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.

I take the view that the proper logical framework in which to do model theory for structures in analysis is continuous logic. For more information on the subject, look up the webpage of Ward Henson.
