Rosser and Mostowski sentences
type: Item
instance of: scholarly article
title: Rosser and Mostowski sentences
zbMATH Open document ID: 0688.03041
DOI: 10.1007/BF01620761
author: Franco Montagna
author: Giovanni Sommaruga
published in: Archive for Mathematical Logic
publication date: 1988-01-01T00:00:00Z
review text: The authors provide a modal framework for the study of Rosser and Mostowski sentences of Peano Arithmetic. The properties of the operator called witness comparison, with which Rosser sentences can be discussed, have been analysed in a modal system by \textit{D. Guaspari} and \textit{R. M. Solovay} [Ann. Math. Logic 16, 81-99 (1979; Zbl 0426.03062)]. In the system, however, it has been shown that there are no fixed points for Rosser formulae. Several extensions of the system, therefore, are defined in this paper by adding new constants to fill the role of the fixed points so that not only Rosser sentences but also Mostowski sentences are discussed uniformly in a more complete modal setting. The main result is a lemma called intersection lemma, from which immediately follow the formalized Rosser theorem, the Friedman-Goldfarb-Harrington's principle, and so on. The arithmetical completeness and semantical completeness by Kripke models for the systems are also proved.
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 4127202
zbMATH Keywords: modal provability operators
zbMATH Keywords: Mostowski sentences
zbMATH Keywords: Peano Arithmetic
zbMATH Keywords: witness comparison
zbMATH Keywords: Rosser sentences
zbMATH Keywords: arithmetical completeness
zbMATH Keywords: semantical completeness
reviewed by: Osamu Sonobe
MaRDI profile type: Publication
cites work: Equivalence relations induced by extensional formulae: classification by means of a new fixed point property
cites work: Q4196401
cites work: Modal logics with several operators and probability interpretations
cites work: A simplification of a completeness proof of Guaspari and Solovay
cites work: Generic generalized Rosser fixed points
cites work: Sentences implying their own provability
cites work: Rosser sentences
cites work: Q3852250
cites work: A generalization of the incompleteness theorem
cites work: Extensions of some theorems of Gödel and Church
cites work: An effective fixed-point theorem in intuitionistic diagonalizable algebras. (The algebraization of the theories which express Theor. IX.)
cites work: Representability of recursively enumerable sets in formal theories
cites work: Q3968927
cites work: Provability interpretations of modal logic
cites work: The provability logics of recursively enumerable theories extending Peano arithmetic at arbitrary theories extending Peano arithmetic
full work available at URL: https://doi.org/10.1007/bf01620761
OpenAlex ID: W2028855113
description: scientific article; zbMATH DE number 4127202