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Euclid's fifth postulate:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Every example (that I've seen) of proving that the parallel postulate in Euclid's five postulates is independent of the others seems to rely on the fact that one can invent a new kind of geometry where the first four postulates hold without the existence of the parallel postulate. Hyperbolic and elliptic geometry are often used as examples. Is this the only known method or approach to prove the parallel postulate's independence? Is discovering these curved geometries as counterexamples the only way?

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  • $\begingroup$ In order to prove that the $5$th postulate is not a consequence of the first four, you have to exhibit a model where the first four hold and the fifth does not. What kind of alternative approach do you have in mind? $\endgroup$ – Jack D'Aurizio Aug 9 '18 at 5:10
  • $\begingroup$ @JackD'Aurizio So that definitely is the only way? $\endgroup$ – FuzzyCat444 Aug 9 '18 at 5:12
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    $\begingroup$ @JackD'Aurizio That's not necessarily true - there could also be a purely proof-theoretic argument (e.g. the proof of the second incompleteness theorem doesn't involve building a nonstandard model). But I don't know of any. $\endgroup$ – Noah Schweber Aug 9 '18 at 5:14
  • $\begingroup$ @JackD'Aurizio The only alternative I could possibly think of is a brute force method of finding every possible theorem that can be proven by the first four postulates and showing none are equivalent to the fifth postulate. This seems impractical or impossible to say the least. So I have no idea about a better approach. $\endgroup$ – FuzzyCat444 Aug 9 '18 at 5:29
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    $\begingroup$ @FuzzyCat444 Since there are infinitely many theorems in any given system, this won't work. In principle there might be some way to analyze the general structure of possible proofs, however, which could shed some light on the picture; I don't know of anything of that type here, but such things exist in proof theory in other contexts. $\endgroup$ – Noah Schweber Aug 9 '18 at 5:37
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I know two ways of proving that something is not provable: - Using the semantic approach, one builds a counter model, i.e. in which all other axioms hold but the parallel postulate fails (this is the usual method for the parallel postulate). - Using the syntactic method, you can show that something is not provable more directly.

Together with Michael Beeson and Pierre Boutry, we have given a syntactic proof of the independence of the parallel postulate in the context of Tarski's axioms : Herbrand's theorem and non-Euclidean geometry, Bulletin of Symbolic Logic, Association for Symbolic Logic, 2015, 21 (2), pp.12

The general idea of the proof is that the parallel postulate (at least some versions of this postulate) allows to construct points which are arbitrarily far from the given points (take two lines which are very close to be parallel), whereas other axioms allows only to double the maximum distance between the points constructed so far.

To explain the difference between a syntactic independence proof and a semantic one, I give an elementary example in the following talk: http://dpt-info.u-strasbg.fr/~narboux/slides/Herbrand-Euclid-vulgarization.pdf

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Are we agreed on what the other four postulates are? As I recall, the first three do not postulate truths at all, but rather things a geometer is supposed to be able to do: 1) draw a straight line between two points, 2) extend a line indefinitely, 3) draw a circle with any given radius. This is the "ruler and compass" basis of Euclidean geometry. Besides the fifth postulate, only the fourth purports to state a truth: all right angles are equal. If Aristotle was right, that you need at least two premises to draw a conclusion, then I don't see how you could deduce the fifth postulate, a statement of fact, from just one other statement of fact, the fourth.

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