Euclid's Parallel Axiom
Nagel and Newman consider the question raised by the parallel postulate to be "...perhaps the most significant development in its long-range effects upon subsequent mathematical history" (p. 9).
The question is: can the axiom that two parallel lines "...will not meet even 'at infinity'" (footnote, ibid) be derived from the other axioms of Euclid's geometry? It was not until work in the nineteenth century by "... Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. ...a proof can be given of the impossibility of proving certain propositions within a given system" (p. 10).
To clarify: Nage and Newman mean "the proposition" to be the statement "Parallel lines will not meet at infinity" and "the given system" is Euclid's axioms of geometry. The proof showed that no proof exists; i.e. "a proof" is impossible.
Read more about this topic: Proof Of Impossibility
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