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A rational number of the form $\frac{a}{2^{n}}$ (with $a,n$ integers) is called dyadic. In the interpretation, restrict to those points which have dyadic coordinates and to those lines which pass through several dyadic points. The incidence axioms, the first three betweenness axioms, and the line seperation property all hold in this dyadic rational plane; show that Pasch's theorem fails. (Hint: The lines $3x+y=1$ and $y=0$ do not meet in this plane.)

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  • $\begingroup$ It might help if you wrote what the question thinks "Pasch's theorem" is, as I have a suspicion it might be referring to what Wikipedia calls Pasch's axiom rather than what it calls Pasch's theorem $\endgroup$ – Henry Feb 16 '16 at 17:34
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Outline: Consider the triangle with vertices $(0,0)$, $(1,0)$, and $(0,3)$. The line $3x+y=1$ meets one side of this triangle at $(0,1)$, but does not meet another side of this triangle.

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