Abstract
Leo Harrington showed that the second-order theory of arithmetic WKL 0 is \({\Pi^1_1}\)-conservative over the theory RCA 0. Harrington’s proof is model-theoretic, making use of a forcing argument. A purely proof-theoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cut-elimination argument.
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References
Avigad J.: Formalizing forcing arguments in subsystems of second-order arithmetic. Ann. Pure Appl. Logic 82, 165–191 (1996)
Avigad, J., Feferman, S.: Gödel’s functional (“Dialectica”) interpretation. In: Buss, S.R. (ed.) Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, vol. 137, pp. 337–405. North Holland, Amsterdam (1998)
Bridges, D.S., Richman, F.: Varieties of Constructive Mathematics, London Mathematical Society Lecture Notes Series, vol. 97. Cambridge University Press, Cambridge (1987)
Buss, S.R.: Bounded arithmetic. Ph.D. thesis, Princeton University, Princeton, New Jersey (1985). A revision of this thesis was published by Bibliopolis (Naples) in 1986
Buss, S.R.: First-order proof theory of arithmetic. In: Buss, S.R. (ed.) Handbook of Proof Theory, vol. 137, pp. 79–147. Elsevier, Amsterdam (1998)
Buss S.R.: An introduction to proof theory. In: Buss, S.R. (ed.) Handbook of Proof Theory, vol. 137, pp. 1–78. Elsevier, Amsterdam (1998)
Fernandes A.: Strict \({\Pi^1_1}\)-reflection in bounded arithmetic. Submitted
Ferreira F.: A feasible theory for analysis. J. Symbolic Logic 59(3), 1001–1011 (1994)
Ferreira F.: A simple proof of Parson’s theorem. Notre Dame J. Formal Logic 46, 83–91 (2005)
Ferreira F., Oliva P.: Bounded functional interpretation. Ann. Pure Appl. Logic 135, 73–112 (2005)
Ferreira, G.: Sistemas de Análise Fraca para a Integração. Ph.D. thesis, Faculdade de Ciências da Universidade de Lisboa (2006), in Portuguese
Friedman H.: Systems of second order arithmetic with restricted induction, I, II (abstracts). J. Symbolic Logic 41, 557–559 (1976)
Kaye, R.: Models of Peano Arithmetic, Oxford Logic Guides, vol. 15. Clarendon Press, Oxford (1991)
Kohlenbach U.: Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization. J. Symbolic Logic 57, 1239–1273 (1992)
Kohlenbach U.: Remarks on Herbrand normal forms and Herbrand realizations. Arch. Math. Logic 31, 305–317 (1992)
Parsons, C.: On a number theoretic choice schema and its relation to induction. In: Kino, A., Myhill, J., Vesley, R.E. (eds.) Intuitionism and Proof Theory, pp. 459–473. North-Holland, Amsterdam (1970)
Sieg W.: Fragments of arithmetic. Ann. Pure Appl. Logic 28, 33–71 (1985)
Sieg W.: Herbrand analyses. Arch. Math. Logic 30, 409–441 (1991)
Simpson S.G.: Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1999)
Takeuti, G.: Proof Theory. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1987)
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Ferreira, F., Ferreira, G. Harrington’s conservation theorem redone. Arch. Math. Logic 47, 91–100 (2008). https://doi.org/10.1007/s00153-008-0080-8
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DOI: https://doi.org/10.1007/s00153-008-0080-8