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The elementary theory of Lindenbaum fixed point algebras is hyperarithmetical. (English) Zbl 0746.03047
Summary: This paper deals with the elementary theory of an algebraic structure, called “The Lindenbaum FPA of PA” which is related to Peano Arithmetic. The main result of this paper is a proof of the interpretability of the true theory of natural numbers with \(+\) and \(\cdot\) in the first-order theory of the Lindenbaum FPA of PA. As a consequence, we prove that the first-order theory of such algebra is not arithmetical, thus improving a result of V. Yu. Shavrukov, stating that the above theory is undecidable [Stud. Logica 50, No. 1, 143-147 (1991; Zbl 0733.03032)].

03F30 First-order arithmetic and fragments
03G05 Logical aspects of Boolean algebras
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)