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