As I know, Tarski had proved his undefinability theorem by $1936$, $5$ years after Gödel's incompleteness theorems had been discovered. I wonder whether his original proof was built upon the work of Gödel or not.
Could the incompleteness theorems have been proved by Tarski by $1936$ if Gödel had not already discovered them?
In principle, yes.
See:
Tarski made clear his indebtedness to Gödel's methods. [...] On the other hand Tarski strongly emphasized the fact that his results were obtained independently.
Tarski did not claim any priority for Gödel's own results. In footnote 2 to the paper Some observations on the concept of omega-consistency and omega-completeness (1933) he wrote:
Already, in the year 1927 (...) I also communicated the example of a consistent and yet not omega-consistent system which I give in the present article in a slightly altered form. Naturally it is not hereby claimed that I already knew then the results later obtained by Gödel or had even foreseen them. On the contrary, I had personally felt that the publication of the work of Gödel was a most exciting scientific event.
Though Tarski saw the similarity of the system $P$ used by Gödel in the paper of 1931 and his own system used in 1933 he adds [...] the remark that (...) the abstract character of the methods used by Gödel renders the validity of his results independent to a high degree of the specific peculiarities of the science investigated.
It is known that Tarski's theorem on undefinability of truth implies the existence of undecidable sentences, hence Gödel's first incompleteness theorem. So one can ask whether Tarski was close to this theorem.
