In my last post on Goldstein’s book, Incompleteness, I said that Goldstein tried to present Goedel’s two theorems from the perspective of Goedel’s own sense of what these theorems ultimately proved. That is, what they proved meta-mathematically. Goldstein is interested in the divergence between the popular image of what Goedel was up to and the fact that Goedle’s incompleteness proof, by insinuating a moment of absolute uncertainty into any formalization of arithmetic, seems to point to the insufficiency of conventionalism, not to affirm it. The uncertainty, you will remember, goes like this: in any formally consistent language adequate for number theory, a., there will be one proposition that one can generate from the axioms of the system the truth or falsity of which can’t be decided by those rules, and b., that the consistency of the system can’t be proved within the system. Now, Goldstein’s major point is to show why Goedel might take his theorems as evidence that there are real
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