Axiom:Density of Betweenness
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Axiom
Let $\mathsf{B}$ be the relation of betweenness.
Let $=$ be the relation of equality.
Then the density of betweenness is expressed by the statement:
- $\forall a, b: \exists c: \neg \left({a = b}\right) \implies \neg \left({c = a}\right)\land \neg \left({c = b}\right) \land \mathsf{B}acb$
where $a, b, c$ are points.
Intuition
Between any two distinct points you can draw a third point, distinct from both.
Sources
- June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $188$ : Axiom $22$