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Can someone explain why this statement is false?

Although 2000 years of efforts to prove the parallel postulate as a theorem in neutral geometry have been unsuccessful, it is still possible that someday some genius will succeed in proving it.

Wangpin Lin
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1 Answers1

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We have a model of hyperbolic geometry where the parallel postulate is false. We have a model of Euclidean geometry where the parallel postulate is true. Both extend the axioms of hyperbolic geometry. If the axioms of neutral geometry are inconsistent, anything is provable, so you could prove the parallel postulate. Otherwise, we have shown that the axioms of neutral geometry are consistent with both, so the parallel postulate cannot be proven.

Ross Millikan
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  • could you please explain what neutral geometry means? and how this shows that the parallel postulate is unprovable? – jg mr chapb May 02 '17 at 04:57
  • Wikipedia calls it absolute geometry but acknowledges the synonym. It is Euclidean geometry without the parallel postulate. – Ross Millikan May 02 '17 at 05:02
  • oh ok. I didn't see that OP used the term though – jg mr chapb May 02 '17 at 05:03
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    @gebra: We have a set of axioms (absolute geometry) and another proposed axiom (parallel postulate). If we can make two models, both in absolute geometry, one of which makes the parallel postulate true and one which makes it false, we can be confident that the parallel postulate is not decidable by the other axioms. – Ross Millikan May 02 '17 at 05:07
  • really, that very conterintuitive – jg mr chapb May 02 '17 at 10:14