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Single Idea 13641

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number ]

Full Idea

'Definitions' of integers as pairs of naturals, rationals as pairs of integers, reals as Cauchy sequences of rationals, and complex numbers as pairs of reals are reductive foundations of various fields.

Gist of Idea

Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 2.1)

Book Ref

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.26


A Reaction

On p.30 (bottom) Shapiro objects that in the process of reduction the numbers acquire properties they didn't have before.