Axiom:Upper Dimensional Axiom

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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:

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

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

Sources

• June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $181, 182$ : Axiom $9$
  Illustration courtesy of Steven Givant.