John R. Steel

An Outline of Inner Model Theory

This paper outlines the basic theory of canonical inner models satisfying large cardinal hypotheses. It begins with the definition of the models, and their fine structural analysis modulo iterability assumptions. It then outlines how to construct canonical inner models, and prove their iterability, in roughly the greatest generality in which it is currently known how to do this. The paper concludes with some applications: genericity iterations, proofs of generic absoluteness, and a proof that the hereditarily ordinal definable sets of L(ℝ) constitute a canonical inner model.

Inner models with many Woodin cardinals

Abstract We extend the theory of “Fine structure and iteration trees” to models having more than one Woodin cardinal.

Complementation in the Turing Degrees

Posner [6] has shown, by a nonuniform proof, that every degree has a complement below 0′. We show that a 1-generic complement for each set of degree between 0 and 0′ can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above ∅′. In the second half of the paper, we show that the complementation of the degrees below 0′ does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a . (This result is independently due to S. B. Cooper.) We end with some open problems.

The maximality of the core model

Our main results are: 1) every countably certified extender that coheres with the core model K is on the extender sequence of K, 2) K computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of K, 4) (joint with W. J. Mitchell) K‖κ is universal for mice of height ≤ κ whenever κ ≥ א2, 5) if there is a κ such that κ is either a singular countably closed cardinal or a weakly compact cardinal, and <ω κ fails, then there are inner models with Woodin cardinals, and 6) an ω-Erdos cardinal suffices to develop the basic theory of K.

Projectively well-ordered inner models

Abstract We show that the reals in the minimal iterable inner model having n Woodin cardinals are precisely those which are Δ n + 2 1 definable from some countable ordinal. (One direction here is due to Hugh Woodin.) It follows that this model satisfies “There is a Δ n + 2 1 well-order of the reals”. We also describe some other connections between the descriptive set theory of projective sets and inner models with finitely many Woodin cardinals.

A weak Dodd-Jensen lemma

We show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice. ?

THE DOMESTIC LEVELS OF K c ARE ITERABLE

We show that the models produced by theKc construction before (if ever) it reaches a non-domestic premouse are all iterable. As a corollary we get thatPFA plus the existence of a measurable cardinal implies the existence of a non-domestic premouse.

Σ31 absoluteness and the second uniform indiscernible

We show that that if every real has a sharp and there are Δ21-definable prewellorderings of ℝ of ordinal ranks unbounded inω2, then there is an inner model for a strong cardinal. Similarly, assuming the same sharps, the Core ModelK is Σ31-absolute unless there is an inner model for a strong cardinal.

Descending Sequences of Degrees

Our unexplained notation is that of Rogers [4]. Let P ⊆ 2 N × 2 N . We call a sequence A n : n ∈ N > of subsets of N a P-sequence iff ∀n(A n +1 = the unique B such that P(A n , B)) . Theorem. Let P ⊆ 2 N × 2 N be arithmetical. Then there is no P-sequence n : n ∈ N> such that ∀n(A′ n +1 ≤ T A n ) . This theorem improves a result of Friedman [2] who showed that for no arithmetical P is there a P -sequence A n : n ∈ N > such that A n + 1 is a code for an ω-model of the relative arithmetic comprehension schema, and A n + 1 is present in the model coded by A n , for all n . Other related results are those of Harrison [3], who showed there is a sequence A n : n ∈ N > such that ∀n n + 1 ≤ T A n >, and of Enderton and Putnam [1], who showed there is no sequence A n : n ∈ N > with ∀n(A′ n + 1 ≤ T A n ) and A 0 hyperarithmetic. Our theorem is closely connected to Godels second incompleteness theorem. Its proof is a recursion theoretic parallel to the proof of Godels theorem. In §2 we draw a version of Godels theorem as a corollary to ours.

Ordinal Definability and Recursion Theory: The Cabal Seminar, Volume III: Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations

Introduction. There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martins conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations. In this paper, we shall give an overview of some work that has been done on Martins conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation. This theorem has interesting corollaries for the theory of universal countable Borel equivalence relations in general. We end with some open problems, and directions for future research. Martins conjecture. Martins conjecture on Turing invariant functions is one of the oldest and deepest open problems on the global structure of the Turing degrees. Inspired by Sacks’ question on the existence of a degree-invariant solution to Posts problem [Sac66], Martin made a sweeping conjecture that says in essence, the only nontrivial definable Turing invariant functions are the Turing jump and its iterates through the transfinite. Our basic references for descriptive set theory and effective descriptive set theory are the books of Kechris [Kec95] and Sacks [Sac90]. Let ≤T be Turing reducibility on the Cantor space ^ω2, and let ≡T be Turing equivalence. Given x ∈^ω2, let x′ be the Turing jump of x. The Turing degree of a real x∈^ω2 is the ≡T equivalence class of x. A Turing invariant function is a function such that for all reals x, y ∈ ^ω2, if x ≡T y, then f(x) ≡T f(y). The Turing invariant functions are those which induce functions on the Turing degrees. With the axiom of choice, we can construct many pathological Turing invariant functions. Martins conjecture is set in the context of ZF+DC+AD, where AD is the axiom of determinacy. We assume ZF+DC+AD for the rest of this section. The results we will discuss all “localize” so that the assumption of AD essentially amounts to studying definable functions assuming definable determinacy, for instance, Borel functions using Borel determinacy.