Abstract:
The spectrum of a first-order sentence is the set of cardinalities of its finite models. Relatively little is known about the subclasses of spectra that are obtained by looking only at sentences with a specific signature. Part of the problem is the fact that there are no natural characterizations of such classes in terms of complexity. In this talk, we study natural subclasses of spectra and their closure properties under simple subdiagonal functions. We show that many natural closure properties turn out to be equivalent to the collapse of potential spectrum hierarchies. We prove all of our results using explicit transformations on first-order structures. We propose that further development of this kind of model-theoretic technique may be necessary to solve some of the long-standing open problems in the theory of spectra. (This is a practice talk for LICS the following week - so any feedback will be appreciated)