Can truth in first order theories be defined without set theory? As far as I understand, we define truth in first order theories like predicate calculus or PA by working in some stronger first order theory like set theory. Is it true? Are there other attempts in defining truth in first order theories without considering stronger first order theories?
I would appreciate your help!
 A: Let's stick to the concrete example of PA.  Tarski's undefinability theorem shows that truth for sentences of PA cannot be defined in PA.
We can look at even simpler sentences, though. For example, if we just look at the truth of $\Sigma^0_1$ sentences, we are essentially asking about whether various Turing machines halt. By the unsolvability of the halting problem, this means that there will not be any computable way to determine which $\Sigma^0_1$ sentences of PA are true.   (The overall Turing degree of the set of true sentences of PA is $0^\omega$, which is very far from being computable.)
So, in whatever semantics we want to use for PA, we will have to go beyond means definable in PA, and the resulting semantics will certainly not be computable. We do not need to use set theory - we could use second-order arithmetic, or another type theory, or topos theory, to define a semantics for PA. All of these will require something that goes beyond PA itself. 
A: Conventionally, we interpret a first-order theory by assigning extensions (sets of objects) to its predicates, and fixing a domain (another set of objects) for its quantifiers to run over. And then we appeal to some modest set-theoretic ideas to do the metatheory for our first-order theories.
But it is a nice question whether we need to invoke sets here. We might wonder: why not directly assign a (monadic) predicate the objects, plural, which satisfy the predicate, rather than something over and above those objects, namely the set of them? Why not treat quantifiers as ranging over some objects, plural, rather than something over and above them, namely the set of them?
Well, to handle a metatheory which deals in objects, plural, we'll need a plural logic. See for example Plural Logic by Alex Oliver and Timothy Smiley: in Ch. 11 they give an account of the semantics of standard first-order logic using a plural metatheory with no commitment to sets. So this can be done. 
It is, however, an interesting question what exactly is gained by the exercise. OK we'll avoid ontological commitment to sets. But we'll still find ourselves tangling with some strongly infinitary ideas. For Oliver and Smiley have to trade in arbitrary sets (whose membership we can't specify) for an unreduced notion of arbitrary properties and arbitrary functions (whose graphs we can't specify); and someone who is worried about the notion of an arbitrary infinite set will argue that we should be equally worried about these substitutes. But that's a debate for another day.
