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An incompleteness theorem for βn-models

  • Carl Mummert (a1) and Stephen G. Simpson (a2)

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.

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[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. 125156.
[2]Enderton, Herbert B. and Friedman, Harvey, Approximating the standard model of analysis, Fundamenta Mathematicae, vol. 72 (1971), no. 2, pp. 175188.
[3]Engström, Fredrik, October 2003, Private communication.
[4]Friedman, Harvey, Subsystems of Set Theory and Analysis, Ph.D. thesis, Massachusetts Institute of Technology, 1967.
[5]Friedman, Harvey, Uniformly defined descending sequences of degrees, this Journal, vol. 41 (1976), pp. 363367.
[6]Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173198.
[7]Löb, M. H., Solution of a problem of Leon Henkln, this Journal, vol. 20 (1955), pp. 115118.
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[10]Simpson, Stephen G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, 1999.
[11]Steel, John R., Descending sequences of degrees, this Journal, vol. 40 (1975). pp. 5961.

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An incompleteness theorem for βn-models

  • Carl Mummert (a1) and Stephen G. Simpson (a2)

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