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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.

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The theory of real closed fields 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. –  Arturo Magidin Jan 24 '12 at 17:19
    
I 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? –  mathNotebook Jan 24 '12 at 17:21
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Thank 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.) –  mathNotebook Jan 24 '12 at 17:25
    
I'll let some our resident logicians do the recommendations, since my acquaintance with the literature in this field is limited. –  Arturo Magidin Jan 24 '12 at 17:29
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@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. –  André Nicolas Jan 24 '12 at 17:37

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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.

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