Completeness of Real Number Arithmetic?

Question

I've recently read that, although Godel Incompleteness holds for the theory of natural numbers, the theory of the real numbers is actually complete. So, why is Godel's Theorem still considered important? Surely, the real number system is the one we use and the theory of the natural numbers is clearly leaving out a lot. Please explain this to me.

Answer 1

Goedel's Incompleteness Theorem is still important because not only arithmetic is susceptible to it, but any theory strong enough to interpret arithmetic. The theory of real closed fields is complete, which implies that it is not strong enough to interpret arithmetic.

You might be surprised, since in your question you clearly consider the theory of real numbers far superior to arithmetic. After all there are more real numbers than natural number. But the "elementary" arithmetic (that is the staff like $1 + 1 = 2$) does not produce incompleteness. Incompleteness comes with quantifiers. And in the structure of real numbers you can't say something like "a proposition holds for all natural numbers". The quantifier $\forall$ automatically quantifies over all real numbers and those extra numbers "spoil the party". If you could define the set of natural numbers in the theory of real closed fields, than it would be incomplete.