What is to define truth in Mathematical Logic?

Question

Tarski's Undefinability Theorem says that arithmetical truth cannot be defined in arithmetic.

What is the meaning of "definition"? Is it formal?

Answer 1

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.

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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.