# Is it possible to prove the independence of the parallel postulate without non-euclidean geometry?

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?

• 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? – Jack D'Aurizio Aug 9 '18 at 5:10
• @JackD'Aurizio So that definitely is the only way? – FuzzyCat444 Aug 9 '18 at 5:12
• @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. – Noah Schweber Aug 9 '18 at 5:14
• @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. – FuzzyCat444 Aug 9 '18 at 5:29
• @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. – Noah Schweber Aug 9 '18 at 5:37