# Can Pasch's axiom be derived from these postulates?

Postulates:

• Given any line, there are points on the line and points not on the line.

• Given two distinct points there exists a unique line passing through those points.

• Given three points on a line, one and only one of them is between the other two.

• Given two points $A$ and $B$ there always exists a point $C$ between $A$ and $B$ and a point $D$ such that $B$ is between $A$ and $D$.

• A line $m$ determines exactly two distinct semi planes, whose intersection is the line $m$.

Pasch's axiom: If a line not going through the vertices of a triangle (here I'm excluding the degenerate case of a triangle formed by three points on the same line) intersects one side, then it intersects another side.

• Please define "semi-plane" Apr 13, 2018 at 13:15
• My foundations instincts are a little rusty, but the "semi-planes" postulate seems like a statement of the Plane Separation Axiom; the PSA is known to be equivalent to Pasch's Axiom, given an appropriate set of other axioms (say, Hilbert's). Off-hand, I couldn't say whether equivalence can be shown with the particular set of axioms you list. What's the context of this problem? A textbook exercise?
– Blue
Apr 13, 2018 at 13:15
• It is indeed a textbook exercise. Thanks for replying, I know now that the answer to my question is yes. Apr 13, 2018 at 13:23