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?
[Math] completeness of the theory of real numbers
logicreference-request
Best Answer
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 $\S 3.8$.
Added: A primary source is
A. Tarski, A Decision Method for Elementary Algebra and Geometry. RAND Corporation, Santa Monica, Calif., 1948.