Why does absolute geometry with the elementary continuity axiom have only two models?
Bulletin Polish Acad. Sci. Math.
MSC: Primary 51F05; Secondary 03B30, 51M05, 51M10
DOI: 10.4064/ba260125-31-5
Published online: 15 June 2026
Abstract
Wanda Szmielew showed in 1959 that $\overline{\mathcal A}$, plane absolute geometry with the elementary continuity axiom schema, has precisely two models: Euclidean planes and hyperbolic planes over real-closed fields. In this note we determine the reason why this surprising result holds. We find that two axioms are responsible for it, the circle axiom and Aristotle’s axiom; both hold in $\overline{\mathcal A}$.
