Summary
In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all trueΠ 02 -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barendregt, H.: The lambda calculus. Amsterdam: North-Holland 1981
Beeson, M.J.: Foundations of constructive mathematics. Berlin Heidelberg New York: Springer 1985
Fourman, M., Scott, D.S.: Sheaves and logic. In: Fourman, M. (ed.) Applications of sheaves. (Lect. Notes Math. vol. 753, pp. 302–401). Berlin Heidelberg New York: Springer 1979
Goodman, N.D.: Relativized realizability in intuitionistic arithmetic of all finite types. J. Symb. Logic43, 23–44 (1978)
de Jongh, D.H.J.: The maximality of the intuitionistic predicate calculus with respect to Heyting's arithmetic. (typed manuscript) Amsterdam 1969
Kleene, S.C., Post, E.L.: The upper semilattice of degrees of recursive unsolvability. Ann Math59, 379–407 (1951)
Leivant, D.: Absoluteness of intuitionistic logic. (thesis) Amsterdam 1975
Odifreddi, P.: Classical recursion theory. Amsterdam: North-Holland 1989
Plisko, V.E.: Some variants of the notion of realizability for predicate formulas. Math. USSR Izv.12, 588–604 (1978)
Plisko, V.E.: Absolute realizability for predicate formulas. Math. USSR Izv.22, 291–308 (1984)
Troelstra, A.S., van Dalen, D.: Constructivism in mathematics. Amsterdam: North-Holland 1988
Troelstra, A.S.: Metamathematical investigation in intuitionistic arithmetic and analysis. (Lect. Notes Math., vol. 344) Berlin Heidelberg New York: Springer 1973
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van Oosten, J. A semantical proof of De Jongh's theorem. Arch Math Logic 31, 105–114 (1991). https://doi.org/10.1007/BF01387763
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01387763