Axiom:Upper Dimensional Axiom

Axiom
Let $a,b,c,\ldots,x,y,z$ be points.

Let $\mathsf{B}$ be the relation of betweenness.

Let $\equiv$ be the relation of equidistance.

Let $=$ be the relation of equality.

0 Dimensions
The upper $0$-dimensional axiom is the assertion:


 * $\forall a,b: a = b$

Intuition
There is only one point, hence the space is at most 0-dimensional.

1 Dimension
The upper $1$-dimensional axiom is the assertion:


 * $\forall a,b,c: \mathsf{B}abc \lor \mathsf{B}bca \lor \mathsf{B}cab$

Intuition
Any three points are collinear.

It follows that the space is at most 1-dimensional.

Be aware that a 0-dimensional space satisfies this axiom.

$n$ Dimensions
Let $n \in \N, n \ge 2$. The upper $n$-dimensional axiom is the assertion:


 * $\forall a,b,c,p: \left({\displaystyle \bigwedge_{1 \le i < j \le n} p_i \ne p_j \land \bigwedge_{i=2}^n ap_1 \equiv ap_i \land \bigwedge_{i=2}^n bp_1 \equiv bp_i \land \bigwedge_{i=2}^n cp_1 \equiv cp_i}\right)$


 * $\implies \left({\mathsf{B}abc \lor \mathsf{B}bca \lor \mathsf{B}cab}\right)$

Intuition
Any three points equidistant from $n$ distinct points are collinear.

In other words, the set of all points equidistant from $n$ distinct points forms a line.

For $n=2$, this might look like:


 * UpperDimensionalAxiom2D.png

and for $n=3$, this might look like:


 * UpperDimensionalAxiom3D.png

As was the case with the upper $1$-dimensional axiom, if $m < n$, then an $m$-dimensional space satisfies the upper $n$-dimensional axiom.

Hence the name upper dimensional axioms, as the axioms effectively give an upper bound on the dimension of the space considered.