Consider an absolute plane (i.e. it satisfies all axioms except the parallel axiom). Let g be a straight line and P a point not in g. Then there is a unique straight line going through P which is perpendicular on g. I asked a question regarding the existence in the same problem here. Now I […]

In Hilbert’s axioms for geometry, the following elements are presented as undefined (meaning “to be defined in a specific model”): point, line, incidence, betweenness, congruence. In deed, when constructing the model for the euclidean plane $\mathbb{E}^2$, for example, one may define a point as an element of $\mathbb{R}^2$, a line as a set $\{(x, y) […]

I have been thinking about the axiomatization of geometry, and I don’t know one thing. Imagine you are defining triangles: Definition [Triangle]: A triangle is a polygon with 3 sides. In this case, is it required to define $3$ (and, by extension, $\mathbb{N}$)? Edit: Maybe I should explain myself more clearly. What I am doing […]

Consider an absolute plane (i.e. it satisfies all axioms except the parallel axiom). Let $g$ be a straight line and $P$ a point not in $g$. Then there is a unique straight line going through $P$ which is perpendicular on $g$. Existence: I try to construct the line perpendicular to $g$ that goes through $P$. […]

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