# Was the idea of encoding meta-mathematics into arithmetic in the air when Gödel proved incompleteness theorems?

I mean, was there anyone who tried (before Gödel) to encode meta-mathematics? Or, was the idea of constructing formal sentences which informally refer to itself completely new when Gödel introduced his work? Or were there some work before which makes such a step quite natural and not surprising?

What I mean, how did he come up with this very idea of arithmetizing logic and constructing self-referential sentences? Was there any effort done before him which makes such a step not very surprising? Or is it better to say that "he was just able to see it where no other mathematician was able to"?

Note that, by "in the air" I mean that there was previous work that indicates that such thing could be fruitful to do or that the idea of encoding things was already thought of by other people (or closely related ideas which could naturally lead to encoding).