# completeness of the theory of real numbers

The theory of natural numbers (such as Peano axioms) is incomplete due to Gödel's incompleteness theorem. But, I heard that the theory of real numbers is complete (edit: not in the sense of Cauchy-sequences, but in terms of statements about real numbers). Does any one have a reference for this?

-
Maybe you are confusing different types of Completeness? Algebraic Completeness means that every cauchy sequence converges. Logical completeness means that every "true" statement is provable. en.wikipedia.org/wiki/Completeness_of_the_real_numbers en.wikipedia.org/wiki/Completeness_(logic)#Logical_completeness – Galois Group Mar 8 '12 at 14:44
It doesn't look to me as if the poster was confusing different kinds of completeness. But if you want to list different kinds, why not also completeness in the sense of Dedekind? – Michael Hardy Mar 8 '12 at 16:35
@MichaelHardy That and several others are listed in the second link I posted . But you are correct. It appears that I was the confused one, not the OP :) – Galois Group Mar 8 '12 at 16:50
Found a related question from mathoverflow mathoverflow.net/questions/65112/… – Memming Mar 17 '12 at 21:13

In mathematical logic you have to be very careful how you phrase things. The "theory of the real numbers" is complete almost by definition, since it is the set of sentences in the language of fields which hold true in $\mathbb{R}$. (Moreover, in this sense the theory of $\mathbb{N}$ is also complete!)
What you want is that the theory of real numbers is decidable, i.e., there is an algorithm to determine whether a given sentence in the language of fields holds in $\mathbb{R}$. This follows from the completeness of the theory of real closed fields, which is a celebrated result of Tarski. Most introductions to model theory discuss this result and give a proof, including these lecture notes of mine: please see $\S 3.8$.