Full Idea
It has been proved (by Tarski) that the real numbers R is a complete theory. But this means that while the real numbers contain the natural numbers, the pure theory of real numbers doesn't contain the theory of natural numbers.
Gist of Idea
The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals
Source
Peter Smith (Intro to Gödel's Theorems [2007], 18.2)
Book Reference
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.157