Hello,

The proofs in logic often use the notion of truth.

Can we ignore the notion of truth, if we add axioms to the Peano's axioms ?

Is it possible to prove Gödel's first incompleteness theorem without the notion of truth ?

The proof of the Gödel's first incompleteness theorem, that I know is: Gödel numbers by $n$ the statements $E_n(i)$ with one argument $i$. "$E_i(i)$ is not provable" is a statement $A(i)$. There is an $n$ such that $A$ is $E_n$. The proof of the theorem says: if $A(n)$ is not $\mathit{true}$, $E_n(n)$ is provable. So $A(n)$ is provable, and $A(n)$ is $\mathit{true}$, contradiction. So $A(n)$ is $\mathit{true}$, and not provable.

If $n$ is the Gödel's number of a statement $Q_n$ (without argument), and if $P(n)$ is the statement saying that $Q_n$ is provable in the formal system of Peano's axioms, we add, for every $n$, the axiom: "$P(n) \implies Q_n$".

So the step "$A(n)$ is provable, implies $A(n)$" in the previous proof, is an axiom.

So we don't use the notion of truth. Do you have references about this subject ?

Thanks in advance.