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New Spectra of Strongly Minimal Theories in Finite Languages Uri Andrews
 

Summary: New Spectra of Strongly Minimal Theories in Finite Languages
Uri Andrews
Abstract
We describe strongly minimal theories Tn with finite languages such that in the chain of
countable models of Tn, only the first n models have recursive presentations. Also, we
describe a strongly minimal theory with a finite language such that every non-saturated
model has a recursive presentation.
1. Introduction
Given an 1-categorical non-0-categorical theory T in a countable language, the
Baldwin-Lachlan theorem [2] says that the countable models of T form an + 1-chain:
M0 M1 . . . M. We define the spectrum of recursive models of T to be
SRM(T) = {i|Mi has a recursive presentation}. The spectrum problem asks "Which
subsets of + 1 can occur as spectra of 1-categorical theories?", and of particular inter-
est is which subsets of + 1 can occur as spectra of strongly minimal theories.
There have been various contributions to the spectrum problem over the years. Many
have been of the form "S is a possible spectrum achieved with a strongly minimal (or
simply 1-categorical) theory". In this paper, the goal is to achieve many of the same
spectra while using a theory in a finite language. This goal has its roots in Herwig,
Lempp, Ziegler [3], where it is shown that {0} is a possible spectrum using only a finite
language. In [1], we show that {} is a possible spectrum using only a finite language.

  

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

 

Collections: Mathematics