TITLE: An (apparently) vicious circle in logic
QUESTION [22 upvotes]: Can someone please help me with this following exercise 4.4 (p. 114) from the Mathematical Logic book of Ebbinghaus et al(this is not homework, but rather something that has been bugging me for a long time, which I now have found in form of an exercise, meaning there must be a definite answer to it):

A reader who has been confused by the discussion in this chapter [I'm
  that reader!] says: "Now I'm completely mixed up. How can ZFC be used
  as a basis for first-order logic, while first-order logic was actually
  needed in order to build up ZFC?" Help such a reader out of his dilemma (Hint: Again be careful in distinguishing between the object and the background level.)

REPLY [11 votes]: In any mathematical study there are two things that have to be clear: the "object" topic that we wish to study and the "background" or "meta" system that we use to study it. We can pick each of these with complete freedom. (Of course, if the background system is too weak, it won't be able to say anything about the object topic, but that's life).
For example, suppose we want to study real numbers as our object topic. 

We could use Euclidean geometry as our background system, so we would end up looking at numbers that can be constructed by ruler and compass.  We would declare some segment $S$ to have length $1$ "by convention" and then other segments would represent real numbers depending on how the ratio of their length to the length of $S$. With more work we could even find ways to represent polynomials in terms of finite arrangements of points on the plane, so we could ask which polynomials can be constructed and which constructible polynomials have constructible roots. 
We could use ZFC, and define real numbers as some sort of sets, and study these particular sets. 
We could work more axiomatically by working in ZFC and studying the theory of a complete Archimedean ordered field. One difference between this and the previous bullet is that if we think of a real number just as an element of some field then it makes no sense to ask whether $5 \in \pi$, because there is no "$\in$" in the language of fields; but if we think of a real number as just some particular set then it does make sense to ask whether $5 \in \pi$.  Similarly, in geometry we could ask whether two numbers (represented by line segments) are parallel. 

The same sort of thing works for any other object theory. For example, if we want to study first-order logic, we could:

We could work with a very weak theory that is only able to manipulate finite binary strings. By appropriate coding techniques we can view certain strings as representing formulas from first-order logic, in the same way that Euclidean geometry uses line segments to represent real numbers.  We can then see what sort of constructions we can do on these strings. 
We could work in ZFC, develop a semantics for first-order logic, and study the relationship between provability and semantics. 

The various ways of studying a single topic each act as "filters" that hide certain aspects of the topic and emphasize other aspects. At the same time, as was seen above, many of the meta systems add their own idiosyncracies as well. Most of the commonly used background systems can be arranged in a linear hierarchy of increasing strength (e.g. PRA < PA < ZF < ZFC), which Simpson has called the "Gödel hierarchy".  Different portions of this hierarchy provide tools to study different aspects of the object topics. 
There is no reason that we cannot use a certain system as a background system to study itself. For example, we can study ZFC within ZFC. This is no more paradoxical than using a wrench to fix the very same casting machine that made that wrench.  Similarly, when we want to study ZFC, we may decide to use syntactic first-order logic as our base system, and when we want to study first-order logic, we may decide to take ZFC as our base system.