An effective fixed-point theorem in intuitionistic diagonalizable algebras

Summary

Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following:

LetT be any first-order theory with a predicate Pr satisfying the canonical derivability conditions, including Löb's property. Then any formula inT built up from the propositional variables q, p₁, ..., p_n, using logical connectives and the predicate Pr, has the same “fixed-points” relative to q (that is, formulas ψ (p₁ ..., p_n) for which for all p₁, ..., p_n ⊢Tϕ(ψ(p₁, ..., p_n), p₁, ..., p_n) ↔(p₁, ..., p_n)) of a formula ϕ^* of the same kind, obtained from ϕ in an effective way.

Moreover, such ϕ^* is provably equivalent to the formula obtained from ϕ substituting with ϕ^* itself all the occurrences of q which are under Pr. In the particular case where q is always under Pr in ϕ, ϕ^* is the unique (up to provable equivalence) “fixedpoint” of ϕ.

Since this result is proved only assumingPr 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.