Yourgrau's appears to be the more accurate account of what Gödel was attempting to accomplish with his two great Incompleteness theorems
Would this be it? From the link:
Finally, Gödel's incompleteness theorem set a permanent limit on our knowledge of the basic truths of mathematics: The complete set of mathematical truths will never be captured by any finite or recursive list of axioms that is fully formal. Thus, no mechanical device, no computer, will ever be able to exhaust the truths of mathematics. It follows immediately, as Gödel was quick to point out, that if we are able somehow to grasp the complete truth in this domain, then we, or our minds, are not machines or computers.