Real numbers and set theory -- extending the neo-Fregean programme beyond arithmetic
type: Item
instance of: scholarly article
title: Real numbers and set theory -- extending the neo-Fregean programme beyond arithmetic
zbMATH Open document ID: 1108.03015
author: Bob Hale
published in: Synthese
publication date: 2006-02-08T00:00:00Z
review text: In previous papers, the author and Crispin Wright showed that the concepts and results of elementary number theory could be established within second-order logic on the basis of what is known as Hume's Principle (which states that, for concepts \(F\) and \(G\), the number of \(F\)s is equal to the number of \(G\)s if and only if there is a one-to-one correspondence between the \(F\)s and the \(G\)s). Hume's Principle is known as an abstraction principle because it specifies the meaning of the identity between new objects `the number of \(F\)s' and `the number of \(G\)s' by means of an equivalence relation between \(F\) and \(G\) (namely, the relation `there is a one-to-one correspondence between the \(F\)s and the \(G\)s'). The philosophical significance of an abstraction principle is that it is not thought to make a non-logical assumption. Thus, the fact that arithmetic can be developed within second-order logic using Hume's Principle would justify Frege's original idea that arithmetic can be based on logic alone. The main purpose of the present paper is to show that the real numbers and their properties can be established within second-order logic by using appropriate abstraction principles. In this direction, the author was inspired by \textit{G. Frege}'s proposal [in his Grundgesetze der Arithmetik. Vol. II. Jena: H. Pohle (1903; JFM 34.0071.05), Part III, \S\S 73,157] that real numbers should be defined as ratios of quantities. The path leading to the desired result is somewhat complicated and can only be sketched very roughly here. A relation \(R\) is said to be quantitative if \(R\) is one of three disjoint relations \(R^<\), \(R^>\), \(R^=\) with the same field such that \(R^<\) and \(R^>\) are converse to each other, \(R^<\) and \(R^>\) are transitive and asymmetric, and \(R^=\) is an equivalence relation. A quantitative relation \(R\) is defined to be strictly quantitative if there is a commutative and associative operation \(\circ\) on certain pairs of el…
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03E99
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 00A30
Mathematics Subject Classification ID: 03B15
zbMATH DE Number: 5005186
zbMATH Keywords: abstraction principle
zbMATH Keywords: neo-Fregean
zbMATH Keywords: real numbers
zbMATH Keywords: quantitative relations
zbMATH Keywords: ratios of quantities
reviewed by: Elliott Mendelson
MaRDI profile type: Publication
full work available at URL: s11229-004-6205-7
OpenAlex ID: W2060857464
DOI: 10.1007/S11229-004-6205-7
description: scientific article; zbMATH DE number 5005186