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From bounded arithmetic to second order arithmetic via automorphisms

Published online by Cambridge University Press:  30 March 2017

Ali Enayat
Affiliation:
American University, Washington DC
Ali Enayat
Affiliation:
American University, Washington DC
Iraj Kalantari
Affiliation:
Western Illinois University
Mojtaba Moniri
Affiliation:
Tarbiat Modares University, Tehran, Iran
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Logic in Tehran , pp. 87 - 113
Publisher: Cambridge University Press
Print publication year: 2006

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References

[AH] F., Abramson and L., Harrington, Models without indiscernibles,The Journal of Symbolic Logic, vol. 43 (1978), no. 3, pp. 572–600.Google Scholar
[Av] J., Avigad, Number theory and elementary arithmetic,Philosophia Mathematica, vol. 11 (2003), no. 3, pp. 257–284.Google Scholar
[Be] J. H., Bennett, On Spectra, Ph.D. dissertation, Princeton University, 1962.
[Bu] S., Buss, First-order proof theory of arithmetic,Handbook of Proof Theory (S., Buss, editor), North-Holland, Amsterdam, 1998, pp. 79–147.
[CK] C. C., Chang and H. J., Keisler, Model Theory, North-Holland, Amsterdam, 1973.
[D] P., D'Aquino, A sharpened version of McAloon's theorem on initial segments of models of IΔ0, Annals of Pure and Applied Logic, vol. 61 (1993), no. 1-2, pp. 49–62.Google Scholar
[DG] C., Dimitracopoulos and H., Gaifman, Fragments of Peano's arithmetic and the MRDP theorem,Logic and Algorithmic, Monograph. Enseign. Math., vol. 30, Univ. Genève, Geneva, 1982, pp. 187–206.
[E-1] A., Enayat, Automorphisms, Mahlo cardinals, and NFU,Nonstandard Models of Arithmetic and Set Theory (A., Enayat and R., Kossak, editors), Contemporary Mathematics Series, vol. 361, American Mathematical Society, Providence, RI, 2004, pp. 37–59.
[E-2] A., Enayat A., Enayat and R., Kossak, editors, Automorphisms of models of bounded arithmetic, to appear.
[E-3] A., Enayat A., Enayat and R., Kossak, editors, Weakly compact cardinals and automorphisms, to appear.
[ER] P., Erdös and R., Rado, A combinatorial theorem,Journal of the London Mathematical Society, vol. 25 (1950), pp. 249–255.Google Scholar
[Fe] U., Felgner, Comparison of the axioms of local and universal choice,Fundamenta Mathematicae, vol. 71 (1971), no. 1, pp. 43–62. (errata insert).Google Scholar
[Fo] T. E., Forster, Set Theory with a Universal Set, Oxford Logic Guides, vol. 31, Oxford University Press, New York, 1995.
[Fr] H., Friedman, Translatability and Relative Consistency, II, Ohio State University, unpublished notes, 1979.
[G] H., Gaifman, Models and types of Peano's arithmetic,Annals of Pure and Applied Logic, vol. 9 (1976), no. 3, pp. 223–306.Google Scholar
[GRS] R., Graham, B., Rothschild, and J., Spencer, Ramsey Theory, John Wiley & Sons, New York, 1980.
[HP] P., Hájek and P., Pudlák, Metamathematics of First-Order Arithmetic, Springer, Berlin, 1993.
[Ho-1] R., Holmes, Strong axioms of infinity in NFU,The Journal of Symbolic Logic, vol. 66 (2001), no. 1, pp. 87–116.Google Scholar
[Jec] T., Jech, Set Theory, Academic Press, New York, 1978.
[Jen] R. B., Jensen, On the consistency of a slight (?) modification of quine's new foundations,Synthese, vol. 19 (1969), pp. 250–263.Google Scholar
[Jo] C., Jockusch, Ramsey's theorem and recursion theory,The Journal of Symbolic Logic, vol. 37 (1972), pp. 268–280.Google Scholar
[Kan] A., Kanamori, The Higher Infinite, Springer-Verlag, Berlin, 1994.
[Kay] R., Kaye, Model-theoretic properties characterizing Peano arithmetic,The Journal of Symbolic Logic, vol. 56 (1991), no. 3, pp. 949–963.Google Scholar
[KKK] R., Kaye, R., Kossak, and H., Kotlarski, Automorphisms of recursively saturated models of arithmetic,Annals of Pure and Applied Logic, vol. 55 (1991), no. 1, pp. 67–99.Google Scholar
