Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Persistently finite theories with hyperarithmetic models


Author: Terrence Millar
Journal: Trans. Amer. Math. Soc. 278 (1983), 91-99
MSC: Primary 03C57; Secondary 03C50
DOI: https://doi.org/10.1090/S0002-9947-1983-0697062-8
MathSciNet review: 697062
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Nerode asked if there could be a complete decidable theory with only finitely many countable models up to isomorphism, such that not all of the countable models were decidable. Morley, Lachlan, and Peretyatkin produced examples of such theories. However, all the countable models of those theories were decidable in $ 0^{\prime}$. The question then arose whether all countable models of such theories had to be, for example, arithmetic. In this paper we provide a negative answer to that question by showing that there are such examples with countable models of arbitrarily high hyperarithmetic degree. It is not difficult to show that any countable model of a hyperarithmetic theory which has only finitely many countable models must be decidable in some hyperarithmetic degree.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03C57, 03C50

Retrieve articles in all journals with MSC: 03C57, 03C50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0697062-8
Article copyright: © Copyright 1983 American Mathematical Society