First PositionResearch Fellow, National University of Singapore
DissertationSets, Models, And Proofs: Topics In The Theory Of Recursive Functions
We prove results in various areas of recursion theory. First, in joint work with Richard Shore, we prove a new jump-inversion result for ideals of recursively enumerable (r.e.) degrees; this defeats what had seemed to be a promising tack on the automorphism problem for the semilattice R of r.e. degrees. Second, in work spanning two chapters, we calibrate the reverse-mathematical strength of a number of theorems of basic model theory, such as the Ryll-Nardzewski atomic-model theorem, Vaught's no-two-model theorem, Ehrenfeucht's three-model theorem, and the existence theorems for homogeneous and saturated models. Whereas most of these are equivalent over RCA0 to one of RCA0 , WKL0 , ACA0 , as usual, we also uncover model-theoretic statements with exotic complexities such as ¬WKL0 ∨ ACA0 and WKL0 ∨ I$\Sigma$0 . 2 Third, we examine the possible weak truth table (wtt) degree spectra of countable first-order structures. We find several points at which the wtt- and Turingdegree cases differ, notably that the most direct wtt analogue of Knight's dichotomy theorem does not hold. Yet we find weaker analogies between the two, including a new trichotomy theorem for wtt degree spectra in the spirit of Knight's.