Euclidean superpythagorean geometry (Q1267320)

In 1972, superpythagorean fields were introduced and studied by R. Elman, T. Y. Lam, and L. Bröcker under the name ``strictly pythagorean fields''. They can be defined as fields in which every half-ordering that does not contain \(-1\) is an ordering. The author first shows that a field \(F\) is superpythagorean iff (i) \((\forall xy)x^2+y^2+1\neq 0\) and (ii) \((\forall x)\) \(x\in F^2\vee-x\in F^2\vee 1+x\in F^2\vee x(1+x)\in F^2\). He then provides an axiom system \(\sum=\{A1,\dots,A12\}\) for plane Euclidean geometry coordinatized by superpythagorean ordered fields. This axiom system requires, besides some order axioms, much less than the axioms of Hilbert's groups I, III, IV. The axioms A1-A6 provide the characterization of the Euclidean planes with bisectable right angles. The conditions (i), (ii) from above can be expressed geometrically by the two axioms A7: ``The circle which has \((1,0)\) and \((a^2+1,0)\) (with \(a\neq 0)\) as endpoints of a diameter does not intersect the \(y\)-axis'' and A8: ``The circle which has \((0,0)\) and \((a^2+ a+1, 0)\) as endpoints of a diameter intersect at least one of the lines \(x=3a\), \(x=-a\), \(x=a^2\), \(x=1\)''. The axioms: A9-A12 are order axioms: A9 and A10 corresponds to the two statements of Hilbert's axiom II.1, A11 tells us that the midpoint of a segment lies between its endpoints, and A12 is a strengthened form of the Pasch axiom. The author proves the theorem: \(M\) is a model of \(\sum\) iff \(M\) is isomorphic to a Cartesian plane over a superpythagorean ordered field.