Is there a first-order-logic for calculus? I just finished a course in mathematical logic where the main theme was first-order-logic and little bit of second-order-logic. Now my question is, if we define calculus as the theory of the field of the real numbers (is it?) is there a (second- or) first-order-logic for calculus? In essence I ask if there is a countable model of calculus.
I hope my question is clear, english is my third language.

The first order theory of the algebraic and order properties of the real numbers is the theory of real closed fields, and you will find various axiomatizations when you follow the link. 
A structure with the first order properties of the real numbers may not satisfy the completeness axiom, which is not first order. For example, the field of hyperreal numbers   has the same first order properties as the field of real numbers, but the set of finite numbers is nonempty and bounded above by any infinite number, yet has no supremum.
