Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A conservation result concerning bounded theories and the collection axiom


Author: Samuel R. Buss
Journal: Proc. Amer. Math. Soc. 100 (1987), 709-715
MSC: Primary 03F30; Secondary 03C62
MathSciNet review: 894442
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Abstract: We present two proofs, one proof-theoretic and one model-theoretic, showing that adding the $ B\Sigma _1^0$-collection axioms to any bounded first-order theory $ R$ of arithmetic yields an extension which is $ \forall \Sigma _1^0$-conservative over $ R$.


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  • [1] Samuel R. Buss, Bounded arithmetic, Studies in Proof Theory. Lecture Notes, vol. 3, Bibliopolis, Naples, 1986. MR 880863
  • [2] Rohit Parikh, Existence and feasibility in arithmetic, J. Symbolic Logic 36 (1971), 494–508. MR 0304152
  • [3] J. B. Paris, Some conservation results for fragments of arithmetic, Model theory and arithmetic (Paris, 1979–1980) Lecture Notes in Math., vol. 890, Springer, Berlin-New York, 1981, pp. 251–262. MR 645006
  • [4] J. B. Paris and L. A. S. Kirby, Σ_{𝑛}-collection schemas in arithmetic, Logic Colloquium ’77 (Proc. Conf., Wrocław, 1977) Stud. Logic Foundations Math., vol. 96, North-Holland, Amsterdam-New York, 1978, pp. 199–209. MR 519815
  • [5] J. P. Ressayre, A conservation result for systems of bounded arithmetic, handwritten notes, 1985.
  • [6] Gaisi Takeuti, Proof theory, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 81, North-Holland Publishing Co., Amsterdam, 1987. With an appendix containing contributions by Georg Kreisel, Wolfram Pohlers, Stephen G. Simpson and Solomon Feferman. MR 882549

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0894442-1
Keywords: Bounded arithmetic, collection axioms, cut elimination, resplendency
Article copyright: © Copyright 1987 American Mathematical Society