Peter Cholak

On the Strength of Ramsey's Theorem for Pairs

We study the proof–theoretic strength and effective content of the infinite form of Ramseys theorem for pairs. Let RT k n denote Ramseys theorem for k –colorings of n –element sets, and let RT n denote (∀ k )RT k n . Our main result on computability is: For any n ≥ 2 and any computable (recursive) k –coloring of the n –element sets of natural numbers, there is an infinite homogeneous set X with X ″ ≤ T 0 ( n ) . Let I Σ n and B Σ n denote the Σ n induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low 2 ) to models of arithmetic enables us to show that RCA 0 + I Σ 2 + RT 2 2 is conservative over RCA 0 + I Σ 2 for Π 1 1 statements and that RCA 0 + I Σ 3 + RT 2 is Π 1 1 -conservative over RCA 0 + I Σ 3 . It follows that RCA 0 + RT 2 2 does not imply B Σ 3 . In contrast, J. Hirst showed that RCA 0 + RT 2 does imply B Σ 3 , and we include a proof of a slightly strengthened version of this result. It follows that RT 2 is strictly stronger than RT 2 2 over RCA 0 .

Computably categorical structures and expansions by constants

Effective model theory is the subject that analyzes the typical notions and results of model theory to determine their effective content and counterparts. The subject has been developed both in the former Soviet Union and in the west with various names (recursive model theory, constructive model theory, etc.) and divergent terminology. (We use “effective model theory” as the most general and descriptive designation. Harizanov [6] is an excellent introduction to the subject as is Millar [13].) The basic subjects of model theory include languages, structures, theories, models and various types of maps between these objects. There are many ways to introduce considerations of effectiveness into the area. The two most prominent derive from starting, on the one hand, with the notion of a theory and its models or, on the other, with just structures. If one begins with theories, then a natural version of effectiveness is to consider decidable theories (i.e., ones with a decidable (equivalently, computable or recursive) set of theorems). When one moves to models and wants them to be effective, one might start with the requirement that the model (of any theory) have a decidable theory (i.e., Th ( ), the set of sentences true in , is decidable). Typically, however, one wants to be able to talk about the elements of the model as well as its theory in the given language. Thus one naturally considers the model as a structure for the language expanded by adding a constant a i , for each element a i of . Of course, one requires that the mapping from the constants to the corresponding elements of be effective (computable). We are thus lead to the following basic definition: A structure or model is decidable if there is a computable enumeration a i of A , the domain of , such that Th( , a i ,) is decidable. (Of course, a i , is interpreted as a i , for each i Є ω .)

Automorphisms of the lattice of Π₁⁰ classes; perfect thin classes and anc degrees

Π1 classes are important to the logical analysis of many parts of mathematics. The Π1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π1 classes) forms an orbit in the lattice of Π 0 1 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π1 classes. We remark that the automorphism result is proven via a ∆3 automorphism, and demonstrate that this complexity is necessary.

Automorphisms of the lattice of recursively enumerable sets: promptly simple sets

We show that for every coinfinite r.e. set A there is a complete r.e. set B such that L* (A) ≃ eff L* (B) and that every promptIy simple set is automorphic (in E*) to a complete set

ON THE DEFINABILITY OF THE DOUBLE JUMP IN THE COMPUTABLY ENUMERABLE SETS

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.

Some orbits for E

Abstract In this article we establish the existence of a number of new orbits in the automorphism group of the computably enumerable sets. The degree theoretical aspects of these orbits also are examined.

Generics for computable Mathias forcing

Abstract We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak n-generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n-generic with n ≥ 2 then it satisfies the jump property G ( n − 1 ) ≡ T G ′ ⊕ ∅ ( n ) . We prove that every such G has generalized high Turing degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real.

Degrees of inferability

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On the orbits of computably enumerable sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion,

Extension theorems, orbits, and automorphisms of the computably enumerable sets

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