TITLE: Are the natural numbers implicit in the construction of first-order logic? If so, why is this acceptable?
QUESTION [18 upvotes]: I have recently been reading about first-order logic and set theory. I have seen standard set theory axioms (say ZFC) formally constructed in first-order logic, where first-order logic is used as an object language and English is used as a metalanguage. 
I'd like to construct first-order logic and then construct an axiomatic set theory in that language. In constructing first-order logic using English, one usually includes a countably infinite number of variables. However, it seems to me that one needs a definition of countably infinite in order to define these variables. A countably infinite collection (we don't want to call it a set yet) is a collection which can be put into 1-1 correspondence with the collection of natural numbers. It seems problematic to me that one appears to be implicitly using a notion of natural numbers to define the thing which then defines natural numbers (e.g. via Von Neumann's construction). Is this a legitimate concern that I have or is there an alternate definition of "countably infinite collection" I should use? If not, could someone explain to me why not? 
I think that one possible solution is to simply assume whatever set-theoretic axioms I wish using clear and precise English sentences, define the natural numbers from there, and then define first-order logic as a convenient shorthand for the clear and precise English I am using. It seems to me that first-order logic is nothing but shorthand for clear and precise English sentences anyway. What the exact ontological status of English is and whether or not we are justified in using it as above are unresolvable philosophical questions, which I am willing to acknowledge, and then ignore because they aren't really in the realm of math.
Does this seem like a viable solution and is my perception of first-order logic as shorthand for clear and precise English (or another natural language) correct?
Thank so much in advance for any help and insight!

REPLY [10 votes]: There are three inter-related concepts:

The natural numbers
Finite strings of symbols
Formulas - particular strings of symbols used in formal logic.

If we understand any one of these three, we can use that to understand all three. 
For example, if we know what strings of symbols are, we can model natural numbers using unary notation and any particular symbol serving as a counter.
Similarly, the method of proof by induction (used to prove universal statements about natural numbers) has a mirror image in the principle of structural induction (used to prove universal statements about finite strings of symbols, or about formulas). 
Conversely, if we know what natural numbers are, we can model strings of symbols by coding them as natural numbers (e.g. with prime power coding).
This gives a very specific argument for the way in which formal logic presupposes a concept of natural numbers. Even if we try to treat formal logic entirely syntactically, as soon as we know how to handle finite strings of symbols, we will be able to reconstruct the natural numbers from them. 
The underlying idea behind all three of the related concepts is a notion that, for lack of a better word, could be described as "discrete  finiteness".   It is not based on the notion of "set", though - if we understand what individual natural numbers are, this allows us to understand individual strings, and vice versa, even with no reference to set theory. But, if we do understand sets of natural numbers, then we also understand sets of strings, and vice versa. 
If you read more classical treatments of formal logic, you will see that they did consider the issue of whether something like set theory would be needed to develop formal logic - it is not. These texts often proceed in a more "finitistic" way, using an inductive definition of formulas and a principle of structural induction that allows us to prove theorems about formulas without any reference to set theory. This method is still well known to contemporary mathematical logicians, but contemporary texts are often written in a way that obscures it. The set-theoretic approach is not the only approach, however.