## INTERMEDIATE LOGICS AND THE DE JONGH PROPERTY (2009)

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@MISC{Jongh09intermediatelogics,

author = {Dick De Jongh and Rineke Verbrugge and Albert Visser},

title = {INTERMEDIATE LOGICS AND THE DE JONGH PROPERTY},

year = {2009}

}

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### Abstract

Abstract. We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property. Dedicated to Petr Hájek, on the occasion of his 70th Birthday 1. Preface The three authors of this paper have enjoyed Petr Hájek’s acquaintance since the late eighties, when a lively community interested in the metamathematics of arithmetic shared ideas and traveled among the beautiful cities of Prague, Moscow, Amsterdam, Utrecht, Siena, Oxford and Manchester. At that time, Petr Hájek and Pavel Pudlák were writing their landmark book Metamathematics of First-Order Arithmetic [HP91], which Petr Hájek tried out on a small group of eager graduate students in Siena in the months of February and March 1989. Since then, Petr Hájek has been a role model to us in many ways. First of all, we have always been impressed by Petr’s meticulous and clear use of correct notation, witness all his different types of dots and corners, for example in the Tarskian ‘snowing’-snowing lemmas [HP91]. But also as a human being, Petr has been a role model by his example of living in truth, even in averse circumstances [Hav89]. The tragic story of the Logic Colloquium 1980, which was planned to be held in Prague and of which Petr Hájek was the driving force, springs to mind [DvDLS82]. Finally, we were moved by Petr’s open-mindedness when coming to terms with a situation that turned out to look disconcertingly unlike the ‘standard model ’ 1. Therefore, in this paper, we would like to pay homage to Petr Hájek. Unfortunately, we cannot hope to emulate his correct use of dots and corners. Instead, we do our best to provide some pleasing non-standard models and non-classical arithmetics. 2.

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Citation Context ...ns in HA is Intuitionistic Predicate Logic. See the extended abstract [Jon70]. de Jongh’s argument uses an ingenious combination of Kripke models and realizability. 1973: Harvey Friedman in his paper =-=[Fri73]-=- gives another proof of de Jongh’s theorem for HA. He provides a single substitution σ mapping the propositional variables to Π 0 2-sentences such that HA ⊢ σ(ϕ) ⇔ IPC ⊢ ϕ. Thus,INTERMEDIATE LOGICS A... |

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Citation Context ...erification of realizability: Markov’s Principle. See David McCarthy’s paper [McC91] for a detailed analysis. Thus we obtain: ∃e ∀σ∈subHA HA + MP ⊢ e r σ(ρ). Hence, a fortiori, ρ ∈ ΛHA+MP+ECT 0 . See =-=[Pli09]-=- for an interesting recent survey of propositional realizability logic. 4. Intermediate Logics and the Finite Model Property Intermediate logics (also called superintuitionistic logics) are the logics... |