Isn't the formulation of Separation shema using finite sets before the term finite is defined? In Jechs book "Set theory" Jech gives in page 8 a formulation of Separation schema as follows:

$Y= \{ u \in X : \phi(u(p_1,...,p_n)) \}$

On the other hand, in page 12, when describing Axion of Infinity, he writes that 

"... we have to define yet the notion of finiteness. The most obvious definition of finiteness uses the notion of a natural number, which is as yet undefined..."

My question is: If we don't have the notion of a natural number, then what is the meaning of the indexes $1,2,...,n$ of $(p_1,...,p_n)$. Isn't $p_1,...,p_n$ a finite set of $n$ constants? How could we use it in page 8 if in page 12 if we don't have the notion of finite sets and natural numbers yet?
Thank you
 A: This is an important point.
Formulas, in particular those that appear in the separation axiom schema, are part of the meta-theory. This meta-theory could be some set theory, where the formulas are encoded as sets, or it could be something as weak as some fragment of Peano arithmetic where the formulas are encoded as natural numbers. But in either case, the natural numbers we refer to when we write a formula already exist for us by the virtue that they are part of the meta-theory.
The notion of finiteness is an internal notion to the universe of sets, it happens within the universe of set theory, not outside. So if you want to say that a set is finite if there is some natural number which can enumerate the elements of the set, you have a problem, since the natural numbers - at that point - are not objects in the universe, they are part of the meta-language and so on. But once you define something which can be perceived as the natural numbers within set theory, you can (to some extent) forget about the "meta-numbers", and define something as finite if it can be enumerated by a natural number.
To drive this point home, if $M$ is a model of set theory, then there is a model $N$ of set theory, which "behaves" just like $M$ in the sense of what statements are true or not, but the collection of objects that $N$ thinks are natural numbers is much larger than $M$ itself. And this shows exactly how different the natural numbers can be between the meta-language and the universe of sets (since $M$ is a universe of set theory in its own eyes).
