Skip to main content
×
×
Home

Forcing in Proof Theory

  • Jeremy Avigad (a1)
Abstract

Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.

Copyright
References
Hide All
[1] Ajtai, Miklos, The complexity of the pigeonhole principle, Proceedings of the IEEE 29th annual symposium on foundations of computer science, 1988, pp. 346355.
[2] Avigad, Jeremy, Formalizing forcing arguments in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 82 (1996), pp. 165191.
[3] Avigad, Jeremy, Interpreting classical theories in constructive ones, The Journal of Symbolic Logic, vol. 65 (2000), pp. 17851812.
[4] Avigad, Jeremy, Algebraic proofs of cut elimination, The Journal of Logic and Algebraic Programming, vol. 49 (2001), pp. 1530.
[5] Avigad, Jeremy, Notes on -conservativity, ω-submodels, and the collection schema, Technical Report CMU-PHIL-125, Carnegie Mellon University, 2001.
[6] Avigad, Jeremy, Saturated models of universal theories, Annals of Pure and Applied Logic, vol. 118 (2002), pp. 219234.
[7] Avigad, Jeremy, Eliminating definitions and Skolem functions in first-order logic, ACM Transactions on Computer Logic, vol. 4 (2003), pp. 402415, Conference version: Proceedings of the 16th annual IEEE symposium on logic in computer science , pp. 139-146, 2001.
[8] Avigad, Jeremy, Weak theories of nonstandard arithmetic and analysis, to appear.
[9] Avigad, Jeremy and Helzner, Jeffrey, Transfer principles in nonstandard intuitionistic arithmetic, Archive for Mathematical Logic, vol. 41 (2002), pp. 581602.
[10] Baker, Theodore, Gill, John, and Solovay, Robert, Relativizations of the P = ? NP question, SIAM Journal of Computing, vol. 4 (1975), pp. 431442.
[11] Beeson, Micfael, Principles of continuous choice and continuity offunctions in formal systems for constructive mathematics, Annals of Mathematical Logic, vol. 12 (1977), pp. 249322.
[12] Beeson, Micfael, Continuity in intuitionistic set theories, Logic colloquium '78 (Mons, 1978), Studies in Logic and Foundations of Mathematics, vol. 97, North-Holland, Amsterdam, 1979, pp. 152.
[13] Beeson, Micfael, Foundations of constructive mathematics, Springer, Berlin, 1985.
[14] Boileau, Andre and Joyal, Andre, La logique des topos, The Journal of Symbolic Logic, vol. 46 (1981), pp. 616.
[15] Brown, Douglas and Simpson, Stepfen, The Baire category theorem in weak subsystems of second-order arithmetic, The Journal of Symbolic Logic, vol. 58 (1993), pp. 557578.
[16] Bucffolz, Wilfried, Ein ausgezeichnetes Modell fur die intuitionistische Typenlogik, Archive for Mathematical Logic, vol. 17 (1975), pp. 5560.
[17] Bucffolz, Wilfried, Feferman, Solomon, Poflers, Wolfram, and Sieg, Wilfried, Iterated inductive definitions and subsystems of analysis: Recent proof-theoretical studies, Lecture Notes in Mathematics, vol. 897, Springer, Berlin, 1981.
[18] Chellas, Brian, Modal logic: an introduction, Cambridge University Press, Cambridge, 1980.
[19] Cholak, Peter, Jockusch, Carl, and Slaman, Theodore, On the strength of Ramsey's theorem for pairs, The Journal of Symbolic Logic, vol. 66 (2001), pp. 155.
[20] Chuaqui, Rolando and Suppes, Patrick, Free-variable axiomatic foundations of infinitesimal analysis: a fragment with finitary consistency proof, The Journal of Symbolic Logic, vol. 60 (1995), pp. 122159.
[21] Cooper, S. Barry, Computability theory, Chapman & Hall / CRC Mathematics, Boca Raton, 2004.
[22] Coquand, Thierry, Constructive topology and combinatorics, Constructivity in computer science (San Antonio, TX, 1991), Lecture Notes in Computer Science, vol. 613, Springer, Berlin, 1992, pp. 159164.
[23] Coquand, Thierry, Computational content of classical logic, Semantics and logics of computation (Cambridge, 1995), Publications of the Newton Institute, vol. 14, Cambridge University Press, Cambridge, 1997, pp. 3378.
[24] Coquand, Thierry, Minimal invariant spaces in formal topology, The Journal of Symbolic Logic, vol. 62 (1997), pp. 689698.
[25] Coquand, Thierry, Two applications of Boolean models, Archive for Mathematical Logic, vol. 37 (1997), pp. 143147.
