The Use of Trustworthy Principles in a Revised Hilbert’s Program

Abstract

After the failure of Hilbert’s original program due to Gödel’s second incompleteness theorem, relativized Hilbert’s programs have been suggested. While most metamathematical investigations are focused on carrying out mathematical reductions, we claim that in order to give a full substitute for Hilbert’s program, one should not stop with purely mathematical investigations, but give an answer to the question why one should believe that all theorems proved in certain mathematical theories are valid.

We suggest that, while it is not possible to obtain absolute certainty, it is possible to develop trustworthy core principles using which one can prove the correctness of mathematical theories. Trust can be established by both providing a direct validation of such principles, which is necessarily non-mathematical and philosophical in nature, and at the same time testing those principles using metamathematical investigations. We investigate three approaches for trustworthy principles, namely ordinal notation systems built from below, Martin-Löf type theory, and Feferman’s system of explicit mathematics. We will review what is known about the strength up to which direct validation can be provided.

References

  1. 1.
    T. Arai, Proof theory of theories of ordinals III: \(\Pi _{1}\) collection. Unpublished notes (1997)Google Scholar
  2. 2.
    T. Arai, A sneak preview of proof theory of ordinals. arXiv:1102.0596v1 [math.LO] (2011). http://arxiv.org/abs/1102.0596
  3. 3.
    P. Dybjer, Program testing and the meaning explanations of intuitionistic type theory, in Epistemology Versus Ontology (Springer, Dordrecht, 2012), pp. 215–241CrossRefGoogle Scholar
  4. 4.
    P. Dybjer, A. Setzer, Induction-recursion and initial algebras. Ann. Pure Appl. Log. 124, 1–47 (2003)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Feferman, A language and axioms for explicit mathematics, in Algebra and Logic, ed. by J. Crossley. Lecture Notes in Mathematics, vol. 450 (Springer, Berlin/Heidelberg, 1975), pp. 87–139. doi:10.1007/BFb0062852Google Scholar
  6. 6.
    S. Feferman, Constructive theories of functions and classes, in Logic Colloquium ’78 Proceedings of the Colloquium Held in Mons, ed. by D.D. Maurice Boffa, K. McAloon. Studies in Logic and the Foundations of Mathematics, vol. 97 (North Holland, Amsterdam, 1979), pp. 159–224Google Scholar
  7. 7.
    S. Feferman, Hilbert’s program relativized: proof-theoretical and foundational reductions. J. Symb. Log. 53(2), 364–384 (1988)MATHMathSciNetGoogle Scholar
  8. 8.
    S. Feferman. What rests on what? The proof-theoretic analysis of mathematics, in Philosophy of Mathematics. Proceedings of the Fifteenth International Wittgenstein-Symposium, Part 1, ed. by J. Czermak (Hölder-Pichler-Tempsky, Vienna, 1993), pp. 147–171. Reprinted in Solomon Feferman: In the light of Logic. Oxford University Press, Oxford, 1998, Ch. 10, 187–208Google Scholar
  9. 9.
    S. Feferman, Why a little bit goes a long way: logical foundations of scientifically applicable mathematics. PSA 2, 442–455 (1993). Reprinted in Feferman: In the light of logic. Oxford University Press, Oxford, 1998, Ch. 14, pp. 284–298Google Scholar
  10. 10.
    S. Feferman, Does reductive proof theory have a viable rationale? Erkenntnis 53(1–2), 63–96 (2000)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    S. Feferman, J.W. Dawson Jr., S.C. Kleene, G.H. Moore, R.M. Solovay, J. van Heijenoort (eds.), Kurt Gödel. Collected Works: Volume II: Publications 1938–1974 (Oxford University Press, Oxford, 1990)Google Scholar
  12. 12.
    D. Fridlender, A proof-irrelevant model of Martin-Löf’s logical framework. Math. Struct. Comput. Sci. 12, 771–795 (2002)MATHMathSciNetGoogle Scholar
  13. 13.
    K. Gödel, Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica 12(3–4), 280–287 (1958). Translation in [11], pp. 240–251Google Scholar
  14. 14.
    E. Griffor, M. Rathjen, The strength of some Martin-Löf type theories. Arch. Math. Log. 33(5), 347 (1994)Google Scholar
  15. 15.
    H. Hesse, Das Glasperlenspiel. Versuch einer Lebensbeschreibung des Magister Ludi Josef Knecht samt Knechts hinterlassenen Schriften. (English: The Glass Bead Game) (Fretz & Wasmuth, Zürich, 1943)Google Scholar
  16. 16.
    R. Kahle, A. Setzer, An extended predicative definition of the Mahlo universe, in Ways of Proof Theory, ed. by R. Schindler. Ontos Series in Mathematical Logic (Ontos Verlag, Frankfurt (Main), 2010), pp. 309–334Google Scholar
  17. 17.
    G. Kreisel, A variant of Hilbert’s theory of the foundations of arithmetic. Br. J. Philos. Sci. IV(14), 107–129 (1953)Google Scholar
  18. 18.
    G. Kreisel, A survey of proof theory. J. Symb. Log. 33(3), 321–388 (1968)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    G. Kreisel, Hilbert’s programme, in Philosophy of Mathematics: Selected Readings, 2nd edn, ed. by P. Benacerraf, H. Putnam (Cambridge University Press, Cambridge, 1983), pp. 207–238. Revised version of article in Dialectica 12(3–4), pp. 346–372, 1958Google Scholar
