Theory spectra and classes of theories

Abstract

We analyze the spectra of theories that are ω \omega -stable, theories whose spectra include almost every degree, and theories with uniformly arithmetical n n -quantifier fragments. We answer a question from Andrews and Miller (2015) by showing that there are ω \omega -stable theories whose spectra are not structure spectra. We show that the spectrum created by Andrews and Knight (2013) is not the spectrum of an ω \omega -stable theory, but is the minimal spectrum of any theory with uniformly arithmetical n n -quantifier fragments. In addition, we give examples of theory spectra that contain almost every degree, including ones that are known not to be structure spectra.