[26] Coquand, Thierry, A Boolean model of ultrafilters, Annals of Pure and Applied Logic, vol. 99 (1999), pp. 231239.
[27] Coquand, Thierry and Hofmann, Martin, A new method ofestablishing conservativity ofclassical systems over their intuitionistic version, Mathematical Structures in Computer Science, vol. 9 (1999), pp. 323333.
[28] Coquand, Thierry and Smith, Jan, An application of constructive completeness, Proccedings of the workshop TYPES '95 (S. Berardi and M. Copo, editors), Lecture Notes in Computer Science, vol. 1158, Springer, 1996, pp. 7684.
[29] Coste, Michel, Lombardi, Henri, and Roy, Marie-Françoise, Dynamical method in algebra: effective Nullstellensätze, Annals of Pure Applied Logic, vol. 111 (2001), pp. 203256.
[30] Dragalin, Albert, Mathematical intuitionism: Introduction to proof theory, Translations of mathematical monographs, American Mathematical Society, 1988.
[31] Fernandes, Antonio, Strict -reflection, Manuscript.
[32] Ferreira, Fernando, A feasible theory for analysis, The Journal of Symbolic Logic, vol. 59 (1994), pp. 10011011.
[33] Fitting, Melvin, Intuitionistic logic, model theory, and forcing, North-Holland, Amsterdam, 1969.
[34] Fourman, Michael, Continuous truth. I. Nonconstructive objects, Logic colloquium '82 (Florence, 1982), Studies in Logic and the Foundations of Mathematics, vol. 112, North-Holland, Amsterdam, 1984, pp. 161180.
[35] Fourman, Michael and Grayson, R. J., Formal spaces, The L. E. J. Brouwer centenary symposium (Noordwijkerhout, 1981), Studies in Logic and the Foundations of Mathematics, vol. 110, North-Holland, Amsterdam, 1982, pp. 107122.
[36] Goldblatt, Robert, The categorial analysis of logic, Topoi, Studies in Logic and the Foundations of Mathematics, vol. 98, North-Holland, Amsterdam, 1984.
[37] Goldblatt, Robert, Logics of time and computation, second ed., CSLI Lecture Notes, vol. 7, Stanford University Center for the Study of Language and Information, Stanford, CA, 1992.
[38] Goldblatt, Robert, Mathematical modal logic: a view of its evolution, Draft available at http://www.mcs.vuw.ac.nz/~rob.
[39] Grayson, R. J., Forcing in intuitionistic systems without power-set, The Journal of Symbolic Logic, vol. 48 (1983), pp. 670682.
[40] Hájek, Petr, Interpretability and fragments of arithmetic, Arithmetic, proof theory, and computational complexity (Peter Clote and Jan Krajíček, editors), Oxford University Press, Oxford, 1993, pp. 185196.
[41] Hayashi, Susumu, A note on bar induction rule, The L. E. J. Brouwer centenary symposium (Noordwijkerhout, 1981), Studies in Logic and the Foundations of Mathematics, vol. 110, North-Holland, Amsterdam, 1982, pp. 149163.
[42] Hilbert, David and Bernays, Paul, Grundlagen der Mathematik, vol. 1, Springer, Berlin, 1934, vol. 2, 1939.
[43] Hirst, Jeffry, Combinatorics in subsystems of second order arithmetic, Ph.D. thesis , The Pennsylvania State University, 1987.
[44] Hughes, G. E. and Cresswell, M. J., A new introduction to modal logic, Routledge, London, 1996.
[45] Jockusch, Carl, Ramsey's theorem and recursion theory, The Journal of Symbolic Logic, vol. 37 (1972), pp. 268280.
[46] Jockusch, Carl and Soare, Robert, classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3356.
[47] Johnstone, Peter, The point of pointless topology, American Mathematical Society. Bulletin. New Series., vol. 8 (1983), pp. 4153.
[48] Johnstone, Peter, The art of pointless thinking: a student's guide to the category of locales, Category theory at work (Bremen, 1990), Research and Exposition in Mathematics, vol. 18, Heldermann, Berlin, 1991, pp. 85107.
[49] Kechris, Alexander, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer, New York, 1995.
[50] Krajíček, Jan, Bounded arithmetic, propositional logic, and complexity theory, Encyclopedia of Mathematics and its Applications, vol. 60, Cambridge University Press, Cambridge, 1995.
[51] Kreisel, Georg, Axiomatizations of nonstandard analysis that are conservative extensions ofclassical systems ofanalysis, Applications ofmodel theory to algebra, analysis, and probability (Luxemburg, W. A. J., editor), Holt, Rinehart, and Winston, 1967, pp. 93106.
