\times$, and $\leq$. This is a stronger condition than obeying the ordered-field axioms. More specifically, one includes additional first-order properties, such as extraction of roots (e.g., existence of a root for every odd-degree polynomial). For example, every number must have a cube root: $(\forall x)(\exists
    y)\;[y^3=x]$, or every positive number have a square root: $(\forall
    x>0)(\exists y)\; [y^2=x]$.

3.  The system could have all the first-order properties of the real number system for statements involving arbitrary relations (regardless of whether those relations can be expressed using $ +, \times$, and $\leq$). For example, there would have to be a sine function that is well defined for infinitesimal and infinite inputs; the same is true for every real function. To do series, one needs a symbol for ${{\mathbb N}}$, so as to define transcendental entities such as $\pi$ or sine. We also introduce function symbols for whatever functions we are interested in working with; say all elementary functions occurring in Leibniz as well as their combinations via composition, differentiation, and integration.

It follows from these examples that the *first order* qualification is connected with the intended domain of discourse, so that any quantifier related to objects outside the domain of discourse 