Axiom:Lower 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 $\mathsf B$ 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 $8$