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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.

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    Important parts of calculus are captured in *Differential Algebra*. There are even nice model-theoretic results.2012-01-27
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    Can we define calculus as the theory of the field of real numbers? Calculus also uses the poset structure of the reals.2012-01-27
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    I have added [model-theory], which I am ~80% sure fits. If someone disagrees feel free to remove it.2012-01-27
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    @ymar: actually one can define the poset structure in purely field-theoretic terms (namely $a \le b$ iff there exists $c$ such that $a + c^2 = b$). See http://en.wikipedia.org/wiki/Real_closed_field .2012-01-27
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    @QiaochuYuan Thanks!2012-01-27
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    @user23823: I am not sure if I agree with your definition of calculus. Shouldn't "calculus" include, at the very least, a notion of the limit of a sequence?2012-01-27
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    Google for "continuous logic"/"continuous model theory".2012-01-30

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