The spectrum Spec(M) of a countable structure M is the set of all Turing degrees of structures isomorphic to M. This topic has been the focus of much research. Here we describe the spectra of countable differentially closed fields of characteristic 0: they are precisely the preimages { d : d’ ∈ Spec(G) } of spectra of arbitrary countable graphs G, under the jump operation. To establish this, we describe the proofs of two theorems: one showing how to build the appropriate differential field K from a given graph G, and the other showing that every low model of the theory DCF₀ is isomorphic to a computable one. The latter theorem (which relativizes, to give the main result above) resembles the famous result of Downey and Jockusch on Boolean algebras, but the proof is different, yielding a Δ₂ isomorphism between the low model and its computable copy; moreover, our first theorem shows that the extension of the result to the low₄ case for Boolean algebras does not hold for DCF₀.

This is joint work by Dave Marker and the speaker. The slides for this talk are available here.

In 1936, only a few years after the incompleteness theorems were proved, Gentzen proved the consistency of Peano arithmetic by using transfinite induction up to the ordinal epsilon_0. I will give a short proof of the result, based on on the simplification introduced by Schutte, and discuss some of the consequences.