Differentially closed fields and universality on a cone

Abstract

The class of all countable differentially closed differential fields $K$ of characteristic $0$ was shown by Marker and the author to be "one jump away" from universal for spectra of structures: for every nontrivial countable structure $\mathfrak M$, there is some $K$ whose spectrum is the preimage under jump of the spectrum of $\mathfrak M$, and conversely, for every $K$, there is such an $\mathfrak M$. We show that the missing jump can be accounted for by adding to the signature of differential fields a predicate describing a certain algebraic transcendence property. The ensuing universality results for differentially closed fields in the new signature include not only spectra of structures, but also many properties related to computable categoricity. However, these latter universality results hold only on the cone above a specific $\Sigma^0_1$ oracle set, whose decidability status remains unknown. Moreover, differentially closed fields simply fail flat-out to be universal for automorphism groups, even non-effectively. We also include a small erratum to an earlier work.