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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
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 17423 | The essence of natural numbers must reflect all the functions they perform [Sicha] |
| 9965 | There couldn't just be one number, such as 17 [Jubien] |
| 13676 | Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are [Shapiro] |
| 8923 | Numbers are identified by their main properties and relations, involving the successor function [MacBride] |
| 17877 | The number series is primitive, not the result of some set theoretic axioms [Almog] |