>From the Rosser, Sokolowski, Kierman, and Rosenberg papers on Goedel's
completeness and consistency theorems:
>> it just says that any system of axioms will not completely describe
the arithmetic system. That is, given any axiomatic system, there are
true statements about arithmetic which can not be proven using those
axioms.
>It's important to note that this came as a shocking surprise, not
simply because of ignorant prejudice, but because Euclidean Geometry
had already been completely axiomatized.
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Goedel set out to solve the paradoxes of transfinite mathematics by constructing meta-models of arithmetic and formed the completeness and consistency theorems in the process.
He had investigated Hilbert and Ackermann's attempt to create a completely formulated foundation as a graduate student under Hans Hahn (University of Vienna) and believed if he could find a means to extend the axioms of finite systems (Peano axioms) to cover infinite or transfinite systems, then the grand project of Hilbert and the formalist school would have jumped its greatest hurdle. His famous paper was a revision of his thesis under Hans Hahn.
But the existence of undecidable propositions (1931) had more comprehensive implications that just those of mathematics. The philosophical thrust was to essentially destroy the idealism, or neo-Platonist belief in the sanctity of the a priori, period. It was a closing parenthesis on an entire branch of analytic philosophy that had developed at least since Kant.
The conference title in which Goedel presented the first of his papers was, "Conference on Epistemology of the Exact Sciences." It was held in Koenigsberg and opened with addresses by Carnap, Heyting and John von Neumann. The journal that published the conference papers was an organ for the Vienna Circle. Other presenters included Heisenberg, Reichenbach, and Neugebauer. Wittenstein's position on the epistemology of mathematics was presented in a lecture by Waismann. The concluding roundtable discussions were chaired by Hans Hahn with Carnap, Heyting, von Neumann and Emmy Noether attending. These were luminaries in mathematics and philosophy and all committed to the grand ideal of a fully rationalized mathematics. They considered mathematics a model for rational thought. Goedel's paper was such a stunning blow, that at least Reichenbach and Carnap didn't feel a thing. Von Neumann on the other hand got it immediately and sent Goedel an independent proof of his own a month later. At this point, von Neumann was working with Hilbert and trying to salvage the Hilbert Ackermann work.
For anybody interested there is a collection of the philosophy papers that were presented at that famous conference by Carnap, Heyting, von Neumann. Other essays follow from Hilbert, Bernays, Brouwer, Russell, Quine, Wittgenstein, and Goedel that help set up the context. See,
_Philosophy of Mathematics, Selected Readings_, Benacerraf P, Putnam H eds. Prentice-Hall, Englewood Cliffs New Jersey, 1964.
Chuck Grimes