# Equidistance is Independent of Betweenness

## Theorem

Let $\mathcal G$ be a formal systematic treatment of geometry containing only:

- The language and axioms of first-order logic, and the disciplines preceding it

- The undefined terms of Tarski's Geometry (excluding equidistance)

- Some or all of Tarski's Axioms of Geometry.

In $\mathcal G$, equidistance $\equiv$ is necessarily an undefined term with respect to betweenness $\mathsf B$.

## Proof

Our assertion is that $\equiv$ cannot be defined in terms of $\mathsf B$.

Seeking a contradiction, assume that it can.

Call this assumption $\left({A}\right)$.

If $\left({A}\right)$ holds, it must hold in all systems.

Let one such system be $\left({\R^2, \mathsf B_1, \equiv_1}\right)$ where:

- $\R^2$ is the cartesian product of the set of real numbers with itself

- $\mathsf B_1$ is a ternary relation of betweenness

- $\equiv_1$ is a quaternary relation of equidistance

Let $\mathcal G$ be the discipline preceding the given discipline, where $\mathcal G$ is as defined above (excluding both $\equiv$ and $\mathsf B$).

Define $\mathsf B_1$ as follows:

Define the following coordinates in the $xy$-plane:

\(\ds a\) | \(=\) | \(\ds \tuple {x_1, x_2}\) | ||||||||||||

\(\ds b\) | \(=\) | \(\ds \tuple {y_1, y_2}\) | ||||||||||||

\(\ds c\) | \(=\) | \(\ds \tuple {z_1, z_2}\) |

where $a, b, c \in \R^2$.

Let:

\(\ds \Delta x_1\) | \(=\) | \(\ds x_3 - x_2\) | ||||||||||||

\(\ds \Delta x_2\) | \(=\) | \(\ds x_2 - x_1\) | ||||||||||||

\(\ds \Delta y_1\) | \(=\) | \(\ds y_2 - y_1\) | ||||||||||||

\(\ds \Delta y_2\) | \(=\) | \(\ds y_3 - y_2\) |

Then:

- $\mathsf{B}abc \dashv \vdash \paren {\Delta x_1 \Delta y_1 = \Delta x_2 \Delta y_2} \land$

- $\paren {0 \le \Delta x_1 \Delta y_1 \land 0 \le \Delta x_2 \Delta y_2}$

Define $\equiv_1$ as follows:

Define the following coordinates in the $xy$-plane:

- $a = \tuple {x_1, x_2}$

- $b = \tuple {y_1, y_2}$

- $c = \tuple {z_1, z_2}$

- $d = \tuple {u_1, u_2}$

where $a, b, c, d \in \R^2$

- $a b \equiv c d \dashv \vdash \paren {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 = \paren {z_1 - u_1}^2 + \paren {z_2 - u_2}^2}$

Now, define the isomorphism $\phi$ on $\left({\R^2, \mathsf B_2, \equiv_2}\right)$ as:

- $\phi: \R^2 \to \R^2$ on $\left({\R^2, \mathsf B_1, \equiv_1}\right), \left({x_1, x_2}\right) \mapsto \left({x_1, 2 x_2}\right)$

Now consider the system:

- $\left({\R^2, \mathsf B_2, \equiv_2}\right)$

where $\mathsf B_2$ and $\equiv_2$ are the relations defined as above, but on the elements in the images of $\mathsf B_1$ and $\equiv_1$, respectively.

Observe that $\mathsf B_1$ and $\mathsf B_2$ coincide, because in:

- $\left({x_1 - y_1}\right) \cdot \left({2 y_2 - 2 z_2}\right) = \left({2 x_2 - 2 y_2}\right) \cdot \left({y_1 - z_1}\right) \land$

- $\left({0 \le \left({x_1 - y_1}\right) \cdot \left({y_1 - z_1}\right)}\right) \land \left({0 \le \left({2 x_2 - 2 y_2}\right) \cdot \left({2 y_2 - 2 z_2}\right)}\right)$

we can simply factor out the $2$ and divide both sides of the equality of inequality by $2$.

But consider the elements:

- $p_1 = \left({0, 0}\right)$

- $p_2 = \left({0, 1}\right)$

- $p_3 = \left({1, 0}\right)$

Observe that $p_1 p_2 \equiv_1 p_1 p_3$:

- $\left({0 - 0}\right)^2 + \left({0 - 1}\right)^2 = \left({0 - 1}\right)^2 + \left({0 - 0}\right)^2$

But $\neg \left({p_1 p_2 \equiv_2 p_1 p_3}\right)$:

- $\left({0 - 0}\right)^2 + \left({0 - 2}\right)^2 \ne \left({0 - 1}\right)^2 + \left({0 - 0}\right)^2$

But both $\left({\R^2, \mathsf B_1, \equiv_1}\right)$ and $\left({\R^2, \mathsf B_2, \equiv_2}\right)$ are both models of $\mathcal G$.

Recall that if $\left({A}\right)$ holds, it must hold in all systems.

But it does not.

Hence $\left({A}\right)$ is false, from Proof by Contradiction.

$\blacksquare$

## Also see

- Betweenness Not Independent of Equidistance, which states that there are models where one
*can*define $\mathsf B$ in terms of $\equiv$.

## Sources

- June 1999: Alfred Tarski and Steven Givant:
*Tarski's System of Geometry*(*Bull. Symb. Log.***Vol. 5**,*no. 2*: pp. 175 – 214) : pp. $199$ – $204$