On the Theory of Self-Reference

  • Wolfgang Rautenberg
Part of the Universitext book series (UTX)


By self-reference we basically mean the possibility of talking inside a theory T about T itself or related theories. Here we can give merely a glimpse into this recently much advanced area of research; see e.g. [Bu]. We will prove Gödel’s second incompleteness theorem, Löb’s theorem, and many other results related to self-reference, while further results are discussed only briefly and elucidated by means of applications. All this is of great interest both for epistemology and the foundations of mathematics.

The mountain we first have to climb is the proof of the derivability conditions for \(\mathrm{PA}\) and related theories in 7.1, and the derivable Σ 1-completeness in 7.2. But anyone contented with leafing through these sections can begin straight away in 7.3; from then on we will just be reaping the fruits of our labor. However, one would forgo a real adventure in doing so, namely the fusion of logic and number theory in the analysis of \(\mathrm{PA}\). For a comprehensive understanding of self-reference, the material of 7.1 and 7.2 (partly prepared in Chapter 6) should be studied anyway.


Modal Logic Preference Order Recursive Function Derivability Condition Chinese Remainder Theorem 
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  1. Bu.
    S. R. Buss (editor), Handbook of Proof Theory, Elsevier 1998.Google Scholar
  2. HB.
    D. Hilbert, P. Bernays, Grundlagen der Mathematik, I, II, Berlin 1934, 1939, 2⟨{ nd}⟩ ed. Springer, Vol. I 1968, Vol. II 1970.Google Scholar
  3. Vi2.
    _________ , An overview of interpretability logic, in Advances in Modal Logic, Vol. 1 (editors M. Kracht et al.), CSLI Lecture Notes 87 (1998), 307–359.Google Scholar
  4. Gö2.
    _________ , Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte f. Math. u. Physik 38 (1931), 173–198, also in [Gö3, Vol. I, 144–195], [Hei, 592–617], [Dav, 4–38].Google Scholar
  5. WR.
    A. Whitehead, B. Russell, Principia Mathematica, I–III, Cambridge 1910, 1912, 1913, 2⟨{ nd}⟩ ed. Cambridge Univ. Press, Vol. I 1925, Vol. II, III 1927.Google Scholar
  6. FS.
    H. Friedman, M. Sheard, Elementary descent recursion and proof theory, Ann. Pure Appl. Logic 71 (1995), 1–47.Google Scholar
  7. Tak.
    G. Takeuti, Proof Theory, Amsterdam 1975, 2⟨{ nd}⟩ ed. Elsevier 1987.Google Scholar
  8. Be4.
    _________ , Parameter free induction and reflection, in Computational Logic and Proof Theory, Lecture Notes in Computer Science 1289, Springer 1997, 103–113.Google Scholar
  9. Si.
    W. Sieg, Herbrand analyses, Arch. Math. Logic 30 (1991), 409–441.Google Scholar
  10. Bar.
    J. Barwise (editor), Handbook of Mathematical Logic, North-Holland 1977.Google Scholar
  11. HP.
    P. Hájek, P. Pudlák, Metamathematics of First-Order Arithmetic, Springer 1993.Google Scholar
  12. Boo.
    G. Boolos, The Logic of Provability, Cambridge Univ. Press 1993.Google Scholar
  13. Ra1.
    W. Rautenberg, Klassische und Nichtklassische Aussagenlogik, Vieweg 1979.Google Scholar
  14. CZ.
    A. Chagrov, M. Zakharyashev, Modal Logic, Clarendon Press 1997.Google Scholar
  15. Ra2.
    _________ , Modal tableau calculi and interpolation, Journ. Phil. Logic 12 (1983), 403–423.Google Scholar
  16. Be1.
    L. D. Beklemishev, On the classification of propositional provability logics, Math. USSR – Izvestiya 35 (1990), 247–275.Google Scholar
  17. Be3.
    _________ , Bimodal logics for extensions of arithmetical theories, J. Symb. Logic 61 (1996), 91–124.Google Scholar
  18. Gor.
    S. N. Goryachev, On the interpretability of some extensions of arithmetic, Mathematical Notes 40 (1986), 561–572.Google Scholar
  19. Be2.
    _________ , Iterated local reflection versus iterated consistency, Ann. Pure Appl. Logic 75 (1995), 25–48.Google Scholar
  20. Ku.
    K. Kunen, Set Theory, An Introduction to Independence Proofs, North-Holland 1980.Google Scholar
  21. So.
    R. Solovay, Provability interpretation of modal logic, Israel Journal of Mathematics 25 (1976), 287–304.Google Scholar
  22. Kra.
    J. Krajíček, Bounded Arithmetic, Propositional Logic, and Complexity Theory, Cambridge Univ. Press 1995.Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikBerlinGermany

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