From Lawvere to Brandenburger-Keisler: Interactive Forms of Diagonalization and Self-reference

  • Samson Abramsky
  • Jonathan Zvesper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7399)

Abstract

Diagonal arguments lie at the root of many fundamental phenomena in the foundations of logic and mathematics. Recently, a striking form of diagonal argument has appeared in the foundations of epistemic game theory, in a paper by Adam Brandenburger and H. Jerome Keisler [11]. The core Brandenburger-Keisler result can be seen, as they observe, as a two-person or interactive version of Russell’s Paradox.

References

  1. 1.
    Abramsky, S.: Retracing Some Paths in Process Algebra. In: CONCUR 1996. LNCS, vol. 1119, pp. 1–17. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  2. 2.
    Abramsky, S.: A Cook’s tour of the finitary non-well-founded sets. In: Artemov, S., Barringer, H., d’Avila Garcez, A., Lamb, L.C., Woods, J. (eds.) We Will Show Them: Essays in Honour of Dov Gabbay, vol. 1, pp. 1–18. College Publications (2005)Google Scholar
  3. 3.
    Abramsky, S.: Coalgebras, Chu spaces, and representations of physical systems. In: 2010 25th Annual IEEE Symposium on Logic in Computer Science, LICS, pp. 411–420. IEEE (2010)Google Scholar
  4. 4.
    Abramsky, S., Jagadeesan, R.: New foundations for the geometry of interaction. Information and Computation 111(1), 53–119 (1994)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, pp. 1–168. Oxford University Press (1994)Google Scholar
  6. 6.
    Abramsky, S., Melliés, P.-A.: Concurrent games and full completeness. In: Proceedings of the Fourteenth International Symposium on Logic in Computer Science, pp. 431–442. IEEE Computer Society Press (1999)Google Scholar
  7. 7.
    Aczel, P., Mendler, N.P.: A Final Coalgebra Theorem. In: Dybjer, P., Pitts, A.M., Pitt, D.H., Poigné, A., Rydeheard, D.E. (eds.) Category Theory and Computer Science. LNCS, vol. 389, pp. 357–365. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  8. 8.
    Barr, M.: Terminal coalgebras in well-founded set theory. Theor. Comput. Sci. 114(2), 299–315 (1993)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Battigalli, P., Siniscalchi, M.: Strong belief and forward-induction reasoning. Journal of Economic Theory 106, 356–391 (2002)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Brandenburger, A., Friedenberg, A., Keisler, H.J.: Admissibility in games. Econometrica 76, 307–352 (2008)MathSciNetMATHGoogle Scholar
  11. 11.
    Brandenburger, A., Jerome Keisler, H.: An impossibility theorem on beliefs in games. Studia Logica 84(2), 211–240 (2006)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Butz, C.: Regular categories and regular logic. Technical Report LS-98-2, BRICS (October 1998)Google Scholar
  13. 13.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications, vol. 93. Cambridge University Press (2003)Google Scholar
  14. 14.
    Harsanyi, J.C.: Games with incomplete information played by ”Bayesian” players, I–III. Part I. The basic model. Management Science 14(3) (1967)Google Scholar
  15. 15.
    Heifetz, A., Samet, D.: Topology-free typology of beliefs. Journal of Economic Theory 82, 324–381 (1998)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kupke, C., Kurz, A., de Venema, Y.: Stone coalgebras. Theor. Comput. Sci. 327(1-2), 109–134 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    William Lawvere, F.: Diagonal arguments and cartesian closed categories. Lecture Notes in Mathematics, vol. 92, pp. 134–145 (1969)Google Scholar
  18. 18.
    Michael, E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71, 152–182 (1951)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Moss, L.S., Viglizzo, I.D.: Final coalgebras for functors on measurable spaces. Inf. Comput. 204(4), 610–636 (2006)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Pacuit, E.: Understanding the Brandenburger-Keisler paradox. Studia Logica 86(3), 435–454 (2007)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Rutten, J.: Universal coalgebra: a theory of systems. Theor. Comput. Sci. 249(1), 3–80 (2000)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Soto-Andrade, J., Varela, F.J.: Self-reference and fixed points: a discussion and an extension of Lawvere’s theorem. Acta Applicandae Mathematicae 2, 1–19 (1984)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    van Oosten, J.: Basic category theory. Technical Report LS-95-1, BRICS (January 1995)Google Scholar
  24. 24.
    Worrell, J.: Terminal sequences for accessible endofunctors. Electr. Notes Theor. Comput. Sci. 19 (1999)Google Scholar
  25. 25.
    Yanofsky, N.S.: A universal approach to self-referential paradoxes and fixed points. Bulletin of Symbolic Logic 9(3), 362–386 (2003)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Samson Abramsky
    • 1
  • Jonathan Zvesper
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordUK

Personalised recommendations