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.
Completeness of Real Number Arithmetic?
5
$\begingroup$
logic
incompleteness
-
5The theory of [real closed fields](http://en.wikipedia.org/wiki/Real_closed_field) *in the first order language of ordered rings* is complete and decidable. This does *not* include statements about families of sets of real numbers, for example, so even the definition of "compact" would escape your theory. The theory in question is somewhat restrictive, and not at all "the real number system" that we use. – 2012-01-24
-
0I realize that godelian incompleteness has consequences for computability. Perhaps, this is why. But, from a purely mathematical standpoint, isn't the completeness of Real Number arithmetic equally important, if not more so? – 2012-01-24
-
1Thank you. That is exactly the sort of answer I wanted. Where can I learn about this? Are there any books that are particularly good (and preferably concise.) – 2012-01-24
-
0I'll let some our resident logicians do the recommendations, since my acquaintance with the literature in this field is limited. – 2012-01-24
-
3@mathNotebook: The completeness, decidability of the theory of real-closed fields is indeed an attractive result. It has the nice consequence that elementary geometry is decidable (via coordinatization). So (if you have a big enough computing budget) there *is* a royal road to geometry. But nice as Tarski's result is, it pales in significance beside the incompleteness results about number theory and relatives. – 2012-01-24
-
0WP has a nice summary of Tarski's result: http://en.wikipedia.org/wiki/Tarski%27s_axioms – 2012-01-24
-
0When you write, "the real number system is the one we use," who do you mean by "we"? Some of us make our living working on the natural numbers. The question of whether there are infinitely many twin primes could be undecidable in Godel's sense. The result on the reals is of no use for this question about primes. – 2012-01-25