TITLE: What was Gödel's real achievement?
QUESTION [47 upvotes]: When I first heard of the existence of Gödel's theorem, I was amazed not just at the theorem but at the fact that the question could be made precise enough to answer: how on earth, even in principle, could one show that it was impossible to prove something in a given system? That doesn't bother me now, and that is not my question.
It seems to me that Gödel's theorem is a combination of at least three amazing achievements, namely these.

Formalizing the notions of proof, model, etc. so that the question could be considered rigorously.
Daring to think that there might be true but unprovable statements in Peano arithmetic.
Thinking of the idea of Gödel numbering and getting the proof to work.

One might think that 3 constitutes two separate achievements, but I think that actually getting the proof to work, though pretty good going, is somehow a technicality once you have had the idea that in principle a proof along those lines might be possible. (I'm not saying I could have done it, but Gödel would have been deeply immersed in these ideas.)
My guess is that pretty well all the credit for 2 and 3 goes to Gödel (except that the idea of diagonal proofs was not invented by him). My question is how much he contributed to 1 as well. Had it occurred to anyone else that it might be possible to think about such questions rigorously, or did an entirely new way of thinking appear pretty well out of nowhere? Popular accounts suggest the latter, but common sense would suggest the former, at least to some extent.

REPLY [4 votes]: If I remember correctly, the notion of a model is already present in Hilbert-Bernays where finite structures are used for proving absolute consistency of some theories, and is probably older. Again, if I remember correctly, Frege, Russel, and Hilbert did have formal systems and the notion of a formal proof. Skolem's construction of a term model (which is now famous as Skolem's Paradox because the set of real numbers of the model turns out to be countable) is in his 1922 paper, where the Godel's completeness theorem is from 1929. In other words, it seems that Skolem did already have all the tools necessary for proving completeness in 1922. It seems that Hilbert had even stated the question of completeness for first-order theories before this date and Godel has learned about this problem in Carnap's logic course in 1928.
Hilbert's 10-th problem from his famous 23 problems asks for an algorithm to decide existence of solutions for Diophantine equations. I think there were attempts after this for understanding what is an algorithm. There were many definitions which came out before Turing's definition which were equivalent to his definition, although they were not philosophically satisfactory, at least Godel did not accept any of them as capturing the intuitive notion of computability before Turing's definition.
Godel's collected works can shed more light on these issues.
EDIT: Also
Solomon Feferman, "Gödel on finitism, constructivity and Hilbert’s program"
http://math.stanford.edu/~feferman/papers/bernays.pdf

Hilbert and Ackermann posed the fundamental problem of the completeness of the first-order predicate calculus in their logic text of 1928; Gödel settled that question in the affirmative in his dissertation a year later. [page 2]
Hilbert introduced first order logic and raised the question of completeness much earlier, in his lectures of 1917-18. According to Awodey and Carus (2001), Gödel learned of this completeness problem in his logic course with Carnap in 1928 (the one logic course that he ever took!). [page 2, footnote]

Martin Davis, "What did Gödel believe and when did he believe it", BSL, 2005

Godel has emphasized the important role that his philosophical views had
played in his discoveries. Thus, in a letter to Hao Wang of December 7,
1967, explaining why Skolem and others had not obtained the completeness
theorem for predicate calculus, ... [page 1]