|WikiProject Mathematics||(Rated Start-class, Low-priority)|
Someone needs to clean this up. This is a mess.
Hyperbolic and Euclidean geometry are both extensions of neutral geometry
The claim that "Its theorems are therefore true in some non-Euclidean geometries, such as hyperbolic geometry, as well as in Euclidean geometry" is misleading: using "some" suggests "not all", but in fact every theorem of neutral geometry is a theorem of both hyperbolic and Euclidean geometry. This follows immediately from the definition of these two theories: hyperbolic geometry is by definition neutral geometry plus the negation of Euclid's parallel postulate, and Euclidean geometry is by definition neutral geometry plus Euclid's parallel postulate. Jesse Alama (talk) 21:09, 13 August 2008 (UTC)
I think there is a result in neutral geometry worth mentioning: Given any line l and any point P not on l, there is at least one line parallel with l crossing P. I think it's proven in Greenberg's book. MammonI.Dumah (talk) 16:00, 24 August 2012 (UTC)
"Absolute geometry is an extension of ordered geometry"
"One can extend absolute geometry...giving rise to...ordered...geometry."
Request for clarification of lede
The lede currently states what absolute geometry is not (it does not assume the parallel postulate), but it does not say what it is. Outside the lede, the article says "Absolute geometry assumes the first four of Euclid's Axioms" -- does this serve as a definition of absolute geometry? I.e., can we put something like the following into the lede: Absolute geometry is defined as a geometry that assumes the first four of Euclid's axioms, and nothing else.? Duoduoduo (talk) 17:37, 15 March 2011 (UTC)
"One can also prove in absolute geometry [...] the Saccheri-Legendre theorem, which states that a triangle has at most 180°."
Seeing as the parallel postulate is equivalent to "a triangle's angles sum to 180°", surely for absolute geometry that isn't Euclidean it would hold that "a triangle's angles sum to less than 180°"? I realise nothing in the quoted statement outright says otherwise (the Saccheri-Legendre theorem is not the Triangle postulate, just a 'superset'), but it seems heavily implied that any absolute geometry can satisfy "a triangle's angles (can) sum to 180°", which is all that's needed.
I'm not a mathematician, though, so I'm going to assume I've just fundamentally misunderstood something here. Can someone clue me in, though? And/or clean up the wording so someone reading the article doesn't get the wrong idea?
Also: "Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry." to my eyes seems to clash with "One can extend absolute geometry by adding different axioms about parallel lines and get incompatible but consistent axiom systems, giving rise to Euclidean, ordered and hyperbolic geometry." Which is an extension of which?
This is actually an equivalent statement. Informally, if our system is complete, then any statement P in its language which is not a direct consequence of the axioms should be false and thus adding it to our axioms leads to inconsistency. Conversely, if it is incomplete, we have a statement P which can be neither proven, nor refuted. Now as neither P nor ~P follows from our axioms, adding P to them should yield a consistent system. — Preceding unsigned comment added by MammonI.Dumah (talk • contribs) 12:47, 24 August 2012 (UTC)