How to define "true" in a formal manner? Title says it all.
My understanding is that axiom system is a set of true propositions which are promises and self-evident. And to prove a proposition, we use deductive way to abbreviate as axioms or axiom based proved theorems or lemmas. From this view, I thought I could think true as axioms itself.
However, while I'm studying mathematical logic, the text book states true without introducing any axioms, but some definitions. They states; for every rule of the system, the following holds: whenever the elements are derivable in a system, they have the property $P$ which holds for every elements derived from a system. (Property $P$ defined as a function $P: X \rightarrow\{\text{true, false}\}$ where $X$ is a set of all strings. Iff the value is $\text{true}$, we say the element has the property $P$ in a system.)
Is this the definition of mathematical true and falsity?
If it's not, how to rigorously define true?
 A: It is not so easy to assert that all axioms of all mathematical theories are "self-evident".
For sure, in mathematics, like all other sciences, "first principles" (i.e.axioms) are assumed to be true (until revision).
Logical inference rules are "designed" to be sound: i.e. to "transfer" truth.
Thus, deductive consequences of our axioms will be true, provided that the axioms are.

Added
If I understand your proposal, after the comments, you start from the set $Z$ of expressions (i.e.well-formed strings of symbols of the language) "generated" with a system; if we interpret system as a "typical" couple : 

axioms + inference rules, 

the set $Z$ of all expressions "generated" with it is what we usually call the set of theorems.
Now you introduce the characteristic function of $Z$, i.e.:

$P_Z(x)=1$ iff $x \in Z$.

Thus, your proposal is to "define" :


$T(x)$ [i.e. $x$ is true] iff $P_Z(x)=1$.


Assuming that my interpretation is correct, we have a couple of problems :
(i) Gödel's Incompleteness Theorem applies to "significant" formal systems, including most of "interesting" mathematical ones :

in any consistent formal system $F$ within which a certain amount of arithmetic can be carried out, there are statements of the language of $F$ which can neither be proved nor disproved in $F$. 

Thus, according to our "natural" insights regarding truth, we have mathematical truths expressible in the system $F$ which are not provable in it, i.e.are the class of theorems of the system $F$ does not "exhaust" all the "true facts" of the system.
(ii) if your system is inconsistent it will "generate" as theorem all well-formed "expressions" of the language. Thus, the set $Z$ of theorems will be "too large". 
