Is there a 'definition' of truth based on sets of true statements? been thinking a fair bit about how to think about truth recently. 
I at one point came up with a deficient theory of truth based on provability, and was directed to Tarski's semantic theory of truth to educate myself. I gave it a look, and I'll give it more of a look in future, but I couldn't help but feel as it it were a bit circular, so while I was reading other ideas for 'defining truth' came into my mind. I just want to ask if one of them is at all valid.
My 'definition' of truth is not really a definition, more of an idea for a workable theory of truth. Say we are working in first order logic. 
Define $P$ to be the collection of all wffs of the predicate calculus that are provable (where you can define provable in some recursive manner).
Posit that there exists a set (meta theoretic collection), $T$, of wffs of the predicate calculus such that:


*

*$P\subseteq T$

*If $p, q$ are wffs such that $p\in T$ and $q$ follows from $p$, then $q\in T$.

*If $\lnot p$ is in $T$, then $p$ is not in $T$.
We call this set the set of true wffs of the propositional calculus.
Now this is not an incredibly complicated idea, and for that reason I know it is most likely deficient in some crucial manner. What I am more interested is... do there exist workable notions of truth which rely on a similar principle, namely to axiomatise the properties of the collection of true statements that we expect to be true. 
Another property, for example, that I thought I could add is that $T$ is minimal with respect to those properties, but I wasn't sure if that would add anything at all.
 A: In first-order logic one can make sense of what it means for a statement to be true with respect to some model. For example, there is a theory $T$ of groups, and a language of groups, and models of $T$ are groups. In any group $G$ one can ask whether a particular statement in the language of groups (for example, "for all $g, h$, it's true that $gh = hg$") is true or false. 
For any group, the set of all statements true in that group is a set of statements satisfying the properties you listed. Moreover, by the completeness theorem, the set of statements true in all groups is precisely the set of provable statements.
However, it's meaningless to ask whether a statement is true, without any further qualifiers; truth only makes sense here with respect to some model. This is even true when your first-order theory is something like Peano arithmetic or ZF set theory; statements are generally true in some models and false in others (meaning they can neither be proven nor disproven), and that's just the way it is. 
A: The point is that in the meta-system you know whether or not a theory $T$ is consistent (otherwise the meta-system is too weak to be a useful meta-system). If it is consistent, then the collection of all provable sentences over $T$ is already the minimal set satisfying your constraints, so there is no reason to introduce your new notion since it is simply $\{ φ : T \vdash φ \}$. And in a sufficiently strong meta-system that can talk about models this is equivalent to $\textbf{Th}\ \textbf{Md}\ T$.
However, you may be interested what happens when we attempt to extend a theory to include its own truth predicate. Kripke's theory of truth is one possible way to do so, once you allow a third truth value, which allows the liar sentence to 'fall into the truth-value gap' and avoid the usual contradiction. Tarski's undefinability theorem shows that the 3-valued logic is crucial for this to work, and also shows that even such a theory $T$ cannot contain a modal operator $True$ such that $True(φ)$ is true if $T \vdash φ$ and is false otherwise.
A: Yes, there has been an enormous amount of work on this. See the article Axiomatic Theories of Truth in the Stanford Encyclopedia of Philosophy for a starting point. 
As another answer points out, in mathematical logic we can often avoid the notion of "truth" simpliciter, and only talk about "truth in a model". This is not the case in philosophy, where the nature of "truth" is a major issue. The questions "What is true?" and "How do we know it's true?" are two of the basic issues philosophers have discussed. 
Alfred Tarski's T-schema is one way of axiomatizing truth. Quoting Wilfrid Hodges from his article Tarski's Truth Definitions:

In 1933 the Polish logician Alfred Tarski published a paper in which he discussed the criteria that a definition of ‘true sentence’ should meet, and gave examples of several such definitions for particular formal languages. In 1956 he and his colleague Robert Vaught published a revision of one of the 1933 truth definitions, to serve as a truth definition for model-theoretic languages. 

In the context of the natural numbers and the formal language of first-order arithmetic, the T-schema goes like this:


*

*An equation $t = s$ between closed terms $t$ and $s$ (i.e. terms with no variables) is true if and only if $t$ and $s$ have the same numerical value.

*A conjunction $\phi \land \psi$ is true if and only if $\phi$ is true and $\psi$ is true

*A disjunction $\phi \land \psi$ is true if and only if $\phi$ is true or $\psi$ is true

*An implication $\phi \to \psi$ is true if and only if $\phi$ is not true or $\psi$ is true

*A negation $\lnot \phi$ is true if and only if $\phi$ is not true

*A universally quantified sentence $(\forall x)\phi(x)$ is true if and only if $\phi(n)$ is true for each natural number $n$

*An existentially quantified sentence $(\exists x)\phi(x)$ is true if and only if $\phi(n)$ is true for some natural number $n$


As Hodges mentions, this defintion was later been adapted to serve as the defintion of "truth in a model", but it was originally intended to capture "truth" simpliciter.  These axioms were intended to capture the minimal class of formulas that should be regarded as "true" (and, indeed, they do characterize exactly the true formulas in any model of arithmetic).  
The axioms above can be changed to handle truth in other formal languages. In many cases, the only part that needs to be changed is the definition of truth for closed terms. 
