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Second order Logic (SL) is required to define the Reals (otherwise they were at most countable). Based on this, SL is involved in the definition of the limit operator, as the 'core' of all Calculus.

Now, was that it with SL? Do we need Second Order Logic in Calculus beyond the definition of the limit operator?

If not, this would be a pretty neat example of encapsulation. Very similar to encapsulation in e.g. software architecture (as far as one is interested in an interdisciplinary perspective).

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  • $\begingroup$ You can do a lot of calculus with finitary arguments coupled with the notion of limit. However, mathematicians (including those who study analysis and/or mathematical logic) like to consider arguments which do other neat and nonfinitary constructions. A lot of this can be done in first order set theory. I am not a student of this area, but if I wanted to be, I would start with H. Jerome Keisler's calculus texts. Gerhard "And Maybe Some Internet Searches" Paseman, 2017.07.10. $\endgroup$
    – Gerhard Paseman
    Commented Jul 10, 2017 at 18:59
  • $\begingroup$ Yes, thx for the link. Already scanned some of my old functional analysis books. However, I'm not actually into this area. My perspective is more the logical one. So, the view from someone of the Calculus community would be extremely helpful. $\endgroup$
    – user462380
    Commented Jul 13, 2017 at 6:49

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