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"I particularly like the analogy with Euclidean geometry. If we remove the parallel postulate, we get absolute geometry, also known as neutral geometry. If afte

"I particularly like the analogy with Euclidean geometry. If we remove the parallel postulate, we get absolute geometry, also known as neutral geometry. If after we remove the parallel postulate, we add a suitable axiom, we get hyperbolic geometry, but if we instead add a different suitable axiom we get elliptic geometry. Every theorem of neutral geometry is a theorem of these three geometries, and more geometries. So a neutral proof is more general." I thought elliptic geometry was not a subset of absolute geometry.

ivanbakel 5y ago
I tried to find a proof online one way or the other, but there seems to be no concrete consensus. It's possible that whether or not elliptic geometry is a subset of absolute geometry is a consequence of the formulation of the axioms - since you can modify any axiom involving the existence of parallel lines using something like "If parallel lines exist, then...". Certainly the way I learned it was that the parallel postulate is the only axiom which distinguishes the elliptic, Euclidean, and hyperbolic geometries.
roywiggins 5y ago
I believe elliptic geometry is a subset, it's spherical geometry that isn't (though it's closely related).
- red_trumpet 5y ago
  Reading the Wikipedia entry about Hilbert's axioms [1], I think spherical geometry is not absolute, because two opposing points lie on many lines connecting the two (think north and south pole, connected by all the meridians). [1] https://en.wikipedia.org/wiki/Hilbert%27s_axioms https://en.wikipedia.org/wiki/Hilbert%27s_axioms
jimhefferon 5y ago
Euclidean has more axioms than absolute (one more). With more axioms you can prove more things (one of them being the axiom itself). So also, elliptic has more axioms than absolute.
jesboat 5y ago
Elliptic geometry is not a subset of absolute geometry (as absolute geometry is typically formalized). The problem isn't with Euclid's other postulates, it's with things that Euclid didn't formalize, specifically, betweenness. In absolute geometry, given any line containing three distinct points, exactly one of the points will be between the other two. There's no sensible way to define this in elliptic geometry. Intuitively, this makes sense; when you have three points on a line in elliptic geometry, you can travel from any of the three points to any other without passing by the third point. It's been a while since I've actively studied this, but my recollection is that all you have to do is replace the axioms for betweenness with axioms for separation, and you get a theory which is suitable for elliptic geometry. (Separation is an arity-4 relationship on points. Intuitively, given any four distinct and collinear points, A and C separate B and D if the order of the points on the line (traveling in either direction) is ABCD.) https://en.wikipedia.org/wiki/Foundations_of_geometry#Elliptic_geometry https://en.wikipedia.org/wiki/Foundations_of_geometry#Ellipt... appears to agree

ivanbakel 5y ago

I tried to find a proof online one way or the other, but there seems to be no concrete consensus. It's possible that whether or not elliptic geometry is a subset of absolute geometry is a consequence of the formulation of the axioms - since you can modify any axiom involving the existence of parallel lines using something like "If parallel lines exist, then...". Certainly the way I learned it was that the parallel postulate is the only axiom which distinguishes the elliptic, Euclidean, and hyperbolic geometries.

roywiggins 5y ago

I believe elliptic geometry is a subset, it's spherical geometry that isn't (though it's closely related).

red_trumpet 5y ago
Reading the Wikipedia entry about Hilbert's axioms [1], I think spherical geometry is not absolute, because two opposing points lie on many lines connecting the two (think north and south pole, connected by all the meridians). [1] https://en.wikipedia.org/wiki/Hilbert%27s_axioms https://en.wikipedia.org/wiki/Hilbert%27s_axioms

jimhefferon 5y ago

Euclidean has more axioms than absolute (one more). With more axioms you can prove more things (one of them being the axiom itself). So also, elliptic has more axioms than absolute.

jesboat 5y ago

Elliptic geometry is not a subset of absolute geometry (as absolute geometry is typically formalized). The problem isn't with Euclid's other postulates, it's with things that Euclid didn't formalize, specifically, betweenness. In absolute geometry, given any line containing three distinct points, exactly one of the points will be between the other two. There's no sensible way to define this in elliptic geometry. Intuitively, this makes sense; when you have three points on a line in elliptic geometry, you can travel from any of the three points to any other without passing by the third point. It's been a while since I've actively studied this, but my recollection is that all you have to do is replace the axioms for betweenness with axioms for separation, and you get a theory which is suitable for elliptic geometry. (Separation is an arity-4 relationship on points. Intuitively, given any four distinct and collinear points, A and C separate B and D if the order of the points on the line (traveling in either direction) is ABCD.) https://en.wikipedia.org/wiki/Foundations_of_geometry#Elliptic_geometry https://en.wikipedia.org/wiki/Foundations_of_geometry#Ellipt... appears to agree