Incompleteness by Rebecca Goldstein

Author:Rebecca Goldstein [Goldstein, Rebecca]
Language: eng
Format: epub
ISBN: 9780393242454
Publisher: W. W. Norton & Company
Published: 0101-01-01T00:00:00+00:00

Von Neumann Takes the Hint

There happened to have been one person present at Königsberg who picked up on the anomalous remark of the young logician, and that was John von Neumann. His appreciation of Gödel’s terse remark is all the more impressive if we consider that von Neumann’s own views were entirely in keeping with Hilbert’s—he had been the designated spokesman in Königsberg for formalism—and that he harbored the sort of strongly positivist bent that would make Gödel’s reference to semantic truth, independent of a formal system, perhaps seem dangerously metaphysical. Nonetheless he buttonholed Gödel after the discussion ended for the day and pumped him for details. Gödel must have told him enough about how he had arrived at his conclusion for von Neumann to take what he heard seriously. He went back to Princeton, to the Institute for Advanced Study, and continued to ponder the astounding pronouncement he’d heard in Königsberg.

Some time in the course of his pondering, von Neumann happened on a remarkable corollary to what Gödel had told him. Von Neumann had seen from what Gödel had told him that Gödel’s proof was conditional: what it says is that if a formal system S of arithmetic is consistent, then it’s possible to construct a proposition, call it G, that’s true but unprovable in that system. So if S is consistent, G is both true and unprovable. Trivially, then, if S is consistent then G is true. Von Neumann had also understood from what Gödel told him that this proof can itself be carried out in a system of arithmetic. (This is the trick that’s accomplished by Gödel numbering.) So if the consistency of S could be proved in S, then G would have been proved in S—since it follows from the consistency of S that G is true. But this contradicts that G is unprovable. The only way out of the contradiction is to deny that S can be formally proved to be consistent within the system of arithmetic. So from Gödel’s result another impossibility follows: it is impossible to formally prove the consistency of a system of arithmetic within that system of arithmetic.

Von Neumann got in touch with Gödel, informing him of this corollary, and Gödel politely told von Neumann that the older man had indeed drawn the correct conclusion, one which Gödel had already rigorously proved. (One can imagine Gödel’s slight crooked grin in imparting this information to the intellectual titan, von Neumann.) This corollary is known as Gödel’s second incompleteness theorem, and though it’s merely a consequence of the first, it’s the one that first received attention, with von Neumann talking it up at the Institute. Hilbert’s program had provided the context for perceiving the significance of Gödel’s second incompleteness theorem. Gödel had proved that Hilbert’s second problem could not be solved: there would never be a finitary formal proof of the consistency of the axioms of arithmetic within the system of arithmetic. There would never be the proof that was to serve as the linchpin for Hilbert’s program.

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