It is well known that, in Peano arithmetic, there exists a formula Theor (x) which numerates the set of theorems. By Gödel's and Löb's results, we have that Theor (˹p˺) ≡ p implies p is a theorem ∼Theor (˹p˺) ≡ p implies p is provably equivalent to Theor (˹0 = 1˺). Therefore, the considered "equations" admit, up to provable equivalence, only one solution. In this paper we prove (Corollary 1) that, in general, if P (x) is an arbitrary formula built (...) from Theor (x), then the fixed-point of P (x) (which exists by the diagonalization lemma) is unique up to provable equivalence. This result is settled referring to the concept of diagonalizable algebra (see Introduction). (shrink)

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. (shrink)

The modal completeness proofs of Guaspari and Solovay (1979) for their systems R and R – are improved and the relationship between R and R – is clarified.

The modal completeness proofs of Guaspari and Solovay for their systems R and R⁻ are improved and the relationship between R and R⁻ is clarified.