 
Summary: A New Spectrum of Recursive Models Using An
Amalgamation Construction
Uri Andrews
September 1, 2010
Abstract
We employ an infinitesignature Hrushovski amalgamation construction to yield
two results in Recursive Model Theory. The first result, that there exists a strongly
minimal theory whose only recursively presentable models are the prime and saturated
models, adds a new spectrum to the list of known possible spectra. The second result,
that there exists a strongly minimal theory in a finite language whose only recursively
presentable model is saturated, gives the second nontrivial example of a spectrum
produced in a finite language.
1 Introduction
Baldwin and Lachlan [2] developed the theory of 1categoricity in terms of strongly
minimal sets. They show, in particular, that for any 1categorical theory in a countable
language which is not 0categorical, the countable models form a chain of elementary
embeddings of length +1: M0 M1 . . . M, where M0 is the prime model and M
is the saturated model. Furthermore, there is a strongly minimal formula such that each
Mi is characterized by the size of a maximal algebraically independent subset realizing
that formula. In particular, the models of a strongly minimal theory are characterized by
