The uniqueness of the fixed-point in every diagonalizable algebra

Summary It is well known that, in Peano arithmetic, there exists a formulaTheor (x) which numerates the set of theorems. By Gödel's and Löb's results, we have that Therefore, the considered «equations» admit, up to provable equivalence, only one solution.In this paper we prove (Corollary 1), that, in general, ifP (x) is an arbitrary formula built fromTheor (x), then the fixed-point ofP (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).The algebraization of the theories which express TheorAllatum est die 3 Augustii 1975