Interpretation # 10

Points:	all dots on a sheet of paper.
Lines:	all horizontal and vertical lines.
Incidence:	PIl if the dot P lies on the line l.

QUESTIONS:

Does this interpretation satisfy the incidence axiom I-1?

Does this interpretation satisfy the incidence axiom I-2?

Does this interpretation satisfy the incidence axiom I-3?

Is this interpretation a model?

Determine whether this interpretation has the elliptic, Euclidean, of hyperbolic parallel property.

ANSWERS:

Does this interpretation satisfy the incidence axiom I-1?

No, there is no line through A and B in the figure. (the existence part of axiom I-1 is not satisfied)

Does this interpretation satisfy the incidence axiom I-2?

Yes, every line in this interpretation has infinitely many points, therefore at least two.

Does this interpretation satisfy the incidence axiom I-3?

Yes, points A, B, and C in the figure are distinct and non-collinear.

Is this interpretation a model?

No, the incidence axiom I-1 is not satisfied.

Determine whether this interpretation has the elliptic, Euclidean, of hyperbolic parallel property.

This interpretation has the Euclidean parallel property. For every horizontal (vertical) line, and point P that is not on the line, there is a unique horizontal (respectively vertical) line through P. Obviously two horizontal (vertical) lines are parallel.