On partial disjunction properties of theories containing Peano arithmetic

  • Taishi Kurahashi


Let \(\varGamma \) be a class of formulas. We say that a theory T in classical logic has the \(\varGamma \)-disjunction property if for any \(\varGamma \) sentences \(\varphi \) and \(\psi \), either \(T \vdash \varphi \) or \(T \vdash \psi \) whenever \(T \vdash \varphi \vee \psi \). First, we characterize the \(\varGamma \)-disjunction property in terms of the notion of partial conservativity. Secondly, we prove a model theoretic characterization result for \(\varSigma _n\)-disjunction property. Thirdly, we investigate relationships between partial disjunction properties and several other properties of theories containing Peano arithmetic. Finally, we investigate unprovability of formalized partial disjunction properties.


Partial disjunction properties Partial existence properties Formal arithmetic Gödel’s incompleteness theorems 

Mathematics Subject Classification

03F30 03F40 


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  1. 1.
    Avron, A.: A note of provability, truth and existence. J. Philos. Logic 20(4), 403–409 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Feferman, S.: Arithmetization of metamathematics in a general setting. Fund. Math. 49, 35–92 (1960)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Friedman, H.: The disjunction property implies the numerical existence property. Proc. Natl. Acad. Sci. USA 72(8), 2877–2878 (1975)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Gentzen, G.: Untersuchungen über das logische Schliessen. Math. Z. 39(2–3), 176–210, 405–431 (1934–1935)Google Scholar
  5. 5.
    Gödel, K.: Zum intuitionistischen Aussagenkalkül. Anzeiger der Akademie der Wissenschaften in Wien 69, 65–66 (1932) (translated in Kurt Gödel, Collected works, vol. 1, pp. 222–225)Google Scholar
  6. 6.
    Grulović, M.Z.: A word on the joint embedding property. Mat. Vesn. 47(203), 39–47 (1995)MathSciNetMATHGoogle Scholar
  7. 7.
    Grzegorczyk, A., Mostowski, A., Ryll-Nardzewski, C.: The classical and the \(\omega \)-complete arithmetic. J. Symb. Logic 23(2), 188–206 (1958)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Guaspari, D.: Partially conservative extensions of arithmetic. Trans. Am. Math. Soc. 254, 47–68 (1979)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hájek, P.: Experimental logics and \(\Pi ^0_3\) theories. J. Symb. Logic 42(4), 515–522 (1977)CrossRefMATHGoogle Scholar
  10. 10.
    Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  11. 11.
    Ignjatović, A., Grulović, M.Z.: A comment on joint embedding property. Period. Math. Hung. 33(1), 45–50 (1996)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jensen, D.C., Ehrenfeucht, A.: Some problems in elementary arithmetics. Fund. Math. 92, 223–245 (1976)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kaye, R.: Models of Peano Arithmetic, Oxford Logic Guides, vol. 15. Oxford Science Publications, New York (1991)MATHGoogle Scholar
  14. 14.
    Kikuchi, M., Kurahashi, T.: Generalizations of Gödel’s incompleteness theorems for \(\Sigma _n\)-definable theories of arithmetic. Rev. Symb. Logic 10(4), 603–616 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kleene, S.C.: On the interpretation of intuitionistic number theory. J. Symb. Logic 10, 109–124 (1945)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kurahashi, T.: Henkin sentences and local reflection principles for Rosser provability. Ann. Pure Appl. Logic 167(2), 73–94 (2016)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lindström, P.: Partially generic formulas in arithmetic. Notre Dame J. Form. Logic 29(2), 185–192 (1988)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lindström, P.: Aspects of Incompleteness. Lecture Notes in Logic, vol. 10, 2nd edn. A K Peters, Natick (2003)Google Scholar
  19. 19.
    Macintyre, A., Simmons, H.: Algebraic properties of number theories. Isr. J. Math. 22(1), 7–27 (1975)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mostowski, A.: On models of axiomatic systems. Fund. Math. 39, 133–158 (1952)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Myhill, J.: A note on indicator-functions. Proc. Am. Math. Soc. 39(1), 181–183 (1973)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Robinson, A.: Forcing in model theory. Actes du Congrés International des Mathématiciens (Nice, 1970), Tome 1, pp. 245–250 (1971)Google Scholar
  23. 23.
    Smoryński, C.: The incompleteness theorems. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 821–865. North-Holland, Amsterdam (1977)CrossRefGoogle Scholar
  24. 24.
    Smoryński, C.: Calculating self-referential statements. Fund. Math. 109(3), 189–210 (1980)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Smoryński, C.: Self-reference and Modal Logic. Universitext. Springer, New York (1985)CrossRefMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Institute of Technology, Kisarazu CollegeKisarazuJapan

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