[@Len04]) but this will not be our concern here. Leibniz famously takes for granted second order logic in formulating his principle governing the *identity of indiscernibles*. While second order logic is possibly part of Leibniz’s metaphysics it is not in any obvious way part of his infinitesimal calculus.

Once we reach topics like Baire category, measure theory, Lebesgue integration, and modern functional analysis, quantification over sets becomes important, but these were not Leibnizian concerns in the kind of analysis he explored.

In fact, the term “first order logic" has several meanings. We can distinguish three levels at which a number system could have first-order properties compatible with those of the real numbers. Note that the real numbers satisfy the axioms of an ordered field as well as a completeness axiom.

1.  An *ordered field* obeys those among the usual axioms of the real number system that can be stated in first-order logic (completeness is excluded). For example, the following commutativity axiom holds: $(\forall x, y)\;[x+y=y+x]$.

2.  A real closed ordered field has all the first-order properties of the real number system, regardless of whether such properties are usually taken as axiomatic, for statements which involve the basic ordered-field relations $+, 