I have been reading this article about Tarski's undefinability theorem:
https://qubd.github.io/files/TarskiUndefinability.pdf
It has a very interesting section on page 4:
This does not really make sense to me. If $\varphi_n$ (a formula with the Godel's number $n$) is a formula with one free variable, then $\forall v_1 (v_1 = \overline{n}\rightarrow \varphi_n) $ seems to be an incorrect notation. It's because $\varphi_n$ is a formula with a free variable -> therefore, it cannot be neither true nor false, and therefore it cannot be a consequent in an implication. Or am I wrong?