Godel's Proof

by Ernest Nagel

In elliptic geometry, for example, Euclid’s parallel postulate is replaced by the assumption that through a given point outside a line no parallel to it can be drawn. Now consider the question: Is the elliptic set of postulates consistent? The postulates are apparently not true of the space of ordinary experience. How, then, is their consistency to be shown? How can one prove they will not lead to contradictory theorems? Obviously the question is not settled by the fact that the theorems already deduced do not contradict each other—for the possibility remains that the very next theorem to be deduced may upset the apple cart. But, until the question is settled, one cannot be certain that elliptic geometry is a true alternative to the Euclidean system, i.e., equally valid mathematically. The very possibility of non-Euclidean geometries was thus contingent on the resolution of this problem.

Well said.