Axiom:Hilbert's Axioms/Connection

Definition

Plane Axioms

\((\text I, 1)\)	$:$							Two distinct points $A$ and $B$ always completely determine a straight line $a$. We write $AB = a$ or $BA = a$.
\((\text I, 2)\)	$:$							Any two distinct points of a straight line completely determine that line; that is, if $AB = a$ and $AC = a$, where $B \ne C$, then also $BC = a$.

Space Axioms

\((\text I, 3)\)	$:$							Three points $A, B, C$ not situated in the same straight line always completely determine a plane $\alpha$. We write $ABC = \alpha$.
\((\text I, 4)\)	$:$							Any three points $A, B, C$ of a plane $\alpha$, which do not lie in the same straight line, completely determine that plane.
\((\text I, 5)\)	$:$							If two points $A, B$ of a straight line $a$ lie in a plane $\alpha$, then every point of $a$ lies in $\alpha$.
\((\text I, 6)\)	$:$							If two planes $\alpha, \beta$ have a point $A$ in common, then they have at least a second point $B$ in common.
\((\text I, 7)\)	$:$							Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane.

Sources

• 1902: David Hilbert: The Foundations of Geometry (translated by E.J. Townsend): $\text I \S 2$. Group $1$: Axioms of connection