1 Prove or disprove the following statement Proivide justifi

1. Prove or disprove the following statement. Proivide justification for your answer.

All models of incidence geometry are also a model of Euclidean Geometry?

I drew a triangle, but I am not sure if it is also a model of Euclidean geometry?

Solution

These undefined terms will be subjected to three axioms, the first
of which is the same as Euclid\'s first postulate.

INCIDENCE AXIOM 1. For every point P and for every point Q not
equal to P there exists a unique line 1incident with P and Q.

INCIDENCE AXIOM 2. For every line !there exist at least two distinct
points incident with I.
INCIDENCE AXIOM 3. There exist three distinct points with the property that no line is incident with all three of them.

This is a statement in the formal system incidence geometry: \"For every line I
and every point P not lying on I there exists a unique line through P
that is parallel to I.\" This statement appears to be correct according to
our drawings (although we cannot verify the uniqueness of the paral-
lelism, since we cannot extend our dashes indefinitely). But what
about our three-point model? It is immediately apparent that no par-
allellines exist in this model: {A, B} meets {B, C} in the point B and
meets {A, C} in the point A; {B, C} meets {A, C} in the point C. (We
say that this model has the elliptic parallel property.)
Thus, we can conclude that no proof of the euclidean parallel postulate
from the axioms of incidence alone is possible; infact, in incidence geometry it
is impossible to prove that parallel lines exist.