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 $8$