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
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.1007/BF02123402
 Save to my reading list
Follow the author(s)
Edit this record
My bibliography
Export citation
Find it on Scholar
Mark as duplicate
Request removal from index
Revision history
Download options
Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 31,359
Through your library
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.

View all 6 references / Add more references

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.

View all 18 citations / Add more citations

Similar books and articles
Added to PP index

Total downloads
18 ( #304,781 of 2,225,164 )

Recent downloads (6 months)
3 ( #181,417 of 2,225,164 )

How can I increase my downloads?

Monthly downloads
My notes
Sign in to use this feature