Real numbers and set theory -- extending the neo-Fregean programme beyond arithmetic
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 elements of the field of \(R\) satisfying the following special condition: For any \(x\) and \(y\), it is possible that \(x \circ y\) exists and the existence of \(x \circ y\) necessarily entails \(x \circ y R^> x\), and, moreover, \(\circ\) should be ``definable without essential reference to any other ordering (i.e., apart from that generated by \(R^>\))''. Then strict quantities will be those definable by Fregean abstraction over a strictly quantitative equivalence relation. Further, a definition similar to that of Eudoxos in Euclid's \textit{Elements} is given for the assertion that the ratio \(a:b\) of quantities \(a\) and \(b\) is equal to the ratio \(c:d\) of quantities \(c\) and \(d\). (Here, the quantities \(a\) and \(b\) can be of a kind different from that of \(c\) and \(d\).) To apply the Eudoxian definition, it is assumed that each kind of quantities \(\Phi\) forms a minimal domain, that is, \(\Phi\) is closed under an additive operation \(\oplus\) satisfying conditions (i) \(\oplus\) is commutative and associative, and (ii) For any \(a\) and \(b\) in \(\Phi\), exactly one of the three statements \(a = b\), \(\exists c(a = b\oplus c)\), \(\exists c(b = a\oplus c)\) holds. A minimal domain satisfying the Archimedean condition (iii) For any \(a\) and \(b\) in \(\Phi\), \(\exists m(ma >b)\) is called a normal quantitative domain. To obtain the reals, we need two additional conditions: (iv) For any \(a, b, c\) in \(\Phi\), \(\exists q\) in \(\Phi\) such that \(a:b = q:c\), (v) Every non-empty property \(P\) on \(\Phi\) that is bounded above has a least upper bound. If we take the collection of ratios on a domain satisfying conditions (i)--(v) and then adjoin 0 and the negative numbers in the usual way, we obtain the complete ordered field of real numbers. However, for this to work, we must know that there is a domain satisfying (i)--(v). To accomplish this, the author uses Hume's Principle to obtain a normal domain \(N^+\). The collection \(R^{N^+}\) of ratios on this domain is a normal domain satisfying (iv). Dedekind's approach then can be used by defining cut-properties on \(R^{N^+}\) and applying an abstraction principle to form the Dedekind cuts in \(R^{N^+}\). These Dedekind cuts satisfy conditions (i)--(v). In a final section, the author discusses the admissibility of the methods he has used in this paper and concludes with a few speculative remarks about a further neo-Fregean development of set theory. For more background about the author's approach to the real numbers, the reader would be well-advised to look at the author's papers ``Reals by abstraction'' [Philos. Math. (3) 8, No. 2, 100--123 (2000; Zbl 0968.03010)] and ``Real numbers, quantities, and measurement'' [Philos. Math. (3) 10, No. 3, 304--323 (2002; Zbl 1034.03505)].
scientific article; zbMATH DE number 5005186