Studia Logica

, Volume 53, Issue 3, pp 389–396 | Cite as

A note on the ω-incompleteness formalization

  • Sergio Galvan


The paper studies two formal schemes related to ω-completeness.

LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where α(x) is a formula with the only free variable x): (a) (for anyn) (⊢ T α(n)), (b) ⊢ T x Pr T (α(x)) and (c) ⊢ T xα(x) (the notational conventions are those of Smoryński [3]). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) ⇒ (c) and (a) ⇒ (b), where α ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) ⇒ (c) is equivalent to ¬ConsT and 2. the schema corresponding to (a) ⇒ (b) is not consistent with 1-CONT. The former result follows from a simple adaptation of the ω-incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma.


Mathematical Logic Formal Theory Base theoryS Computational Linguistic Formal Extension 
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  1. [1]
    S. Galvan,Introduzione ai teoremi di incompletezza Angeli, Milano 1992.Google Scholar
  2. [2]
    J. Y. Girard,Proof Theory and Logical Complexity, Vol. I, Bibliopolis, Napoli 1987.Google Scholar
  3. [3]
    C. Smoryński,The Incompleteness Theorems, in J. Barwise (ed.),Handbook of Mathematical Logic North Holland, Amsterdam 1977, pp. 821–865.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Sergio Galvan
    • 1
  1. 1.Institute of PhilosophyUniversity of VeronaVeronaItaly

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