Differentially closed fields and universality on a cone

TL;DR

This paper extends the universality of differentially closed fields by adding a predicate for algebraic transcendence, achieving broader spectrum universality on a cone but not for automorphism groups.

Contribution

It introduces a new signature for differential fields that accounts for the missing jump, expanding universality results to spectra and properties related to computable categoricity.

Findings

Universality for spectra of structures is achieved on a cone with the new signature.

Universality for automorphism groups does not hold, even non-effectively.

Results depend on an unknown decidability status of a specific oracle set.

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.