Cardinality logics. I: Inclusions between languages based on ``exactly''
type: Item
instance of: scholarly article
title: Cardinality logics. I: Inclusions between languages based on ``exactly''
zbMATH Open document ID: 0653.03023
DOI: 10.1016/0168-0072(88)90010-3
published in: Annals of Pure and Applied Logic
publication date: 1988-01-01T00:00:00Z
review text: In this paper the author develops a logic which includes the notion of cardinal. To do this the author uses different kinds of variables. Variables of type 0 are reserved for elements, variables of type 1 are used for sets (in case they are allowed, then a generalization of second order logic is obtained), variables of type 2 are cardinals. Variables of the next type are used to speak about ``the cardinality of the set of all cardinals less than a given cardinal''. This process is iterated. Cardinals are Scott-cardinals and the author avoids reliance on the Axiom of Choice. The author investigates the expressive power of his languages and introduced severals kinds of hierarchies of languages. A main tool are Ehrenfeucht-games. Some questions remain open. The operator ``exactly'' is used to express that something has a given cardinality.
Mathematics Subject Classification ID: 03C80
Mathematics Subject Classification ID: 03C85
Mathematics Subject Classification ID: 03E15
zbMATH DE Number: 4064971
zbMATH Keywords: higher order logic
zbMATH Keywords: cardinality
zbMATH Keywords: Scott-cardinals
zbMATH Keywords: Ehrenfeucht-games
author: Harold T. Hodes
reviewed by: Martin Weese
MaRDI profile type: Publication
full work available at URL: 0168-0072(88)90010-3
OpenAlex ID: W1975412541
cites work: Q5181658
cites work: Cardinality logics. Part II: Definability in languages based on ‘exactly’
cites work: Where do the natural numbers come from?
cites work: Q4189277
description: scientific article; zbMATH DE number 4064971