Feinstrukturtheorie und Innere Modelle

Abstract

The workshop Fine Structure Theory and Inner Models, organised by Ronald Jensen (Berlin), Menachem Magidor (Jerusalem) and Ralf Schindler (Münster) was held April 30th - May 6th, 2006. It was attended by most of the leading researchers in the area.

Fine structure theory was initiated by the first organizer, R. Jensen, in the 70ies. It has been exploited ever since for producing a series of spectacular results in set theory. One such is Jensen's Covering Lemma for Gödel's constructible hierarchy, $L$, which says, put informally, that the universe $V$ of all sets either resembles $L$ to a large extent or else is very different from $L$. Later on, various people (most of which were participants of this workshop) proved versions of the Covering Lemma for larger inner models.

The main goal of “Fine structure theory and inner model theory” is to construct fine structural inner models of set theory, i.e., definable transitive proper class-sized models of the standard axiom system $\mathsf{ZFC}$ of set theory, which reflect the large cardinal structure of the universe, but at the same time admit a fine structure that makes it possible to analyze them in great detail and prove various combinatorial properties in them. Other applications of such inner models are consistency strength investigations, and they can be used as a tool for proving implications which don't mention inner models at all, but for which no “direct” proof is in sight.

A large cardinal concept is one such that $\mathsf{ZFC}$ cannot prove that there is an incarnation of it. Our area is the key tool for uncovering the large cardinal structure which is implicit in many (not only set theoretic) hypotheses. In fact, often a given statement which doesn't mention large cardinals at all and a statement about the existence of models with large cardinals turn out to be two sides of the same coin. Breathtaking results by Martin, Steel, Woodin, and others in the 80ies and 90ies have shown that the large cardinal concept of a Woodin cardinal is a crucial one here.

The main issues of this area are the following.

• Fine structure theory. This is a general theory of the definability over the structures that form the building blocks of the inner models one wants to construct. In most cases, these structures are premice, that is, models constructed from sequences of extenders which code fragments of elementary embeddings. The existence of such embeddings is the essence of the crucial large cardinal concepts.

• Iterability. That a (well-founded) structure be iterable means that we can keep taking ultrapowers of it (i.e., decoding the elementary embedding coded by some extender on the sequence of the structure, along with the target model, which, by elementarity, is again a model constructed relative to a sequence of extenders) without ever producing non-well-founded structures. In fact, what one needs for iterability is an iteration strategy for the given structure. The iterability of a premouse is the key property one needs in order to choose the next building block in the construction of an inner model in a canonical way. This eventually makes the resulting inner model definable in some reasonable way. Also, without iterability we wouldn't know how to prove key (fine structural) first order properties which we require of our premice and which are then inherited by the inner model we are about to construct. For instance, the fact that the (generalized) continuum hypothesis holds in the inner models we construct relies on iterability. It is important here to isolate criteria for the iterability of a premouse which are not too strong so that sufficiently many iterable premice can be shown to exist.

• The model construction. The construction of an inner model is done by recursion on its “building blocks”. In order to verify that the construction doesn't trivialize one has to prove that sufficiently many premice meet the iterability criterion one works with. Also, one wants to show that a Covering Lemma holds for the inner model which was built.

• Applications of inner models. Woodin's core model induction makes use of “locally defined” inner models which are used for verifying inductively that (sufficiently iterable) models of $\mathsf{ZFC}$ plus there are such-and-such many Woodin cardinals exist. This induction can therefore be used for showing that a given (say, combinatorial) statement implies that definable sets of reals are determined. It turns out that in order to organize such an induction properly, one has to construct a new kind of “hybrid” premice which are constructed not only relative to a sequence of extenders, but also relative to iteration strategies for certain structures.

The conference had 16 participants. 13 talks were given, and they covered both pure and applied parts of inner model theory. Because this was a gathering of true specialists, there was no need for overview-style talks and we could concentrate on issues which are at the focus of current research. The talks came with intriguing results, but also with promising new perspectives for upcoming research. We had very lively discussions.

It was a fruitful workshop, and many of the ideas which were exchanged are sure to be further elaborated in the near future.