Single Idea 13657

[catalogued under 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic]

Full Idea

Few theorists consider first-order arithmetic to be an adequate representation of even basic number theory.

Gist of Idea

First-order arithmetic can't even represent basic number theory

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 5 n28)

Book Reference

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.132


A Reaction

This will be because of Idea 13656. Even 'basic' number theory will include all sorts of vast infinities, and that seems to be where the trouble is.

Related Idea

Idea 13656 Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]