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NJ Wildberger says in this video that there's a logical flaw in Euclid's construction of the triangle, that you're not really able to know (apart from the picture) if the circles intersect. He also provided no further comments about that. So, how's that?

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up vote 4 down vote accepted

As the speaker says: How do we know that the circles intersect? The axiom system in Euklid's Elements is in fact not strong enough to show that these circles intersect. Essentially, there is missing a notion of "between" and completeness, which has been corrected later by Hilbert. To see the problem immediately, note that you can do Euklidean geometry in $\mathbb Q^2$.

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Could you explain which notion of "between" in the situation of Euclid would help guarantee that the circles do intersect, and which additional axiom is involved? – Marc van Leeuwen Apr 20 '13 at 15:10
The mention to Hilbert's axioms was quite helpful, I've searched for it, here it is. From here I just need to research a little. – Voyska Apr 20 '13 at 15:38
@Hagen I can't see the problem with the euclidean geometry in $\mathbb{Q}^2$ – Voyska Apr 22 '13 at 13:19
@GustavoBandeira The circle around $(0,0)$ through $(1,0)$ and the circle around $(1,0)$ through $(0,0)$ "should" intersect in $(\frac12,\frac{\sqrt3}{2})$, but that opint is not there in $\mathbb Q^2$. – Hagen von Eitzen Apr 22 '13 at 13:53
@MarcvanLeeuwen I added "completeness" as that is at least as important, I guess. – Hagen von Eitzen Apr 22 '13 at 13:55

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