The finite inseparability of the first-order theory of diagonalisable algebras
type: Item
instance of: scholarly article
title: The finite inseparability of the first-order theory of diagonalisable algebras
zbMATH Open document ID: 0542.03024
DOI: 10.1007/BF00403334
published in: Studia Logica
publication date: 1982-01-01T00:00:00Z
review text: In this paper, the author proves the finite inseparability of the first order theory of diagonalizable algebras (shortly: DAs). DAs constitute the algebraic analogue of the modal logic of provability, and can be considered as an algebraic approach to the study of the provability predicate for Peano Arithmetic or related theories. Smoryński's proof uses a result of \textit{Yu. L. Ershov} and \textit{M. A. Taj'tslin}, who proved the finite inseparability of the theory of distributive lattices [Algebra Logika, Sem. 2, No.5, 37-41 (1963; Zbl 0192.056)]. The undecidability of the first order theory of diagonalizable algebras was proved by the reviewer by direct means [Stud. Logica 39, 355-359 (1980; Zbl 0463.03016)]. Of course, Smoryński's result is stronger (it implies the undecidability of the first order theory of finite DAs); however, it does not solve the problem of the decidability or undecidability of the Lindenbaum DA of Peano Arithmetic.
Mathematics Subject Classification ID: 03D35
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03G05
zbMATH DE Number: 3861138
zbMATH Keywords: finite inseparability
zbMATH Keywords: first order theory of diagonalizable algebras
zbMATH Keywords: modal logic of provability
zbMATH Keywords: provability predicate for Peano Arithmetic
author: Craig Smoryński
MaRDI profile type: Publication
cites work: Representation and duality theory for diagonalizable algebras. (The algebraization of theories which express Theor. IV.)
cites work: The undecidability of the first-order theory of diagonalizable algebras
cites work: Q4068699
cites work: Fixed point algebras
cites work: Provability interpretations of modal logic
cites work: Sentences true in all constructive models
full work available at URL: https://doi.org/10.1007/bf00403334
OpenAlex ID: W2048339526
description: scientific article; zbMATH DE number 3861138