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Minimum models of analysis1

  • J. R. Shilleto (a1)

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Although there is no smallest ω model [6], Putnam and Gandy independently proved about 1963 that the class of ramified analytical sets, as defined by Cohen [3], form the smallest β model of analysis [2], [8]. This paper will consider other restricted classes of models, namely the βn models for integers n > 1 [10], and prove under appropriate assumptions the existence of minimum such models. In fact we shall construct the minimum βn model in a fashion similar to the procedure yielding the class of ramified analytical sets, but adding at each stage a segment of the sets in those already obtained (for n a fixed integer > 1).

Roughly speaking a βn model is an ω model absolute for n-set-quantifier assertions about its subsets of natural numbers. A β model is simply a β1 model. See Enderton and Friedman [4] for a further investigation of βn models.

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1

This paper presents the work of the author's doctoral dissertation submitted to the University of California at Berkeley in February 1969.

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[1]Addison, J. W., Some consequences of the axiom of constructibility, Fundamenta Mathematicae, vol. 46 (1959), pp. 337357.
[2]Boyd, R., Hensel, G., and Putnam, H., A recursion-theoretic characterization of the ramified analytical hierarchy, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 3762.
[3]Cohen, Paul J., A minimal model for set theory, Bulletin of the American Mathematical Society, vol. 69 (1963), pp. 537540.
[4]Enderton, H. B. and Friedman, Harvey, Approximating the standard model of analysis, forthcoming.
[5]Gödel, Kurt, The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set-theory, Annals of Mathematical Studies, no. 2, second edition, Princeton Univ. Press, Princeton, N.J., 1940.
[6]Gandy, R. O., Kreisel, G., and Tait, W. W., Set existence, Bulletin de l' Academie Polonaise des Sciences, Serie des Sciences, vol. 8 (1960), pp. 577582.
[7]Grzegorczyk, A., Mostowski, A., and Ryll-Nardzewski, C., The classical and the ω-complete arithmetic, this Journal, vol. 23 (1958), pp. 188200.
[8]Mostowski, A., Formal system of analysis based on an infinitistic rule of proof, Infinististic methods, Proceedings of the Symposium on the Foundations of Mathematics, Pergamon Press, Oxford, 1961, pp. 141166.
[9]Rogers, Hartley Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[10]Shilleto, J. R., The arithmetic of standard models of ZF, Notices of the American Mathematical Society, vol. 15 (1968), p. 549.
[11]Shoenfield, J. R., The problem of predicativity, Essays on the foundations of mathematics, Magnes Press, Jerusalem, 1961, pp. 132139.

Minimum models of analysis1

  • J. R. Shilleto (a1)

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