, Volume 50, Issue 2–3, pp 309–332 | Cite as

Nonmonotonicity in (the Metamathematics of) Arithmetic

  • Karl-Georg Niebergall


This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic “logic” through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, Uδ are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second half, these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by Uδ -closed theories and Uδ -fixed points. The last section contains results on V-closed theories which are specific for V.


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  1. Alchourrón, C. E., P. Gärdenfors, and D. Makinson: 1985, ‘On the Logic of Theory Change: Partial Meet Contraction and Revision Functions’, The Journal of Symbolic Logic 50, 510-530.Google Scholar
  2. Clark, K. L.: 1978, ‘Negation as Failure’, in Gallaire and Minker (eds.), Logic and Databases, Plenum, New York, pp. 293-322.Google Scholar
  3. Feferman, S.: 1962, ‘Transfinite Recursive Progressions of Axiomatic Theories’, The Journal of Symbolic Logic 27, 259-316.Google Scholar
  4. Hájek, P. and P. Pudlak: 1993, Metamathematics of First-Order Arithmetic, Springer, Berlin.Google Scholar
  5. Halpern, J.: 1993, ‘A Critical Reexamination of Default Logic, Autoepistemic Logic and Only Knowing’, in Gottlob, Leitsch, and Mundici (eds.), Computational Logic and Proof Theory, Lecture Notes in Computer Science 713, Springer, Berlin.Google Scholar
  6. Kent, C. F.: 1973, ‘The Relation of A to Prov⌌A⌍ in the Lindenbaum Sentence Algebra’, The Journal of Symbolic Logic 38, 295-298.Google Scholar
  7. Marek, W. and M. Truszczynski: 1993, Nonmonotonic Logics; Context-Dependent Reasoning, Springer, Berlin.Google Scholar
  8. McCarthy, T.: 1994, ‘Self-Reference and Incompleteness in a Non-monotonic Setting’, Journal of Philosophical Logic 23, 423-449.Google Scholar
  9. McDermott, D. and J. Doyle: 1980, ‘Non-monotonic Logic I’, Artificial Intelligence 13, 41-72.Google Scholar
  10. Moore, R. C.: 1985, ‘Semantical Considerations on Nonmonotonic Logic’, Artificial Intelligence 25, 75-94.Google Scholar
  11. Niebergall, K.-G.: 1996, Zur Metamathematik nichtaxiomatisierbarer Theorien, dissertation, CIS, München.Google Scholar
  12. Reiter, R.: 1980, ‘A Logic for Default Reasoning’, Artificial Intelligence 13, 81-132.Google Scholar
  13. Rescher, N.: 1973, The Coherence Theory of Truth, Clarendon Press, Oxford.Google Scholar
  14. Shapiro, S. (ed.): 1985, Intensional Mathematics, North-Holland, Amsterdam.Google Scholar
  15. Smorynski, C.: 1985, Self-Reference and Modal Logic, Springer, Berlin.Google Scholar
  16. Tarski, A.: 1956, Logic, Semantics, Metamathematics, Clarendon Press, Oxford.Google Scholar
  17. Ursini, A.: 1978, ‘On the Set of ‘Meaningful’ Sentences of Arithmetic’, Studia Logica 37, 237-241.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Karl-Georg Niebergall
    • 1
  1. 1.Institut für Philosophie, Logik und TheorieUniversität MünchenMünchenGermany

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