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Looking at the rules of projective planes the rule indicates:

There exists a set of four points, no three of which belong to the same line.

But I'm wondering why should there be a set of 4 points? I'm asking what could have gone wrong if we had this rule instead:

There exists 3 points not all on a same line.

Ali1S232
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1 Answers1

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This rule is meant to exclude degenerate cases like the near pencil.

http://mathworld.wolfram.com/Near-Pencil.html

The nearpencil consists of $n+1$ points $O,A_1,\dots,A_n$. The lines are given by the sets $\{A_1,\dots,A_n\}$, $\{O,A_1\}$, $\{O,A_2\}$, ..., $\{O,A_n\}$.

One can think of such a near pencil as a projective space of dimension 1 (a line) and a projective space of dimension 0 (a point) where all possible connections between both are made.

Myself
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  • may I ask what's wrong with near-pencils? I mean what kind of conjectures would invalidate if we consider them to be projective geometries? – Ali1S232 May 13 '13 at 15:17
  • For instance, the fact that the number of points on each line is constant. – Myself May 15 '13 at 10:11