Abstract
The aim of this paper is to describe from a semantic perspective the problem of conservativity of classical first-order theories over their intuitionistic counterparts. In particular, we describe a class of formulae for which such conservativity results can be proven in case of any intuitionistic theory T which is complete with respect to a class of T-normal Kripke models. We also prove conservativity results for intuitionistic theories which are closed under the Friedman translation and complete with respect to a class of conversely well-founded Kripke models. The results can be applied to a wide class of intuitionistic theories and can be viewed as generalization of the results obtained by syntactic methods.
Highlights
IntroductionWe say that a classical theory is conservative over its intuitionistic counter-part with respect to Γ if both theories prove exactly the same formulae of this class
Let Γ be a class of formulae of some first-order language
A typical example of a conservativity result states that Peano Arithmetic (PA) is Π2-conservative over Heyting Arithmetic (HA)
Summary
We say that a classical theory is conservative over its intuitionistic counter-part with respect to Γ if both theories prove exactly the same formulae of this class. A typical example of a conservativity result states that Peano Arithmetic (PA) is Π2-conservative over Heyting Arithmetic (HA) It can be proven in several ways. The so-called (Godel-Gentzen) negative translation together with the Godel functional interpretation of HA or proof theoretic analysis of HA can be used This fact can be proven by means of the negative translation and the so-called Friedman translation. The aim of this paper is to describe conservativity of classical first-order theories over their intuitionistic counterparts from a semantic perspective. We consider properties of a class of Kripke models for a given intuitionistic theory that are sufficient to prove conservativity results.
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