonical": for instance, the set of reals of such models must be definable. Also, it is expected that when the universe itself is complicated, for instance, Proper Forcing Axiom (PFA) holds or that it has large cardinals, then the inner models produced via such universal constructions have large cardinals. To test the universality of the constructions, then, we apply them in various situations such as under large cardinals, PFA or under other combinatorial statements known to have large cardinal strength, and see if the resulting models indeed have significant large cardinals. Determining the consistency strength of PFA, which is the forward direction of the PFA Conjecture stated below, has been one of the main applications of the inner model theory and many partial results have been obtained (for instance, see Theorem 3.32 and Theorem 3.33). The reverse direction of the PFA Conjecture is a well-known theorem due to Baumgartner.Conjecture 0.1 (The PFA Conjecture) The following theories are equiconsistent.1. ZFC+PFA. ZFC+"there is a supercompact cardinal".Recently, in [54], Viale and Weiss showed that if one forces PFA via a proper forcing then one needs to start with a supercompact cardinal. This result is a strong evidence that the conjecture must be true.One approach to the inner model program has been via descriptive set theory. Section 3.8 contains some details of such an approach. Remark 3.34 summarizes this approach in more technical terms then what follows. The idea is to use the canonical structure of models of determinacy to squeeze large cardinal strength out of stronger and stronger theories from t...

show abstract

Section: The Inner Model Problemmentioning

confidence: 99%

Descriptive inner model theory

Sargsyan¹

2013

Bull. symb. log

View full text Add to dashboard Cite

show abstract

“…There are many excellent resources for the introduction of the inner model theory, the inner model program and its development. We refer the reader to [16, 22, 26, 29, 33, 37, 43].…”

Section: Introductionmentioning

confidence: 99%

On -Strongly Measurable Cardinals

Ben-Neria

Hayut

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

We prove several consistency results concerning the notion of $\omega $ -strongly measurable cardinal in $\operatorname {\mathrm {HOD}}$ . In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(\kappa ) = \kappa $ , that every successor of a regular cardinal is $\omega $ -strongly measurable in $\operatorname {\mathrm {HOD}}$ .

show abstract

“…In particular, the ground axiom holds in many of the canonical inner models of large cardinal assumptions, such as the Dodd-Jensen core model K DJ , the model V = L[µ], and also the Jensen-Steel core model K, provided that there is no inner model with a Woodin cardinal. The reason is that these inner models are definable in a way that is generically absolute, see Jensen-Steel [JS13] and Mitchell [Mit12] (one uses the hypothesis of no inner models with a Woodin cardinal in the case of K), and so they have no non-trivial ground models.…”

Section: Introductionmentioning

confidence: 99%

Inner-Model Reflection Principles

et al. 2019

View full text Add to dashboard Cite

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement ϕ(a) in the first-order language of set theory is true in the set-theoretic universe V , then it is also true in a proper inner model W V . A stronger principle, the ground-model reflection principle, asserts that any such ϕ(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π 2 -conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. 2010 Mathematics Subject Classification. Primary 03E45; Secondary 03E35, 03E55, 03E65. Key words and phrases. Inner-model reflection principle, ground-model reflection principle. This article grew out of an exchange held by some of the authors on the Mathematics.StackExchange site in response to an inquiry posted by the first-named author concerning the nature of width-reflection in comparison to height-reflection [Bar16]. Commentary concerning this paper can be made at http://jdh.hamkins.org/inner-model-reflection-principles.

show abstract

Inner Models for Large Cardinals

Cited by 3 publications

References 71 publications

Descriptive inner model theory

Descriptive inner model theory

On -Strongly Measurable Cardinals

Inner-Model Reflection Principles

Contact Info

Product

Resources

About