The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...

Abstract. In this paper we consider three potential simplifications to a result of Solovay’s concerning the Turing degrees of nonstandard models of arbitrary completions of first-order Peano Arithmetic (PA). Solovay characterized the degrees of nonstandard models of completions T of PA, showing that they are the degrees of sets X such that there is an enumeration R ≤T X of an “appropriate ” Scott set and there is a family of functions (tn)n∈ω, ∆ 0 n(X) uniformly in n, such that lim s→ ∞ tn(s) is an index for T ∩ Σn and for all s, tn(s) is an index for a subset of T ∩ Σn. The simplifications under consideration are attempts to restrict the families of functions (tn)n∈ω that appear in Solovay’s result, known henceforth as Solovay families. We show that none of these potential simplifications may be made, by proceeding as follows. First, we construct a nonstandard model A of PA such that there is no Solovay family (tn)n∈ω for Th(A) relative to A in which all the functions tn are constant. Second, for each k we construct a nonstandard model A of PA such that there is no Solovay family (tn)n∈ω for Th(A) relative to A in which all the functions tn change values at most k many times. Third, we construct a nonstandard model A of PA such that there is no Solovay family (tn)n∈ω for Th(A) relative to A with a computable function f such that for all n, tn(s) changes values at most f(n) times. Our constructions answer three questions asked by Julia Knight [7]. Our solutions make use of several consistency results that seem to be of independent interest. 1.

Abstract. We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. We discuss the main open questions in the area. 1.

A basic aim of logic is to consider what axioms are used in proving various theorems of mathematics. This thesis will be concerned with such issues applied to a particular area of mathematics: combinatorics. We will consider two widely known groups of proof methods in combinatorics, namely, probabilistic methods and methods using linear algebra. We will consider certain applications of such methods, both of which are significant to Ramsey theory. The systems we choose to work in are various theories of bounded arithmetic. For the probabilistic method, the key point is that we use the weak pigeonhole principle to simulate the probabilistic reasoning. We formalize various applications of the ordinary probabilistic method and linearity of expectations, making partial progress on the Local Lemma. In the case of linearity of expectations, we show how to eliminate the weak pigeonhole principle by simulating the derandomization technique of “conditional probabilities.” We consider linear algebra methods applied to various set system theorems. We formalize some theorems using a linear algebra principle as an extra axiom. We also show how weaker results can be attained by giving alternative proofs that avoid linear algebra, and thus also avoid the extra axiom. We formalize upper and lower Ramsey bounds. For the lower bounds, both the probabilistic methods and the linear algebra methods are used. We provide a stratification of the various Ramsey lower bounds, showing that stronger bounds can be proved in stronger theories. A natural question is whether or not the axioms used are necessary. We provide “reversals” in a few cases, showing that the principle used to prove the theorem is in fact a consequence of the theorem (over some base theory). Thus this work can be seen as a (humble) beginning in the direction of developing the Reverse Mathematics of finite combinatorics.

. For various proper inclusions of classes of groups C ae D, we find a group H 2 D and a first--order sentence ' such that H j= ' but no G 2 C satisfies '. The classes we consider include the classes of finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely Z p oZfor some prime p, which is the only f.g. group satisfying an appropriate first--order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(N;+; \Theta) can be interpreted in the theory of the class finitely presented groups and other classes of groups. 1.

Working in the base theory of PA - exp, we show that for all n #, the bounding principle for # n -formulas (B# n ) is equivalent to the induction principle for # n -formulas (I# n ). This partially answers a question of J. Paris; see Clote and Kraj cek (1993).

This class will be dealing with the intersection of randomness and computability theory, or how the tools of randomness can be applied to computability-theoretic problems. This class will not cover the history of the subject, which can be