[52] Kunen, Kenneth, Set theory: an introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.
[53] Kuroda, S., Intuitionistische Untersuchungen der formalistischen Logik, Nagoya Mathematical Journal, vol. 2 (1951), pp. 3547.
[54] Lerman, Manuel, Degrees of unsolvability: local and global theory, Perspectives in Mathematical Logic, Springer, Berlin, 1983.
[55] Lubarsky, Robert S., IKP and friends, The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 12951322.
[56] Lane, Saunders Mac and Moerdijk, Ieke, Sheaves in geometry and logic, Springer, New York, 1992.
[57] Mathias, A. R. D., The strength of Mac Lane set theory, Annals of Pure and Applied Logic, vol. 110 (2001), pp. 107234.
[58] Moerdijk, Ieke, A modelfor intuitionistic non-standard arithmetic, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 3751.
[59] Moerdijk, Ieke and palmgren, Erik, Minimal models of Heyting arithmetic, The Journal of Symbolic Logic, vol. 62 (1997), pp. 14481460.
[60] Moerdijk, Ieke and Reyes, Gonzalo, Models for smooth infinitesimal analysis, Springer, New York, 1991.
[61] Moore, Gregory, The origins of forcing, Logic colloquium '86 (Hull, 1986), Studies in Logic and the Foundations of Mathematics, vol. 124, North-Holland, Amsterdam, 1988, pp. 143173.
[62] Nelson, Edward, Internal set theory: anew approach to nonstandard analysis, Bulletin of the American Mathematical Society, vol. 83 (1977), pp. 11651198.
[63] Okada, Mitsuhiro, A uniform semantic prooffor cut-elimination and completeness of various first and higher order logics, Theoretical Computer Science, vol. 281 (2002), pp. 471498.
[64] Palmgren, Erik, Constructive sheaf semantics, Mathematical Logic Quarterly, vol. 43 (1997) , pp. 321327.
[65] Palmgren, Erik, Developments in constructive nonstandard analysis, this Bulletin, vol. 4 (1998), pp. 233272.
[66] Palmgren, Erik, An effective conservation result for nonstandard analysis, Mathematical Logic Quarterly, vol. 46 (2000), pp. 1723.
[67] Riis, Søren, Making infinite structures finite in models of second order bounded arithmetic, Arithmetic, proof theory, and computational complexity (Prague, 1991), Oxford Logic Guides, vol. 23, Oxford University Press, New York, 1993, pp. 289319.
[68] Sacks, Gerald, Higher recursion theory, Perspectives in Mathematical Logic, Springer, Berlin, 1990.
[69] Sambin, Giovanni, Some points in formal topology, Theoretical Computer Science, vol. 305 (2003), pp. 347408.
[70] Seetapun, David and Slaman, Theodore A., On the strength of Ramsey's theorem, Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 570582.
[71] Shoenfield, Josepf, Mathematical logic, Association for Symbolic Logic, Urbana, IL, 2001, Reprint of the 1973 second printing.
[72] Simpson, Stepfen, Subsystems of second-order arithmetic, Springer, Berlin, 1998.
[73] Simpson, Stephen and Smith, Rick, Factorization of polynomials and induction, Annals of Pure and Applied Logic, vol. 31 (1986), pp. 289306.
[74] Simpson, Stephen, Tanaka, Kazuyuki, and Yamazaki, Takeshi, Some conservation results on weak König's lemma, Annals of Pure Applied Logic, vol. 118 (2002), pp. 87114.
[75] Sommer, Richard and Suppes, Patrick, Finite models of elementary recursive nonstandard analysis, Notas De la Sociedad de Matematica de Chile, vol. 15 (1996), pp. 7395.
[76] Takeuti, Gaisi, Two applications of logic to mathematics, Publications of the Mathematical Society of Japan, vol. 13, Iwanami Shoten and Princeton University Press, 1978.
[77] Takeuti, Gaisi, Proof theory, second ed., North-Holland, Amsterdam, 1987.
[78] Takeuti, Gaisi and Yasumoto, Masahiro, Forcing on bounded arithmetic. II, The Journal of Symbolic Logic, vol. 63 (1998), pp. 860868.
[79] Troelstra, A. S., Realizability, Handbook of proof theory, Studies in Logic and the Foundations of Mathematics, vol. 137, North-Holland, Amsterdam, 1998, pp. 407473.
[80] Troelstra, A. S. and Van Dalen, Dirk, Constructivism in mathematics: An introduction, vol. 1, North-Holland, Amsterdam, 1988.
[81] Weyl, Hermann, Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis, Veit, Leipzig, 1918, Second edition, 1932.
[82] Wilkie, Alex and Paris, Jeff, On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261302.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed