I hoped someone would make that connection! This one is actually sound but there is a closely related limitative result, the undefinability of truth (attributed to tarski) which uses a “liar sentence” like the “liar set” of Russell’s paradox: “this sentence is not true”. Of course, liar sentence have been known since ancient times, but it was only in the 20th century when we could give them a mathematical interpretation, rather than a purely logical one.

This means that there is no mathematical definition of what is true about the natural numbers, but there are still definitions of other things, and we can still quantify over those definitions.