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The Uniqueness of the Fixed-Point in Every Diagonalizable Algebra (The Algebraization of the Theories Which Express Theor; VIII)

Claudio Bernardi
Studia Logica: An International Journal for Symbolic Logic
Vol. 35, No. 4 (1976), pp. 335-343
Published by: Springer
Stable URL: http://www.jstor.org/stable/20014824
Page Count: 9
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The Uniqueness of the Fixed-Point in Every Diagonalizable Algebra (The Algebraization of the Theories Which Express Theor; VIII)
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Abstract

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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