Axiom:Axiom of Continuity

First form
This is more properly an collection of axioms rather than a single axiom.

Let $a,b,x,y,$ be points.

Let $\mathsf{B}$ be the relation of betweenness.

Let $\alpha, \beta$ be first-order formulas.

Second form
Let $a,b,x,y,$ be points.

Let $\mathsf{B}$ be the relation of betweenness.

Let $X,Y$ be point sets.

This axiom asserts:


 * $\left[{\exists a : \forall x,y : \left({x \in X \land y \in Y}\right)\implies \mathsf{B}axy}\right] \implies \left[{\exists b : \forall x,y : \left({x \in X \land y \in Y}\right)\implies \mathsf{B}xby}\right]$

Note that the second form of the Axiom of Continuity quantifies over sets. As such, it cannot be expressed as a first-order statement.