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In Projective Geometry, Fano's Axiom says:

The three diagonal points of a complete quadrangle are never collinear.

I would like to prove this from more basic Axioms within three-dimensional Projective Space. The theorem of Desargues is non-trivial in plane geometry, but can be proven from basic axioms within three-dimensional geometry; could the same be true for Fano's Axiom? If not, are there nice (equivalent) alternatives?

Edit: by “more basic” I mean intuitively more basic, which makes it a somewhat subjective question of course.

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You can't prove Fano's axiom from 3-dimensional geometry because the projective plane over the field $F_2$ with two elements does not satisfy Fano's axioms. Recall that the projective plane can be defined starting with 3-dimensional space over $F_2$ by a suitable equivalence relation.

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  • $\begingroup$ I admit that the formulation of my question was not quite clear. What I meant to ask is: could Fano's axiom be replaced by (i.e. proven from) intuitively more basic axioms in three-dimensional space? So in particular spaces over $F_2$ would be excluded by these axioms. $\endgroup$ – Larry Sep 28 '17 at 10:27
  • $\begingroup$ Yes, the axiom is that the characteristic of the field has to be different from 2. $\endgroup$ – Mikhail Katz Sep 28 '17 at 11:59
  • $\begingroup$ I was hoping for somewhat more synthetic axioms – similar to Fano's Axiom, but more elementary. $\endgroup$ – Larry Sep 28 '17 at 15:42
  • $\begingroup$ The existence of a betweenness relation (a la Hilbert) is probably enough to guarantee this (though this needs to be checked) but that would be a much stronger condition. $\endgroup$ – Mikhail Katz Sep 28 '17 at 15:53

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