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]