Axiom:Axiom of Continuity

Axiom Schema
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.

Let $\alpha$ contain no free occurances of $a,b,y$.

Let $\beta$ contain no free occurances of $a,b,x$.

This axiom asserts:


 * $\exists a: \forall x,y: \left({ \alpha \land \beta \implies \mathsf{B}axy }\right)$


 * $\implies \exists b: \forall x,y: \left({ \alpha \land \beta \implies \mathsf{B}xby }\right)$

Alternate 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 this alternate form of the Axiom of Continuity assumes the existence of sets. As such, it cannot be expressed in the framework of first-order logic.

Intuition
Consider an infinite line.

Consider it as consisting of a left side and a right side, where the transition is any place you care to choose.

No matter what place you choose, that place of transition is a point.

Also see

 * Dedekind's Theorem