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.

Fields can be studied in the first-order signature $(0,1,+,\times, =)$. For the reals, you could move to a larger language for ordered fields. It is very common for model theorists to study fields in this way. 
By the Lowenheim-Skolem theorem, any first-order theory with an infinite model has a model that is countably infnite. This includes the set of all statements in the language of fields that are true in $\mathbb{R}$. It also includes ZFC. 
