Combining Type Theory and Untyped Set Theory

  • Chad E. Brown
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4130)


We describe a dependent type theory with proof irrelevance. Within this framework, we give a representation of a form of Mac Lane set theory and discuss automated support for constructing proofs within this set theory. One of the novel aspects of the representation is that one is allowed to use any class (in the set theory) as a type (in the type theory). Such class types allow a natural way of representing partial functions (e.g., the first and second operators on the class of Kuratowski ordered pairs). We also discuss how automated search can be used to construct proofs. In particular, the first-order prover Vampire can be called to solve a challenge problem (the injective Cantor Theorem) which is notoriously difficult for higher-order automated provers.


Type Theory Class Type Simple Type Automate Reasoning Logical Framework 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abel, A., Coquand, T., Norell, U.: Connecting a logical framework to a first-order logic prover. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Andrews, P.B., Bishop, M.: On sets, types, fixed points, and checkerboards. In: Miglioli, P., Moscato, U., Ornaghi, M., Mundici, D. (eds.) TABLEAUX 1996. LNCS, vol. 1071, pp. 1–15. Springer, Heidelberg (1996)Google Scholar
  3. 3.
    Andrews, P.B., Bishop, M., Brown, C.E.: System description: TPS: A theorem proving system for type theory. In: CADE 2000. LNCS, vol. 1831, pp. 164–169. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Avron, A.: Formalizing set theory as it is actually used. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) MKM 2004. LNCS, vol. 3119, pp. 32–43. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Belinfante, J.G.F.: Computer Proofs in Gödel’s class theory with equational definitions for composite and cross. Journal of Automated Reasoning 22, 311–339 (1999)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Boyer, R., Lusk, E., McCune, W., Overbeek, R., Stickel, M., Wos, L.: Set theory in first-order logic: Clauses for Gödel’s axioms. Journal of Automated Reasoning 2, 287–327 (1986)MATHCrossRefGoogle Scholar
  7. 7.
    Cantone, D., Zarba, C.G., Ruggeri-Cannata, R.: A tableau-based decision procedure for a fragment of set theory with iterated membership. Journal of Automated Reasoning 34(1), 49–72 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dowek, G.: Collections, sets and types. Mathematical Structures in Computer Science 9(1), 109–123 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Farmer, W.M.: Stmm: A set theory for mechanized mathematics. J. Autom. Reasoning 26(3), 269–289 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gordon, M.J.C.: Set theory, higher order logic or both? In: von Wright, J., Harrison, J., Grundy, J. (eds.) TPHOLs 1996. LNCS, vol. 1125, pp. 191–201. Springer, Heidelberg (1996)Google Scholar
  11. 11.
    Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the Association for Computing Machinery 40(1), 143–184 (1993)MATHMathSciNetGoogle Scholar
  12. 12.
    Lambek, J., Scott, P.: Introduction to Higher Order Categorial Logic. Cambridge University Press, Cambridge (1986)Google Scholar
  13. 13.
    Lane, S.M.: Mathematics, Form and Function. Springer, Heidelberg (1986)Google Scholar
  14. 14.
    Meng, J.: Integration of interactive and automatic provers. In: Carro, M., Correas, J. (eds.) Second CologNet Workshop on Implementation Technology for Computational Logic Systems (2003),
  15. 15.
    Nordström, B., Petersson, K., Smith, J.: Martin-löf’s type theory. In: Abramsky, S., et al. (eds.) Handbook of Logic in Computer Science, vol. 5, Oxford University Press, Oxford (2000)Google Scholar
  16. 16.
    Paulson, L.C.: Set Theory for Verification: II. Induction and Recursion 15(2), 167–215 (1995)MATHMathSciNetGoogle Scholar
  17. 17.
    Pfenning, F., Schürmann, C.: System Description: Twelf–A Meta-Logical Framework for Deductive Systems. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 202–206. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  18. 18.
    Quaife, A.: Automated Development of Fundamental Mathematical Theories. Kluwer Academic Publishers, Norwell (1992)MATHGoogle Scholar
  19. 19.
    Reed, J.: Proof irrelevance and strict definitions in a logical framework. Technical Report 02-153, School of Computer Science, Carnegie Mellon University (2002)Google Scholar
  20. 20.
    Riazanov, A., Voronkov, A.: Vampire 1.1 (system description). In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 376–380. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  21. 21.
    Turner, R.: Type inference for set theory. Theor. Comput. Sci. 266(1-2), 951–974 (2001)MATHCrossRefGoogle Scholar
  22. 22.
    Wiedijk, F.: Is ZF a hack? Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics. Journal of Applied Logic 4 (to appear, 2006)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Chad E. Brown
    • 1
  1. 1.Universität des SaarlandesSaarbrückenGermany

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