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Single Idea 13676

[from 'Foundations without Foundationalism' by Stewart Shapiro, in 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers ]

Full Idea

The main problem of characterizing the natural numbers is to state, somehow, that 0,1,2,.... are all the numbers that there are. We have seen that this can be accomplished with a higher-order language, but not in a first-order language.

Gist of Idea

Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)

Book Reference

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.246