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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Every low Boolean algebra is isomorphic to a recursive one


Authors: Rod Downey and Carl G. Jockusch
Journal: Proc. Amer. Math. Soc. 122 (1994), 871-880
MSC: Primary 03C57; Secondary 03D30, 03D45, 06E99
MathSciNet review: 1203984
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Abstract: It is shown that every (countable) Boolean algebra with a presentation of low Turing degree is isomorphic to a recursive Boolean algebra. This contrasts with a result of Feiner (1967) that there is a Boolean algebra with a presentation of degree $ \leq 0' $ which is not isomorphic to a recursive Boolean algebra. It is also shown that for each n there is a finitely axiomatizable theory $ {T_n}$ such that every $ {\text{low}_n}$ model of $ {T_n}$ is isomorphic to a recursive structure but there is a $ {\text{low}_{n + 1}}$ model of $ {T_n}$ which is not isomorphic to any recursive structure. In addition, we show that $ n + 2$ is the Turing ordinal of the same theory $ {T_n}$, where, very roughly, the Turing ordinal of a theory describes the number of jumps needed to recover nontrivial information from models of the theory. These are the first known examples of theories with Turing ordinal $ \alpha $ for $ 3 \leq \alpha < \omega $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1203984-4
PII: S 0002-9939(1994)1203984-4
Article copyright: © Copyright 1994 American Mathematical Society