[KM] R., Kaye and D., MacPherson, Automorphisms of First-Order Structures, Oxford University Press, New York, 1994.
[Ki-1] L., Kirby, Initial Segments of Models of Arithmetic, Ph.D. thesis, University of Manchester, 1977.
[Ki-2] L., Kirby, Ultrafilters and types on models of arithmetic,Annals of Pure and Applied Logic, vol. 27 (1984), no. 3, pp. 215–252.Google Scholar
[KP] L., Kirby and J., Paris, Initial segments of models of Peano's axioms,Set Theory and Hierarchy Theory, V, Lecture Notes in Mathematics, vol. 619, Springer, Berlin, 1977, pp. 211–226.
[Kr] J., Krajı cek, Bounded Arithmetic, Propositional Logic, and Complexity Theory, Cambridge University Press, Cambridge, 1995.
[Ku] K., Kunen, Some applications of iterated ultrapowers in set theory,Annals of Pure and Applied Logic, vol. 1 (1970), pp. 179–227.Google Scholar
[MS] R., Mac Dowell and E., Specker, Modelle der Arithmetik,Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959), Pergamon, Oxford, 1961, pp. 257–263.
[Mile] J., Mileti, The canonical ramsey theorem and computability theory, to appear.
[Mill] G., Mills, A model of Peano arithmetic with no elementary end extension,The Journal of Symbolic Logic, vol. 43 (1978), no. 3, pp. 563–567.Google Scholar
[Mo-1] A., Mostowski, Models of second order arithmetic with definable Skolem functions,Fundamenta Mathematicae, vol. 75 (1972), no. 3, pp. 223–234.Google Scholar
[Mo-2] A., Mostowski, Errata to “Models of second order arithmetic with definable Skolem functions”,Fundamenta Mathematicae, vol. 84 (1974), no. 2, p. 173.Google Scholar
[Pa] R., Parikh, Existence and feasibility in arithmetic,The Journal of Symbolic Logic, vol. 36 (1971), pp. 494–508.Google Scholar
[Pu-1] P., Pudlák, A definition of exponentiation by a bounded arithmetical formula,Commentationes Mathematicae Universitatis Carolinae, vol. 24 (1983), no. 4, pp. 667–671.Google Scholar
[Pu-2] P., Pudlák, Cuts, consistency statements and interpretations,The Journal of Symbolic Logic, vol. 50 (1985), no. 2, pp. 423–441.Google Scholar
[Q] W. V. O, Quine, New Foundations for Mathematical Logic,The American Mathematical Monthly, vol. 44 (1937), no. 2, pp. 70–80.Google Scholar
[Ra] R., Rado, Note on canonical partitions,The Bulletin of the London Mathematical Society, vol. 18 (1986), no. 2, pp. 123–126.Google Scholar
[Re] J.-P., Ressayre, Nonstandard universes with strong embeddings, and their finite approximations,Logic and Combinatorics (S., Simpson, editor), Contemporary Mathematics, vol. 65, Amer. Math. Soc., Providence, RI, 1987, pp. 333–358.
[Sc] J., Schmerl, Automorphism groups of models of Peano arithmetic,The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1249–1264.Google Scholar
[Si-1] S., Simpson, Review of [Fe] and [Mo-1], The Journal of Symbolic Logic, vol. 38, pp. 652– 653.
[Si-2] S., Simpson, Subsystems of Second Order Arithmetic, Springer, Berlin, 1999.
[Sm] C., Smory nski, Nonstandard models and related developments,Harvey Friedman's Research on the Foundations of Mathematics (L. A., Harrington et al., editors), North-Holland, Amsterdam, 1985, pp. 179–229.
[Sol] R., Solovay, The consistency strength of NFUB, preprint available at Front for the Mathematics ArXiv, http://front.math.ucdavis.edu.
[Som-1] R., Sommer, Transfinite Induction and Hierarchies Generated by Transfinite Recursion within Peano Arithmetic, Ph.D. thesis, University of California, Berkeley, 1990.
[Som-2] R., Sommer, Transfinite induction within Peano arithmetic,Annals of Pure and Applied Logic, vol. 76 (1995), no. 3, pp. 231–289.Google Scholar
[V] A., Visser, Categories of theories and interpretations, this volume.
[W] H., Wang, Popular Lectures on Mathematical Logic, Dover Publications, New York, 1993.

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