Relation between Metalanguage and Object Language  first,I think we can avoid set theory to bulid the first order logic ,  by the operation of the finite string.but I have  The following questions:
How does "meta-logic" work. I don't really know this stuff yet, but from what I can see right now, meta-logic proves things about formal languages and logics in general. But does it use some logic to do so? Like if I want to prove that two formal languages are equivalent in some respect, aren't I presupposing a "background" formal language? And won't my choice of a "background" (meta) language affect what I can and can't demonstrate? For example,  what logic was Godel using when he proved his famous theorems? Was it a bivalent one? A three valued logic? etc
In short,I'm still not sure how reasoning about all possible formal languages work. For example, suppose I say something of the form "for all formal theories, F, if F has property X, then F must have property Y". If I wanted to prove something like that, how does such very general reasoning work? What I mean is that in such a proof, what kind of logic would be employed (for example, would it be a two valued logic?), and does the choice of logic affect the outcome? Do logicians agree on some kind of meta-meta logic, which they use to reason about absolutely everything? Or do they just choose their favorite one?
if metalogic is just predicate logic,It seems circular to me! we build the theory of predicate logic by using predicate logic?For example, in proving some theorem in the object language we seem to assume that it is already correct (in the metalanguage). Or defining some connective in the object language, we use that connective in the metalanguage to do so. It's like they're saying "Alright guys! We are going to prove a bunch of stuff about logic! Oh, by the way, you have to take all this stuff we are about to prove for granted, but don't worry, that's just the "metalanguage"." Something about this seems wrong to me. Maybe I have misunderstood?
 A: 
If I want to prove that two formal languages are equivalent in some respect, aren't I presupposing a "background" formal language? 

Yes -- but the distinction between object language and meta language can be studied carefully. This is an important part of proof theory, as well as (in a more modern context) of the theory of programming languages.  Let me quote from Olivier Danvy's entry on "Self-interpreter" in the appendix to Jean-Yves Girard's Locus Solum:

Overall, a computer system is constructed inductively as a (finite) tower of interpreters, from the micro-code all the way up to the graphical user interface. Compilers and partial evaluators were invented to collapse interpretive levels because too many levels make a computer system impracticably slow. The concept of meta levels therefore is forced on computer scientists: I cannot make my program work, but maybe the bug is in the compiler? Or is it in the compiler that compiled the compiler? Maybe the misbehaviour is due to a system upgrade? Do we need to reboot? and so on. Most of the time, this kind of conceptual regression is daunting even though it is rooted in the history of the system at hand, and thus necessarily finite.

In this view (in contrast to the views expressed in some of the other answers), the meta language does not have any special status: it is distinguished from the object language by its role rather than by its character.  In particular, a meta language $L_1$, used to interpret some object language $L_2$, may itself be the object language of some interpretation in $L_0$.
On the other hand, often both mathematicians and computer scientists are interested in keeping the meta language as minimalistic as possible, simpler than the object language in some sense.  A paradigmatic example is Gentzen's cut-elimination argument, which proves the consistency of first-order Peano arithmetic (PA).  If the meta language for this proof is taken to be ZFC, then the result seems vacuous, since ZFC already includes PA (indeed is much more complicated than PA).  However, in fact only a very small logical fragment of ZFC is needed to formalize the proof, namely PRA + $\epsilon_0$ (primitive recursive arithmetic plus transfinite induction up to $\epsilon_0$).  Although PRA + $\epsilon_0$ is not included in PA (so that Gentzen's theorem does not contradict Gödel's), neither is PA included in PRA + $\epsilon_0$, since the latter only allows (transfinite) induction on quantifier-free statements.  This is what prevents Gentzen's argument from being "circular", and the sense in which it reduces a statement about the object language to a "simpler" meta language.
A: The issue of language/metalanguage and logic/metalogic seems easy to grasp after a careful study on modern math log textbooks (like Shoenfiled) but it can have (for me) interesting "philosophical" aspects.
Please, take a look to Russell & Whithead Intro to their monumental book Principia Mathematica (written 100 years ago): it is a masterpiece of modern mathematics ... but they have the same difficulty with object/meta-logic (see their "explanation" of Modus Ponens).
Logic (and mathematics ?) are "language games" : so, we need language (as a tool) to speak about language (an human activity, i.e.an object of the world, to be studied) and we need logic (as a tool) to reason about logic (a mathematical object).
The issue (and I think is a big one) comes from the "foundational" aspect that logic receive in the framework of scientific activities. If "fundational" means : start from scratch and build every layer on top of the preceeding one ... I think that last century philosophical debate show us that it is NOT possible to start from scratch at all.
A: There are two roles for metalanguage: First, to avoid contradictions like the liar paradox, because in the liar paradox we have a statement that speaks about itself so it does not respect the hierarchy language-metalanguage.
The other role is to allow us to speak freely and to use theorems of the language as meta-theorems. So if we use the scheme of deduction using the principle of excluded middle, we know that we are using a meta-theorems, but this is just a way to use the corresponding theorem in the object language without repetition. So for example the metatheorem:
 "If the negation of a proposition A does not hold, then A holds"
 can be replaced by the theorem in the object language " $\neg \neg A \Rightarrow A$".
