# Is this axiomatization of affine plane categorical?

First I'll give some definitions.

Hilbert's plane axioms of incidence: We consider a set $$P$$ (plane) and a family $$\mathcal{L}$$ (a family of lines) with axioms:

1. For any two distinct points $$a,b$$ there exists exactly one line $$L$$ such that $$a,b\in L$$.
2. For any line $$L$$ there exist two distinct points $$a,b$$ such that $$a,b\in L$$.
3. There exist three distinct points not lying on one line.

Next let $$B_{\mathbb{R}}\subset\mathbb{R}\times\mathbb{R}\times\mathbb{R}$$ be standard (strict) betweenness relation on $$\mathbb{R}$$ i.e. $$B_{\mathbb{R}}(abc):\iff\left(a

My definition of affine plane is as follows:

We say that $$(P,\mathcal{L},B)$$ is an affine plane whenever

1. $$(P,\mathcal{L})$$ is a model of incidence axioms.
2. $$B(abc)$$ implies that $$a,b,c$$ are collinear.
3. For any line $$L$$: $$(L,B|_{L\times L\times L})$$ is isomorphic to $$(\mathbb{R},B_{\mathbb{R}})$$.
4. Pasch's axioms holds.
5. For any line $$L$$ and point $$p\notin L$$ there is exactly one line parallel to $$L$$ passing through $$p$$.

Question:

Is this definition standard or equivalent to standard definitions of affine plane in terms of lines and ternary betweenness relation? And what is more important to me: Is this axiomatization categorical?

I had an idea to prove that my affine plane is isomorphic to $$\mathbb{R}^2$$. The idea was to pick two intersecting lines $$L_1,L_2$$ and isomorphisms $$\xi_1:L_1\rightarrow\mathbb{R},\xi_2:L_2\rightarrow\mathbb{R}$$ and to map any point $$p$$ on a plane to $$(\xi_1(\pi_1(p)),\xi_2(\pi_2(p)))\in\mathbb{R}^2$$ where $$\pi_1,\pi_2$$ are parallel projections to $$L_1,L_2$$. I think I know how to prove that this mapping is a bijection, yet I don't know how to prove it preserves betweenness. Not even sure it does.

• The set of planes is not used in your axioms of incidence. – Julien Narboux Mar 7 at 17:06
• Which version of Pasch do you assume ? is it a version which implies that the dimension of the space is two ? Otherwise, I don't see in your list any axiom forcing the dimension 2. – Julien Narboux Mar 7 at 17:09
• These incidence axioms don't involve the set of planes, just one plane $P$ which is actually the whole (2-dimensional) space. And yes, it is exactly Pasch axiom which forces the geometry to be 2-dimensional. – Kulisty Mar 7 at 20:05