Re countable v. uncountable models: True, but nitpicking. There are lots of models of axiom sets that are incredibly larger than the smallest ones, but one typically studies one of the smaller models. The second part is a round-about way of saying that real closed fields are decidable. Real closed fields are pretty interesting because they incorporate Sturm theory, which enables counting & locating polynomial roots. I was shocked, shocked, when I learned that you didn't need all of the baggage of "real" real numbers (including infinitesimal calculus) in order to study polynomials. I considered this to be laziness on the part of my math teachers, that they used calculus mumbo-jumbo, when algebraic mumbo-jumbo worked just fine for this case. At 02:36 PM 2/15/2010, Andy Latto wrote:

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Clearly the elements of real closed fields are a countable set.

Not true. Any field that satisfies the axioms of a real closed field is a real closed field. The smallest such is countable, but there are uncountable ones too, like the reals. From a logician's point of view, the interesting thing about the real-closed-field axioms is that they are categorical; any statement you can make in a language which talks about elements, +, and *, but not sets of elements, is either true of all real closed fields or false for all of them.