ABSTRACT
Go¨del first presented his incompleteness theorem at a conference on “Epistemology of the exact sciences” in 1930. Go¨del was then not yet 25 years old, so he is one of a considerable number of mathematicians and logicians who have made major discoveries at an early age. Among those present was the Hungarian mathematician John von Neumann, three years Go¨del’s senior, one of the great names of twentieth-century mathematics and the subject of various anecdotes about his remarkable powers of quick apprehension. It appears that he was the one participant at the conference who immediately understood Go¨del’s proof. At this point Go¨del had not yet arrived at his second incompleteness theorem, and his proof of the first incompleteness theorem was not applicable to PA, but only to somewhat stronger theories. His proof did, however, establish that assuming a theory S to which the proof applied to be consistent, it follows that the Go¨del sentence G for S is unprovable in S. Reflecting on this after the conference, von Neumann realized that the argument establishing the implication “if S is consistent, then G is not provable in S” can be carried out within S itself. But then, since G is equivalent in S to “G is not provable in S,” it follows that if S proves the statement ConS expressing “S is consistent” in the language of S, S proves G, and hence is in fact inconsistent. Thus, the second incompleteness theorem follows: if S is consistent, ConS is not provable in S. By the time von Neumann had discovered this and written
to Go¨del about it, Go¨del had himself already made the same discovery and included it in his recently accepted 1931 paper.
