$v$”, with $u,v$ being free variables.

Note that we use FOL in a different sense from that used in formalizing Zermelo–Fraenkel set theory (ZFC).

When we seek hyperreal proxies, following the pioneering work of [@He48] and [@Ro66], for Leibniz’s procedural moves, the theory of *real closed fields* at level (2) is insufficient and we must rely upon level (3).

Thus, Leibniz’s series of Subsection \[s11\] is expressible in $\text{FOL}+\square$ at level (3) but FOL level (2) does not suffice since $\pi$ is not algebraic. Similarly, examples in Subsection \[s22\] and Subsection \[s23\] need symbols for unspecified functions which are not available at level (2).

Modern frameworks {#two}
=================

Based on the examples of Section \[one\], we would like to consider the following question:

> *Which modern mathematical framework is the most appropriate for interpreting Leibnizian infinitesimal calculus?*

The frameworks we would like to consider are

1.  a Weierstrassian (or “epsilontic”) framework in the context of what has been called since [@Sto83] an *Archimedean* continuum, satisfying Euclid V.4 (see Section \[one\]), namely the real numbers exclusively; and

2.  a modern framework exploiting infinitesimals such as the hyperreals, which could be termed a *Bernoullian* continuum since Johann Bernoulli was the first to 