is qualified as not a first order one. It could be added however that all mathematical objects are, generally speaking, (represented by suitable) sets from the set theoretic standpoint, and hence all mathematical quantifiers are first-order with respect to the background set universe (superstructure).

The point with level (3) is that instead of quantifying over sequences or functions, we relate to each individual sequence or function, and make sure that it has an analogue in the extended domain. Such an analogue of $f$ is sometimes referred to as the *natural extension* of $f$. Then we can say something about the extension of every standard object in our system, e.g., function, without ever being able to assert anything about *all* functions. Thus, the product rule for differentiation is proved for the assortment of functions chosen in item (3) above.

An alternative to the multitude of functional symbols would be to add a countable list of variables $u,v,w,\ldots$ meant to denote unspecified functions. The idea is to avoid quantifying over such variables, and use them as merely free variables. Then, for example, the product rule is the following statement: “if $u,v$ are differentiable functions then the Leibniz rule holds for $u$ and 