TITLE: Set theoretic concepts in first order logic
QUESTION [8 upvotes]: I have been reading introductory texts on first order logic (for example, Leary&Kristiansen). All of them used concepts that I have heard in set theory courses - ordered pairs, functions, bijections, isomorphism and so on.
I have read a lot of material in Math.StackExchange on set theory and first order logic and their interplay. I understand that we cannot define mathematics from nothing - we have to have primitive concepts. My problem is with understanding which are primitive concepts and which are not. Of course, different books might take these to be different, but still - maybe there are commonly accepted principles and notions that noone doubts. 
For example, I am willing to accept that strings exist, that they can be glued together or separated, also I am willing to accept recursion and induction. I am also willing to accept counting numbers (which could as well be infinite: I, II, III, ...). 
Question 1: As far as I have read and understood - sets in first order logic are different than those in set theory. But how so? At first I thought that it is because sets in first order logic are finite by definition and are basically just collections of finite terms, strings, and so on. Then, paradoxes that arise in set theory due to infinities do not arise in logic. But on the other hand, we use counting numbers and then, for example, the number of terms can be infinite. 
Question 2: Are sets, ordered pairs, functions, bijections - primitive notions (by primitive notion I understand concept that is not defined) in first order logic (at least in the one that most of the mathematicians use)?
Question 3: If sets, ordered pairs, functions are indeed primitive notions then are they different from set theoretical definitions? If yes, then in which way? If no, then why define these concepts once again in set theory if we had them in the language of set theory anyway? 
Question 4: If sets, ordered pairs, functions are not primitive notions in first order logic then how are they defined? 
I would appreciate any comments and discussions about this topic.

REPLY [3 votes]: It is worth adding to the other answers that in fact, while it is conventional to talk about sets when giving the metatheory of first-order logic, it just isn't necessary. (A lot of set talk in mathematics is unnecessary overkill.)

Instead of talking of consequence as a relation between a set of wffs (premisses) and a wff (the conclusion), we can treat it as a relation between wffs (plural) and a wff.
Instead of talking of the domain of quantification as one thing, a set of objects, we can take the quantifiers as ranging over things, plural.
Instead of talking of the interpretation of a predicate as a set of things (the extension of the predicate), we can talk of the things (plural) that satisfy the predicate.

And so it goes (fine print: take 'plural' to cover, as necessary, the zero and singular cases). 
It is convenient and familiar to trade in plural talk for set talk; but it isn't necessary. We can take the plurals seriously (indeed sometimes we have to -- e.g. we can't take the domain of quantification of set theory to be a set, for the familiar reason that there is no set of all sets). And if we like, we can theorise about our familiar first order logic in a formal metatheory with a plural logic, without invoking  sets at all.