Ramsey Properties of Countably Infinite Partial Orderings

A partial ordering ℙ is chain-Ramsey if, for every natural number n and every coloring of the n-element chains from ℙ in finitely many colors, there is a monochromatic subordering ℚ isomorphic to ℙ. Chain-Ramsey partial orderings stratify naturally into levels. We show that a countably infinite partial ordering with finite levels is chain-Ramsey if and only if it is biembeddable with one of a canonical collection of examples constructed from certain edge-Ramsey families of finite bipartite graphs. A similar analysis applies to a large class of countably infinite partial orderings with infinite levels

A Basis Theorem for Perfect Sets

We show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair M ⊂ N of models of set theory implying that every perfect set in N has an element in N which is not in M

On Turing Reducibility

We show that the transitivity of pointwise Turing reducibility on the recursively enumerable sets of integers cannot be proven in P - + I# 1 , first order arithmetic with induction limited to # 1 predicates. We produce a example of intransitivity in a nonstandard model of P - +I# 1 by a finite injury priority construction

Classes and Minimal Degrees

Theorem: There is a non-empty \Pi 0 1 class of reals, each of which computes a real of minimal (Turing) degree. Corollary: WKL ` "there is a minimal Turing degree". This answers a question of H. Friedman and S. Simpson. 1 Introduction We show there is a non-empty \Pi 0 1 class of reals, each member of which (Turing) computes a real of minimal Turing degree. Equivalently, there is an infinite recursive binary tree T , such that every infinite branch through T computes a real of minimal degree. Jockusch and Simpson [1980] considered this question in the form, "Does every degree of a complete extension of Peano Arithmetic bound a minimal degree?" They gave a partial answer, showing that (in a precise sense) almost all degrees of complete extensions of Peano Arithmetic bound minimal degrees. Our result shows that in fact all of them do. A major part of the interest of these issues lies in their relation to the following question, posed by Harvey Friedman and Simpson: "Does the subtheor..

The Sacks Density Theorem and S_2-Bounding

The Sacks Density Theorem (Sacks 1964) states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P \Gamma +B \Sigma 2 . The proof has two components: a lemma that in any model of P \Gamma +B \Sigma 2 , if B is recursively enumerable and incomplete then I \Sigma 1 holds relative to B and an adaptation of Shore's (1976) blocking technique in ff-recursion theory to models of arithmetic. 1 Introduction Proofs using the priority method are the trademark of recursion theory. In this paper, we continue the line of inquiry in which we use subsystems of first order arithmetic to calibrate priority methods and the theorems in whose proofs they appear. In the hierarchy of Groszek and Slaman (unpublished), we classify priority constructions according to the syntactic complexity of the outcomes in its most complicated families of strategies. In a \Pi 1 -priority construction the strategies have outcomes that are descri..