Geodesic
Determinateness and the Pasch Axiom
Canadian mathematical bulletin, Tome 16 (1973) no. 2, pp. 159-160

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Let E be the (nonelementary) plane Euclidean geometry without the Pasch axiom. (The Pasch axiom says that a line cutting one side of a triangle must also cut another side. A full list of axoms for E is given in [5].) E satisfies in particular the full second-order continuity axiom.Szczerba [5] has recently shown using a Hamel basis for the reals over the rationals that there exists a model of E not satisfying the Pasch axiom. It is natural to ask whether the axiom of choice plays an essential role in the proof. It will turn out that it does. 
Adler, Andrew. Determinateness and the Pasch Axiom. Canadian mathematical bulletin, Tome 16 (1973) no. 2, pp. 159-160. doi: 10.4153/CMB-1973-027-1
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