Bachmann's Axioms
Undefined Terms: motion and line
Axioms: (Not including axioms
and definitions from group theory)
Ax1: The set of motions is a group.
Ax2: Lines are motions.
Ax3: Lines are involutions.
Ax4: Every motion is a product of lines.
Def: For any involutions a,b, if ab=ba and
a<>b then we write a|b.
Def: A POINT is a product of two lines that is an involution.
Def: If P is a point and b an involution, then P is ON b iff P|b.
Def: If a,b are lines, then a is PERPENDICULAR to b iff a|b.
Let P,Q be points and a,b,c,d lines in the
following.
Ax5: For any P,Q, there is b such that P|b and Q|b
Ax6: For all P and b there exists c such that P|b and c|b.
Ax7: For all P,Q and a,b,c, if P|b and Q|b and P|c and Q|c then either P=Q or
b=c.
Ax8: For all P and a,b,c, if (P|a and P|b and P|c) or (d|a and d|b and d|c) then
abc is a line.
Any model of this system is called a
Bachmann geometry.
[Source: Henle, Modern Geometries, 2nd
edition, ISBN:0-13-03231306]
|
