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:
 * $\ds \forall a, b, c, p_1, \cdots, p_n: \paren {\bigwedge_{1 \mathop \le i \mathop < j \mathop \le n} \map \neg {p_i = p_j} \land \bigwedge_{i \mathop = 2}^n a p_1 \equiv a p_i \land \bigwedge_{i \mathop = 2}^n b p_1 \equiv b p_i \land \bigwedge_{i \mathop = 2}^n c p_1 \equiv cp_i}$


 * $\implies \paren {\mathsf B abc \lor \mathsf B bca \lor \mathsf B cab}$

where:
 * $a, b, c, p_i$ are points
 * $\ds \bigwedge$ denotes the general conjunction operator.

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.

Also see

 * Lower Dimensional Axiom