Given the following system Points S12345 Lines 123 14 15 24

Given the following system:

Points: S={1,2,3,4,5}
Lines: {1,2,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}
Planes: {1,2,3,4}, {1,2,3,5}, {2,3,4,5}, {1,4,5},

which of Hilbert’s incidence axiom(s) does not hold? Explain why not

Solution

axioms that doesn\'t hold ---

(1) If two points A, B of a line a lie in a plane , then every point of a lies in .

Since plane {1,2,3,4} contain point 1, and line {1,5} contain point 1 but plane doesn\'t contain point 5 while it must be contain.

(2) For every three points A, B, C which do not lie in the same line, there exists no more than one plane that contains them all.

Since {2,3,4} do not lie in a same line. although there are two planes ( plane {1,2,3,4} and plane {2,3,4,5} ) which contain all them which is not allowed by axioms.