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### Single Idea 10610

#### [catalogued under 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers]

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