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A New Spectrum of Recursive Models Using An Amalgamation Construction
 

Summary: A New Spectrum of Recursive Models Using An
Amalgamation Construction
Uri Andrews
September 1, 2010
Abstract
We employ an infinite-signature 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 non-trivial example of a spectrum
produced in a finite language.
1 Introduction
Baldwin and Lachlan [2] developed the theory of 1-categoricity in terms of strongly
minimal sets. They show, in particular, that for any 1-categorical theory in a countable
language which is not 0-categorical, 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

  

Source: Andrews, Uri - Department of Mathematics, University of Wisconsin at Madison

 

Collections: Mathematics