  20. 20.
    P. Martin-Löf, Intuitionistic Type Theory (Bibliopolis, Naples, 1984)MATHGoogle Scholar
  21. 21.
    P. Martin-Löf, Truth of a proposition, evidence of a judgement, validity of a proof. Synthese 73, 407–420 (1987)MathSciNetCrossRefGoogle Scholar
  22. 22.
    P. Martin-Löf, On the meaning of the logical constants and the justification of the logical laws. Nord. J. Philos. Log. 1(1):11–60 (1996). Short course given at the meeting Teoria della Dimostrazione e Filosofia della Logica, organized in Siena, 6–9 April 1983, by the Scuola di Specializzazione in Logica Matematica of the Università degli Studi di Siena. Available from http://www.hf.uio.no/filosofi/njpl/vol1no1/meaning/meaning.html
  23. 23.
    P. Martin-Löf, An intuitionistic theory of types, in Twenty-Five Years of Constructive Type Theory, ed. by G. Sambin, J. Smith (Oxford University Press, Oxford, 1998), pp. 127–172. Reprinted version of an unpublished report from 1972Google Scholar
  24. 24.
    P. Martin-Löf, The Hilbert-Brouwer controversy resolved? in One Hundred Years of Intuitionism (1907–2007), ed. by M. Atten, P. Boldini, M. Bourdeau, G. Heinzmann. Publications des Archives Henri Poincaré, Publications of the Henri Poincaré Archives (Birkhäuser, Basel, 2008), pp. 243–256Google Scholar
  25. 25.
    B. Nordstrom, K. Petersson, J. Smith, Martin-löf’s type theory, in Handbook of Logic in Computer Science: Logic and Algebraic Methods, vol. 5, ed. by S. Abramsky, D.M. Gabbay, T. Maibaum (Oxford University Press, Oxford, 2001), pp. 1–38Google Scholar
  26. 26.
    B. Nordström, K. Petersson, J.M. Smith, Programming in Martin-Löf’s Type Theory. An Introduction. (Oxford University Press, Oxford, 1990). Book out of print. Online version available via http://www.cs.chalmers.se/Cs/Research/Logic/book/
  27. 27.
    E. Palmgren, On universes in type theory, in Twenty Five Years of Constructive Type Theory, ed. by G. Sambin, J. Smith (Oxford University Press, Oxford, 1998), pp. 191–204Google Scholar
  28. 28.
    E. Palmgren, V. Stoltenberg-Hansen, Domain interpretations of Martin-Löf’s partial type theory. Ann. Pure Appl. Log. 48, 135–196 (1990)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    A. Ranta, Type-Theoretical Grammar (Oxford University Press, Oxford, 1995)Google Scholar
  30. 30.
    M. Rathjen, The strength of Martin-Löf type theory with a superuniverse, part I. Arch. Math. Log. 39(1), 1–39 (2000)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    M. Rathjen, The strength of Martin-Löf type theory with a superuniverse, part II. Arch. Math. Log. 40(3), 207–233 (2001)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    M. Rathjen, The constructive Hilbert program and the limits of Martin-Löf type theory. Synthese 147, 81–120 (2005)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    M. Rathjen, An ordinal analysis of parameter free \(\Pi _{2}^{1}\)-comprehension. Arch. Math. Log. 44(3), 263–362 (2005)MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    A. Setzer, Proof theoretical strength of Martin-Löf type theory with W-type and one universe, Ph.D. thesis, Universität München, 1993. http://www.cs.swan.ac.uk/~csetzer
  35. 35.
    A. Setzer, Well-ordering proofs for Martin-Löf type theory. Ann. Pure Appl. Log. 92, 113–159 (1998)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    A. Setzer, Ordinal systems, in Sets and Proofs, ed. by S. Barry Cooper, J. Truss (Cambridge University Press, Cambridge, 1999), pp. 301–331CrossRefGoogle Scholar
  37. 37.
    A. Setzer, Ordinal systems part 2: one inaccessible, in Logic Colloquium ’98, ed. by S. Buss, P. Hajek, P. Pudlak. ASL Lecture Notes in Logic 13 (Peters Ltd., Natick, 2000), pp. 426–448Google Scholar
  38. 38.
    A. Setzer, Universes in type theory Part II – autonomous Mahlo. Ann. Pure Appl. Log. (2011, submitted). http://www.cs.swan.ac.uk/~csetzer/articles/modelautomahlomain.pdf
  39. 39.
    A. Setzer, Coalgebras as types determined by their elimination rules, in Epistemology Versus Ontology, ed. by P. Dybjer, S. Lindström, E. Palmgren, G. Sundholm. Logic, Epistemology, and the Unity of Science, vol. 27 (Springer, Dordrecht, 2012), pp. 351–369. doi:10.1007/978-94-007-4435-6_16
  40. 40.
    Stanford Encyclopedia of Philosophy, Hilbert’s program (2003). http://plato.stanford.edu/entries/hilbert-program/#1
  41. 41.
    W.W. Tait, Finitism. J. Philos. 78(9), 524–546 (1981)CrossRefGoogle Scholar
  42. 42.
    A. Tasistro, Substitution, record types and subtyping in type theory, with applications to the theory of programming, Ph.D. thesis, Department of Computing Science, University of Gothenburg, Gothenburg, 1997Google Scholar
  43. 43.
    W.H. Woodin, The tower of Hanoi, in Truth in Mathematics, ed. by H.H.G. Dales, G. Oliveri (Oxford University Press, Oxford, 1998), pp. 329–351Google Scholar
  44. 44.
    R. Zach, Hilbert’s program then and now, in Philosophy of Logic, ed. by D. Jacquette (North-Holland, Amsterdam, 2007), pp. 411–447CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceSwansea UniversitySwanseaUK

Personalised recommendations