# Axiom:Upper Dimensional Axiom

This article needs to be tidied.Please fix formatting and $\LaTeX$ errors and inconsistencies. It may also need to be brought up to our standard house style.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Tidy}}` from the code. |

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## 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

## 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.