The thing about mathematics and logic is that logic went from being a branch of philosophy to a branch of mathematics and once logic was formalized it could then be described in terms of mechanical calculation. The proof of the halting problem took less than a page in my college text on Language, Automata and Machines and some version of the proof of Gödel's Incompleteness Theorem take little more space. But that's now that the properties of computation are well established and can be hand-waved away.

So a lot of what Kurt Gödel did was follow a fairly inevitable progression before anyone else. And it is one of those ironies the result, that the-provable and the truth are far apart, did not please such a brilliant idealist.

Edit: From wikipedia: "Mathematical logic emerged in the mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics (Ferreirós 2001, p. 443). "Mathematical logic, also called 'logistic', 'symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the last [nineteenth] century with the aid of an artificial notation and a rigorously deductive method."[3] Before this emergence, logic was studied with rhetoric, with calculationes,[4] through the syllogism, and with philosophy. "

https://en.wikipedia.org/wiki/Mathematical_logic#History