13 research outputs found
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
On the jumps of degrees below an recursively enumerable degree
We consider the set of jumps below a Turing degree, given by JB(a) = {x(1) : x <= a}, with a focus on the problem: Which recursively enumerable (r.e.) degrees a are uniquely determined by JB(a)? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high(2) r.e. degree a is determined by JB(a), then R cannot have a nontrivial automorphism. We then defeat the strategy-at least in the form presented-by constructing pairs a(0), a(1) of distinct r.e. degrees such that JB(a(0)) = JB(a(1)) within any possible jump class {x : x' = c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity
Structural properties of the local Turing degrees
In this thesis we look at some properties of the local Turing Degrees, as a partial
order. We first give discussion of the Turing Degrees and certain historical results,
some translated into a form resembling the constructions we look at later.
Chapter 1 gives a introduction to the Turing Degrees, Chapter 2 introduces the
Local Degrees. In Chapter 3 we look at minimal Turing Degrees, modifying some
historical results to use a priority tree, which we use in chapter 4 to prove the new
result that every c.e. degree has the (minimal) meet property. Chapter 5 uses
similar methods to establish existence of a high 2 degree that does not have the meet
property
Definability in the Recursively Enumerable Degrees
this paper was presented to the Association at its annual meeting in Madison, March 1996 in a lecture given by the second author
Definability In The Recursively Enumerable Degrees
this paper was presented to the Association at its annual meeting in Madison, March 1996 in a lecture given by the second author
Parameter Definability in the Recursively Enumerable Degrees
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..