An effective fixed-point theorem in intuitionistic diagonalizable algebras

Studia Logica 35 (4):345 - 361 (1976)
Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following: Let T be any first-order theory with a predicate Pr satisfying the canonical derivability conditions, including Löb's property. Then any formula in T built up from the propositional variables $q,p_{1},...,p_{n}$ , using logical connectives and the predicate Pr, has the same "fixed-points" relative to q (that is, formulas $\psi (p_{1},...,p_{n})$ for which for all $p_{1},...,p_{n}\vdash _{T}\phi (\psi (p_{1},...,p_{n}),p_{1},...,p_{n})\leftrightarrow \psi (p_{1},...,p_{n})$ ) of a formula $\phi ^{\ast}$ of the same kind, obtained from φ in an effective way. Moreover, such $\phi ^{\ast}$ is provably equivalent to the formula obtained from φ substituting with $\phi ^{\ast}$ itself all the occurrences of q which are under Pr. In the particular case where q is always under Pr in φ, $\phi ^{\ast}$ is the unique (up to provable equivalence) "fixed-point" of φ. Since this result is proved only assuming Pr to be canonical, it can be deduced that Löb's property is, in a sense, equivalent to Gödel's diagonalization lemma. All the results are proved more generally in the intuitionistic case
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References found in this work BETA
Helena Rasiowa (1974). An Algebraic Approach to Non-Classical Logics. Warszawa, Pwn - Polish Scientific Publishers.

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Citations of this work BETA
Leo Esakia, Mamuka Jibladze & Dito Pataraia (2000). Scattered toposes. Annals of Pure and Applied Logic 103 (1-3):97-107.
Franco Montagna (1987). Iterated Extensional Rosser's Fixed Points and Hyperhyperdiagonalizable Algebras. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (4):293-303.

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