An effective fixed-point theorem in intuitionistic diagonalizable algebras

Studia Logica 35 (4):345 - 361 (1976)
Abstract
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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DOI 10.1007/BF02123402
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References found in this work BETA
An Algebraic Approach to Non-Classical Logics.Helena Rasiowa - 1974 - Warszawa, Pwn - Polish Scientific Publishers.
The Mathematics of Metamathematics.Helena Rasiowa - 1963 - Warszawa, Państwowe Wydawn. Naukowe.
The Mathematics of Metamathematics.Helena Rasiowa & Roman Sikorski - 1967 - Journal of Symbolic Logic 32 (2):274-275.

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Citations of this work BETA
The Modal Logic of Provability. The Sequential Approach.Giovanni Sambin & Silvio Valentini - 1982 - Journal of Philosophical Logic 11 (3):311 - 342.
Some Remarks on the Algebraic Structure of the Medvedev Lattice.Andrea Sorbi - 1990 - Journal of Symbolic Logic 55 (2):831-853.
Scattered toposes.Leo Esakia, Mamuka Jibladze & Dito Pataraia - 2000 - Annals of Pure and Applied Logic 103 (1-3):97-107.

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