Leo Harrington

A Glimm-Effros dichotomy for Borel equivalence relations

A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in the Polish group action case when additionally the induced equivalence relation is Fσ. It is the purpose of this paper to extend the Glimm-Effros dichotomy to the very general context of an arbitrary Borel equivalence relation (not even necessarily induced by a group action). Despite the totally classical descriptive set-theoretic nature of our result, our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability (or recursion) theory on the integers.

Analytic determinacy and 0

Martin [12] has shown that the determinacy of analytic games is a consequence of the existence of sharps. Our main result is the converse of this: Theorem. If analytic games are determined, then x 2 exists for all reals x . This theorem answers question 80 of Friedman [5]. We actually obtain a somewhat sharper result; see Theorem 4.1. Martin had previously deduced the existence of sharps from 3 − Π 1 1 -determinacy (where α − Π 1 1 is the αth level of the difference hierarchy based on − Π 1 1 see [1]). Martin has also shown that the existence of sharps implies 2 − Π 1 1 -determinacy. Our method also produces the following: Theorem. If all analytic, non-Borel sets of reals are Borel isomorphic, then x* exists for all reals x . The converse to this theorem had been previously proven by Steel [7], [18]. We owe a debt of gratitude to Ramez Sami and John Steel, some of whose ideas form basic components in the proofs of our results. For the various notation, definitions and theorems which we will assume throughout this paper, the reader should consult [3, §§5, 17], [8], [13] and [14, Chapter 16]. Throughout this paper we will concern ourselves only with methods for obtaining 0 # (rather than x # for all reals x ). By relativizing our arguments to each real x , one can produce x 2 .

Recursively Presentable Prime Models

It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable ℵ 1 -categorical theory. The language of a theory T is denoted by L ( T ). All languages will be presumed countable. An x-type of T is a set of formulas with free variables x , which is consistent with T and which is maximal in this property. A formula with free variables x is complete if there is exactly one x -type containing it. A type is principal if it contains a complete formula. A countable model of T is prime if it realizes only principal types. Vaught has shown that a complete countable theory can have at most one prime model up to isomorphism. If T is a decidable theory, then the decision procedure for T equips L ( T ) with an effective counting. Thus the formulas of L ( T ) correspond to integers. The integer a formula φ ( x ) corresponds to is generally called the Godel number of φ ( x ) and is denoted by ⌜ φ ( x )⌝. The usual recursion theoretic notions defined on the set of integers can be transferred to L ( T ). In particular a type Γ is recursive with index e if {⌜ φ ⌝.; φ ∈ Γ} is a recursive set of integers with index e .

The d.r.e. degrees are not dense

Abstract By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n -r.e. degrees ( n ⩾2) and of the ω-r.e. degrees.

On the determinacy of games on ordinals

Let λ be an ordinal and A ⊆ λ^ω x λ^ω. As usual we associate with it the game G(A;λ): I II I and II alternatively ξ0 η0 play ξ0, η0, ξ1,η1,.... from λ; ξ1 I wins iff (ξ, ή) eA. η1 ξ ή.

Fundamentals of forking

On expose les faits fondamentaux de la theorie de bifurcation de Shelah. On donne des preuves courtes et directes des proprietes de base et des equivalences entre les diverses definitions de bifurcation

The Δ₃⁰-automorphism method and noninvariant classes of degrees

A set A of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = ({We}e∈ω ,⊆). Most previously known automorphisms Φ of the structure E of sets were effective (computable) in the sense that Φ has an effective presentation. We introduce here a new method for generating noneffective automorphisms whose presentation is ∆3, and we apply the method to answer a number of long open questions about the orbits of c.e. sets under automorphisms of E. For example, we show that the orbit of every noncomputable (i.e., nonrecursive) c.e. set contains a set of high degree, and hence that for all n > 0 the well-known degree classes Ln (the lown c.e. degrees) and Hn = R −Hn (the complement of the highn c.e. degrees) are noninvariant classes. Department of Mathematics, University of California at Berkeley, Berkeley, California 94720 E-mail address: leo@math.berkeley.edu Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637-1538 E-mail address: soare@math.uchicago.edu World Wide Web address: http://www.Cs.uchicago.edu/∼soare

An exposition of Shelah's ``main gap'': counting uncountable models of

On considere la possibilite que la conjecture de Morley peut eventuellement etre demontree en donnant plus ou moins explicitement toutes les fonctions spectre possibles K(≥κ 1 )→I(T,K) avec chaque possibilite se conformant a la condition de Morley

omega

It is shown that every ∑^0_3 set of reals which contains reals of arbitrarily high Turing degree in the hyperarithmetic hierarchy contains reals of every Turing degree above the degree of Kleenes O. As an application it is shown that every Turing degree above the Turing degree of Kleenes O is a minimal cover.

-stable and superstable theories

We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: let is the Turing degree of a set J ≥T0″}. Let such that is upward closed in . Then there is an ℒ(A) property such that if and only if there is an A where A ≡T F and . A corollary of this is that, for all n ≥ 2, the highn () computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martins Invariance Conjecture.