The Development of Self-Reference: Löb’s Theorem

  • C. Smoryński


Social processes operate on the sciences and social history must be understood in order to understand the history of science. This simple truth has transcended mere axiomatic truth; it has transcended mere fashion; today, the social treatment is de rigueur in the history of science and the history of computing. In the history of mathematics, however, the social emphasis is not so firmly entrenched. Historians who discuss the history of mathematics certainly steep their discussions in social contexts, but the history of mathematics is quite often written by mathematicians or, at least, mathematical educators, and mathematicians generally do not care about the overall societal background. Modern mathematics has largely divorced itself from the sciences and has become something of an intellectual sport, a game of ideas. Now, mathematicians are somewhat social and love anecdotes about fellow mathematicians, but this is as far as their interest in social history goes: Where the sciences inject history into their textbooks and discuss the growth of ideas, the history in most calculus texts limits itself to paragraph-length biographies, and one of the most successful textbooks on the history of mathematics is highly anecdotal, its author having additionally published no fewer than three books consisting of nothing but anecdotes. When the mathematician wants to discuss history (i.e., history of mathematics) and he wants to discuss it seriously and will not settle for mere anecdotes, it is not the social forces that created a need for a given type of mathematics that he writes about—mathematics has the odd habit of developing long before it is useful (i.e., before it is needed by society at large); it is not a simple chronology that he lists—despite all one reads about how the mathematician is like a fish out of water when he is not doing mathematics, he is not as stupid as all that; it is the development of the subject that interests him: What was the key idea? How did it come about? etc.


Modal Logic Proof Theory Derivability Condition Completeness Theorem Modal Language 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Van Bentham, J. (1978), Four paradoxes, J. Philos. Logic 7, 49–72.MathSciNetGoogle Scholar
  2. Bernardi, C. (1975), The fixed point theorem for diagonalizable algebras, Studia Logica 34, 239–251.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Bernardi, C. (1976), The uniqueness of the fixed point in every diagonalizable algebra, Studia Logica 35, 335–343.zbMATHCrossRefMathSciNetGoogle Scholar
  4. Boolos, G. (1979), The Unprovability of Consistency, Cambridge University Press, Cambridge, UK.zbMATHGoogle Scholar
  5. Carlson, T. (1985), Modal logics with several operators and provability interpretations, Israel J. Math., to appear.Google Scholar
  6. Feferman, S. (1960), Arithmetization of metamathematics in a general setting, Fund. Math. 49, 35–92.zbMATHMathSciNetGoogle Scholar
  7. Friedman, H. McAloon, K. and Simpson, S. (1982), A finite combinatorial principle which is equivalent to the 1-consistency of predicative analysis, in G. Metakides (ed.), Patras Logic Symposion, North-Holland, Amsterdam.Google Scholar
  8. Gödel, K. (1931), Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsh. Math. Phys. 38, 173–198.CrossRefMathSciNetGoogle Scholar
  9. Henkin L. (1952), Problem, J. Symbolic Logic 17, 160.Google Scholar
  10. Hilbert, D. and Bernays, P. (1939), Grundlagen der Mathematik II, Springer-Verlag, Berlin.zbMATHGoogle Scholar
  11. Hofstadter, D. (1979), Gödel, Escher and Bach, Harvester Press, Sussex, UK.Google Scholar
  12. Kreisel, G. (1953), On a problem of Henkin’s, Proc. Netherlands Acad. Sci. 56, 405–406.zbMATHMathSciNetGoogle Scholar
  13. Kreisel, G. (1965), Mathematical logic, in T.L. Saaty (ed.), Lectures on Modern Mathematics III, Wiley, New York.Google Scholar
  14. Kreisel, G. (1968), Notes concerning the elements of proof theory, Lecture Notes, UCLA.Google Scholar
  15. Kreisel, G. Review of Macintyre and Simmons (1973), Zentralblatt 288.02018.Google Scholar
  16. Kreisel, G. and Takeuti, G (1974), Formally self-referential propositions for cut-free classical analysis and related systems, Diss. Math. 118.Google Scholar
  17. Löb, M.H. (1954), Solution of a problem by Leon Henkin, Proc. of the ICM 1954 II, 405–406, North-Holland, Amsterdam.Google Scholar
  18. Löb, M.H. (1955), Solution of a problem of Leon Henkin, J. Symbolic Logic 20, 115–118.zbMATHCrossRefGoogle Scholar
  19. Macintyre, A. and Simmons, H. (1973), Gödel’s diagonalization technique and related properties of theories, Colloq. Math. 28, 165–180.zbMATHMathSciNetGoogle Scholar
  20. Montagna, F. (1978), On the algebraization of a Feferman’s predicate, Studia Logica 37, 221–236.zbMATHCrossRefMathSciNetGoogle Scholar
  21. Montagna, F. (1984), The modal logic of provability, Notre Dame J. Formal Logic 25, 179–189.zbMATHCrossRefMathSciNetGoogle Scholar
  22. Montague, R. (1963), Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability, Acta Phil. Fennica 16, 153–167.zbMATHMathSciNetGoogle Scholar
  23. Sambin, G. (1974), Un estensione del theorema di Löb, Rend. Sent. Mat. Univ. Padova 52, 193–199.MathSciNetGoogle Scholar
  24. Sambin, G. (1976), An effective fixed-point theorem in intuitionistic diagonalizable algebras, Studia Logica 35, 345–361.zbMATHCrossRefMathSciNetGoogle Scholar
  25. Simpson, S. (1983), Review of Paris, J. Symbolic Logic 48, 482–483.CrossRefMathSciNetGoogle Scholar
  26. Smoryński, C. (1979), Calculating self-referential statements I: explicit calculations, Studia Logica 38, 17–36.zbMATHCrossRefMathSciNetGoogle Scholar
  27. Smoryński, C. (1981a), Review of Boolos (1979), J. Symbolic Logic 46, 871–873.CrossRefGoogle Scholar
  28. Smoryński, C. (1981b), Fifty years of self-reference in arithmetic, Notre Dame J. Formal Logic 22, 357–374.zbMATHCrossRefMathSciNetGoogle Scholar
  29. Smoryński, C. (1985), Self-Reference and Modal Logic, Springer-Verlag, New York.zbMATHGoogle Scholar
  30. Smoryński, C. (1987), Quantified modal logic and self-reference, Notre Dame J. Formal Logic 28, 356–370.zbMATHCrossRefMathSciNetGoogle Scholar
  31. Solovay, R.M. (1976), Provability interpretations of modal logic, Israel J. Math. 25, 287–304.zbMATHCrossRefMathSciNetGoogle Scholar
  32. Solovay, R.M. (1985), Explicit Henkin sentences, J. Symbolic Logic 50, 91–93.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  • C. Smoryński
    • 1
  1. 1.WestmontUSA

Personalised recommendations