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I am not a specialist in Logic (my field is Functional Analysis), so excuse me my ignorance.

I suppose there must be texts where Calculus is presented as a structure in the sense of Model theory. I mean, one can construct a first-order language $\mathcal L$, where functional symbols are usual algebraic operations + symbols of elementary functions like $\sin$, $\cos$, $x^y$ in intuitively clear sense (in the case where the domain of the function does not coincide with $\mathbb R$ or $\mathbb R^2$, the corresponding symbol should be understood as a relation symbol, not as a functional symbol). And after that one can construct an $\mathcal L$-structure with $\mathbb R$ as a universe. If we add two supplementary operations, a formal derivative and a formal integral into the language $\mathcal L$, then this construction would be a formal definition of Calculus (I believe).

So I want to ask,

can anybody recommend me a paper or a book where Calculus is described in this way?

I am asking this for teaching, not for research, and I would appreciate any help, not necessarily references, but any advice as well. Thank you.

Sergei Akbarov
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  • I believe Pete L. Clark has some notes where he introduces integration axiomatically, if that interests you. – Git Gud Oct 28 '15 at 10:34
  • I am dreaming of reading them. :) – Sergei Akbarov Oct 28 '15 at 10:38
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    Realize, though, that the language you describe extends first-order logic in a significant way, so standard model theoretic results may not apply. Your formal derivative and formal integral operators are a new kind of creature: each takes a variable and a term, binds the variable and creates a new term. That's more a 2nd-order than a 1st-order notion. – BrianO Oct 28 '15 at 10:45
  • Brian, yes, I understand that these supplementary operations, derivative and integral, are something new in constructions of that kind. However, I believe somebody considered this already, and can clarify the details... – Sergei Akbarov Oct 28 '15 at 10:52
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    For a rigorous presentation that do not consider derivative and integration, you can see Stephen Simpson, Subsystems of Second Order Arithmetic (2nd ed - 2009), Development of Mathematics within Subsystems of $Z_2$ : Ch.II. – Mauro ALLEGRANZA Oct 28 '15 at 10:58
  • @BrianO, if we narrow the list of elementary functions, will it be still possible to present this as a 1st-order notion? Say, if only polynomials will be elementary functions, will it be correct? – Sergei Akbarov Oct 28 '15 at 18:16
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    (Yes, +1 for the reference to Subsystems of Second Order Arithmetic. You can see there how basic and derived notions of analysis can be developed in a 2nd-order system, though the focus of the book goes well beyond that.) Certainly you won't want distinct function constants for each polynomial, even with only rational coefficients – that's clearly a bad approach. So polynomials have to be terms, built up from variables, constants, $+$ and $\times$, and it's those things you want to apply the derivative & integration ops to. So it seems that those operators are inevitably lambda-like. – BrianO Oct 28 '15 at 19:27
  • Even if this is inevitably a 2nd order construction, there must be a "scientific explanation" of the observed picture. :) Is it possible that nobody tried to look at Calculus from this point of view? – Sergei Akbarov Oct 28 '15 at 19:58
  • It seems your idea is intermediate in strength. It exceeds first order logic; it can be interpreted in 2nd order logic, but it doesn't quantify over sets or functions. I'd say a lot of people have looked at formalizing calculus, going back to Principia Mathematica (not Newton's!). Arguably, the "natural" setting is 2nd or even 3rd order logic. Simpson's book shows how you can stay low in the type hierarchy by using a pairing function and much ingenuity. I wouldn't be surprised if other approaches have appeared in mid-20th C issues of Journal of Symbolic Logic. – BrianO Oct 28 '15 at 20:45

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