Recursive spectra of flat strongly minimal theories

TL;DR

This paper classifies the possible recursive spectra of model complete strongly minimal theories with flat pregeometries, narrowing down the unresolved cases to four specific sets.

Contribution

It provides a complete classification of the recursive spectra for a broad class of strongly minimal theories with flat pregeometries, identifying the exact possible forms.

Findings

Recursive spectra are either of the form [0,α) with α in ω+2, or [0,n]∪{ω} for n in ω, or {ω}, or subsets of {0,1,2}.

Four specific spectra remain unresolved in the classification.

The results extend previous classifications and narrow the scope of open problems.

Abstract

We show that for a model complete strongly minimal theory whose pregeometry is flat, the recursive spectrum (SRM($T$)) is either of the form $[0,\alpha)$ for $\alpha\in \omega+2$ or $[0,n]\cup\{\omega\}$ for $n\in \omega$, or $\{\omega\}$, or contained in $\{0,1,2\}$. Combined with previous results, this leaves precisely 4 sets for which it is not yet determined whether each is the spectrum of a model complete strongly minimal theory with a flat pregeometry.