What is to define truth in Mathematical Logic? Tarski's Undefinability Theorem says that arithmetical truth cannot be defined in arithmetic.
What is the meaning of "definition"? Is it formal?
 A: Yes, it is.
According to :

TARSKI UNDEFINABILITY THEOREM (1933) : The set $\# \text {Th} \mathfrak N$ [i.e. the set of Gödel-numbers of sentences true in $\mathfrak N$] is not definable in $\mathfrak N$.

Here the concept of definability is defined as follows :

Consider a structure $\mathfrak A$ and a formula $\varphi$ whose free variables are among $v_1,\ldots, v_n$. We can construct the $n$-ary relation on $|\mathfrak A|$

$\{ \ \langle a_1,\ldots,a_n \rangle \vDash _{\mathfrak A} \varphi [a_1,\ldots, a_n] \ \}$.

Call this the $n$-ary relation $\varphi$ defines in $\mathfrak A$.
In general, a $n$-ary relation on $|\mathfrak A|$ is said to be definable in $\mathfrak A$ iff there is a formula (whose free variables are among $v_1,\ldots, v_n$) that defines it there.


This means that we cannot "manufacture" a formula $\varphi$ in first-order language of arithmetic with one free variable such that $\varphi$ holds (in the "standard model" of arithmetic) of exactly those natural numbers that are the Gödel numbers of the true sentences of arithmetic.
