Gödel's Proof (Classic Reprint)

Excerpt from Gödel's Proof

The axiomatic development of geometry made a powerful impression upon thinkers throughout the ages; for the relatively small number of axioms carry the whole weight of the inexhaustibly numerous prop ositions derivable from them. Moreover, if in some way the truth of the axioms can be established - and, indeed, for some two thousand years most students be lieved without question that they are true of space both the truth and the mutual consistency of all the theorems are automatically guaranteed. For these rea sons the axiomatic form of geometry appeared to many generations of outstanding thinkers as the model of scientific knowledge at its best. It was natural to ask, therefore, whether other branches of thought besides geometry can be placed upon a secure axiomatic foun dation. However, although certain parts of physics were given an axiomatic formulation in antiquity. By Archimedes), until modern times geometry was the only branch of mathematics that had what most students considered a sound axiomatic basis.

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