# Axiom:Lower Dimensional Axiom

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## Axiom

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

Let $\mathsf B$ denote the relation of betweenness.

Let $\equiv$ be the relation of equidistance.

Let $=$ be the relation of equality.

## $1$ Dimension

The **lower $1$-dimensional axiom** is the assertion:

- $\exists a, b: \map \neg {a = b}$

### Intuition

There are two points, hence the space is at least $1$-dimensional.

## $2$ Dimensions

The **lower $2$-dimensional axiom** is the assertion:

- $\exists a, b, c: \neg \mathsf B abc \land \neg \mathsf B bca \land \neg \mathsf B cab$

### Intuition

There are three points that are *not* collinear.

It follows that the space is at least $2$-dimensional.

## $n$ Dimensions

Let $n \in \N, n \ge 3$.

The **lower $n$-dimensional axiom** is the assertion:

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

- $\land \paren {\neg \mathsf B abc \land \neg \mathsf B bca \land \neg \mathsf B cab}$

where:

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

### Intuition

There exist $n - 1$ (pairwise) distinct points.

There are also three points $a, b, c$.

It is possible to set up these points such that all of $a, b, c$ are equidistant from the $n - 1$ points and yet $a, b, c$ are *not* collinear.

In other words, the set of all points equidistant from of $n - 1$ distinct points is *not* a line.

These axioms effectively give a lower 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. $180, 181$ : Axiom $8$