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An incompleteness theorem for βn-models

Published online by Cambridge University Press:  12 March 2014

Carl Mummert  and
Stephen G. Simpson
Carl Mummert
Affiliation:
Department of Mathematics, Pennsylvania State University, State College, PA 16802, USA, URL: http://www.math.psu.edu/mummert/, E-mail: [email protected]
Stephen G. Simpson
Affiliation:
Department of Mathematics, Pennsylvania State University, State College, PA 16802, USA, URL: http://www.math.psu.edu/simpson/, E-mail: [email protected]
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Abstract.

Let n be a positive integer. By a βn-model we mean an ω-model which is elementary with respect to formulas. We prove the following βn-model version of Gödel's Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a βn-model of S, then there exists a βn-model of S + “there is no countable βn-model of S”. We also prove a βn-model version of Löb's Theorem. As a corollary, we obtain a βn-model which is not a βn+1-model.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 69 , Issue 2 , June 2004 , pp. 612 - 616
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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