Descriptive inner model theory

Abstract

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...

Perception, Intuition, and Reliability

“…In particular, Woodin's original proof of the consistency of AD ℝ + ‘Θ regular’ – one of the strongest determinacy axioms studied up to that point – required once more the existence of a large cardinal that to many set theorists did not seem to be sufficiently far removed from an inconsistent principle . This concern turned out to be unfounded when Sargsyan () brought the required assumption down to a Woodin limit of Woodin cardinals, extending the elaborate machinery that had been built around determinacy axioms, forcing and canonical models for large cardinals…”

“…It is instructive to consider the informal discussion on p. 4 of Sargsyan () following Corollary 0.4 of why it had been “natural to conjecture, as Woodin did, that AD ℝ +‘Θ regular’ is very strong” in the light of the teleological structure of evidence.…”

AXIOM I0 AND HIGHER DEGREE THEORY

“…Besides the classical fine structure models, recent developments in descriptive inner model theory (see for example Sargsyan's survey paper [36]) suggest a much advanced and daring path of investigation -looking into higher degrees in the HODs of determinacy models, as determinacy gives a whole family of canonical models -the ones given by Solovay hierarchy. Assume AD``V " LpPpRqq, it's believed that HOD is a canonical model.…”

“…Assume AD``V " LpPpRqq, it's believed that HOD is a canonical model. Although it's still an open question whether AD``V " LpPpRqq proves that HOD is a fine structure model, HOD of AD``V " LpPpRqq models are believed to be fine structure models at least up to Θ " def suptα | Df : R onto ÝÝÑ αu (see Steel [45] for LpRq and Sargsyan [36] §3 for a discussion on general AD`models). Based on this understanding, the first test question would be Question 2.25.…”

Chang’s Conjecture, Generic Elementary Embeddings and Inner Models for Huge Cardinals

“…Attempts to prove the existence of fine structural inner models of supercompact cardinals have all foundered on the rocks of the "iterability problem." (We refer readers to [20][21][22] for information on the current state of the art. )…”