the extreme value theorem (EVT) and the intermediate value theorem (IVT), one notices that the proofs involve procedures that are not easily encoded in first order logic. These 19th century results (due to suitable combinations of Bolzano, Cauchy, and Weierstrass) arguably fall outside the scope of Leibnizian calculus, as do infinitesimal foundations for differential geometry as developed in [@NK], [@KKN].

There are axioms in FOL for a real closed field $F$ (e.g., real algebraic numbers, real numbers, hyperreal numbers, Conway numbers). One of these axioms formalizes the fact that IVT holds for odd degree polynomials $F[x]$. In fact, one needs infinitely many axioms like $(\forall a,b,c) (\exists x) [x^3+ax^2+bx+c=0]$. Meanwhile, IVT in its full form is equivalent to the continuity axiom for the real numbers [@Bl15].

What does “first-order” mean exactly? {#FOL}
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The adjective ‘first-order’ as we use it entails limitations on quantification over sets (as opposed to elements). Now Leibniz really did not have much to say about properties of sets in general in the context of his infinitesimal calculus, and even declared on occasion that infinite totalities don’t exist, as mentioned above. Note that Leibniz arguably did exploit second-order logic in areas outside infinitesimal calculus (see [@Len87], 