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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: mummert@math.psu.edu
Stephen G. Simpson
Affiliation:
Department of Mathematics, Pennsylvania State University, State College, PA 16802, USA, URL: http://www.math.psu.edu/simpson/, E-mail: simpson@math.psu.edu
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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

REFERENCES

[1]Blass, Andreas R., Hirst, Jeffry L., and Simpson, Stephen G., Logical analysis of some theorems of combinatorics and topological dynamics, in [9], 1987, pp. 125–156.CrossRefGoogle Scholar
[2]Enderton, Herbert B. and Friedman, Harvey, Approximating the standard model of analysis, Fundamenta Mathematicae, vol. 72 (1971), no. 2, pp. 175–188.CrossRefGoogle Scholar
[3]Engström, Fredrik, October 2003, Private communication.Google Scholar
[4]Friedman, Harvey, Subsystems of Set Theory and Analysis, Ph.D. thesis, Massachusetts Institute of Technology, 1967.Google Scholar
[5]Friedman, Harvey, Uniformly defined descending sequences of degrees, this Journal, vol. 41 (1976), pp. 363–367.Google Scholar
[6]Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198.CrossRefGoogle Scholar
[7]Löb, M. H., Solution of a problem of Leon Henkln, this Journal, vol. 20 (1955), pp. 115–118.Google Scholar
[8]Shilleto, J. R., Minimum models of analysis, this Journal, vol. 37 (1972), pp. 48–54.Google Scholar
[9]Simpson, S. G. (editor), Logic and Combinatorics, Contemporary Mathematics, American Mathematical Society, 1987.CrossRefGoogle Scholar
[10]Simpson, Stephen G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, 1999.CrossRefGoogle Scholar
[11]Steel, John R., Descending sequences of degrees, this Journal, vol. 40 (1975). pp. 59–61.Google Scholar