In three-dimensional space, can Playfair's Axiom:
Given a line $a$ and a point $P$ not in $a$, there is at most one line in $P$ parallel to $a$.
be “replaced by“ the following axiom?
Given a plane $\alpha$ and a point $P$ not in $\alpha$, there is one and only one plane in $P$ parallel to $\alpha$.
To make the question precise, can Plaifair's Axiom be proved from the following set of axioms?
- If a point and a plane have a line in common, the point must lie in the plane.
- Any two distinct points have one and only one line in common.
- If two distinct planes have a point in common, they have one and only one line in common.
- Any line and any point not in this line have one and only one plane in common.
- Any plane and any line not in the plane have at most one point in common.
- Given a plane and a point not in the plane, there is one and only one plane in the point having no point in common with the given plane.
- In every line lie at least two points.
- There exist two lines having no plane in common.
where to lie in is a symmetric binary relation on a set partitioned by the three sorts Point, Line, Plane, “$A$ and $B$ have $C$ in common“ is short for “$A$ lies in $C$ and $B$ lies in $C$“, and lines $a$ and $b$ are parallel iff they have a plane but no point in common.
My first guess was yes, but now I'm starting to think Axiom 6 is weaker than Playfair's Axiom, since I can't think of a way to prove it. In case anyone is wondering, this question is not from a book or anything but arose quite naturally while trying to find nice axioms characterizing three-dimensional affine space.