The uniqueness of the fixed-point in every diagonalizable algebra

Studia Logica 35 (4):335 - 343 (1976)
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)
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DOI 10.2307/20014824
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References found in this work BETA
M. H. Lob (1955). Solution of a Problem of Leon Henkin. Journal of Symbolic Logic 20 (2):115-118.
George Grätzer (1982). Universal Algebra. Studia Logica 41 (4):430-431.

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