To formulate the hyperbolic parallel postulate for the hyperbolic 2 dimensional (plane) is easy:
- Given any line ''L'' and point ''P'' not on ''L'', there are at least two distinct lines passing through ''P'' which do not intersect ''L''.
But now for 3 (and higher) dimensional geometry?
I found the rather horrible:
- "There exists a plane and a line in the plane, and a point in the plane not on the line, such that trough the point at least two lines can be drawn in the plane that do not intersect the given line" (Ramsay & Richtmeyer, " Introduction to hyperbolic geometry" , Springer 1991, page 130, Axiom 7b)
Hardly readable and 4 mentions of "plane" are there no more succinct formulations?
Preferably one that doesn't mention " a line in an earlier defined plane " which is an axiomatic rather complicated construction.
Here's one straightforward formulation of the parallel postulate: "Given a point in the space, and a co-line not incident to that point, there is a unique co-line incident to the point which does not meet the given co-line."
By 'co-line' here I mean a $(n-1)$-dimensional subspace; in the case where the 'space' is the normal 2d plane this is just the traditional parallel postulate. In three dimensions, this is the statement that given a plane and a point not on that plane, there's another plane passing through the given point and parallel to the given plane; etc.
Transferring from this postulate into a statement of its negation then follows just as it does with the traditional parallel postulate; for instance, you can say that 'given a point in the space, and a co-line not incident with the given point, there exists more than one co-line incident with the given point which does not meet the given co-line.'