The logic of arithmetical hierarchy
type: Item
instance of: scholarly article
title: The logic of arithmetical hierarchy
zbMATH Open document ID: 0804.03045
DOI: 10.1016/0168-0072(94)90063-9
author: Giorgie Dzhaparidze
published in: Annals of Pure and Applied Logic
publication date: 1995-01-24T00:00:00Z
review text: Formulas of the propositional modal language with the unary modal operators \(\square\), \(\Sigma_ 1, \mathbb{B}_ 1, \Sigma_ 2, \mathbb{B}_ 2\), etc. are considered as schemata of sentences of arithmetic (PA), where \(\square A\) is interpreted as (a formalization of) ``\(A\) is PA- provable'', \(\Sigma_ nA\) as ``\(A\) is PA-equivalent to a \(\Sigma_ n\)- sentence'' and \(\mathbb{B}_ n A\) as ``\(A\) is PA-equivalent to a Boolean combination of \(\Sigma_ n\)-sentences''. We give an axiomatization and show decidability of the sets of the modal formulas which are schemata of: (1) PA-provable, (2) true arithmetical sentences.
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 516623
zbMATH Keywords: provability logic
zbMATH Keywords: arithmetic complexity
zbMATH Keywords: arithmetical hierarchy
zbMATH Keywords: PA- provability
zbMATH Keywords: axiomatization
zbMATH Keywords: decidability
zbMATH Keywords: true arithmetical sentences
Wikidata QID: Q56144458
MaRDI profile type: Publication
full work available at URL: 0168-0072(94)90063-9
OpenAlex ID: W1982073522
cites work: Q4196401
cites work: Q3199395
cites work: Sentences implying their own provability
cites work: Q4250049
cites work: The relation of A to Prov ˹A˺ in the Lindenbaum sentence algebra
cites work: Self-reference and modal logic
cites work: Provability interpretations of modal logic
description: scientific article; zbMATH DE number 516623