ON THE SIMPLICITY OF AN AXIOM SYSTEM FOR PLANE EUCLIDEAN GEOMETRY

The author has provided in [2] an axiom system in Tarski's [7] first order language LBD for plane Euclidean geometry coordinatized by Euclidean ordered fields, a theory we shall denote by S2. All the axioms were statements which, when written in prenex form, contained at most 5 variables. It was stated in [2] that there is no axiom system for E2, all of whose axioms have, when written in prenex form, fewer than 5 variables. The proof given in [2] for this fact is flawed. The aim of this note is to provide a valid proof of it. Let, as in [2], T := Cn{{y> \ £ £2 fl L4, <p is written in prenex form}), where L4 stands for the language that contains the same symbols as LBD, except that there are not countably many, but only 4 individual variables. In the proof given in [2], we stated that T C Cn(T{ U S). This is false, since the circle axiom may also be expressed by a 4-variable sentence, namely (Va6c)(3d) [B(abc) da = db A ad = ac], therefore Szczerba's [6] model of independence for the Pasch axiom is not a model of T. The result is nevertheless true, i. e. T ^ S2, that is the simplicity degree of S2 is indeed 5. The idea of the proof is to show that T is a subtheory of a certain plane geometry in which the congruence relation is not transitive. The model for this geometry with a non-transitive congruence relation is the plane over the field of real numbers, with the usual affine and order structures, but with a congruence relation that is strictly included in the usual congruence relation of the Cartesian plane over the reals. Let C2(K) : = x =1) be the Cartesian plane over the real numbers, i. e. B and D will have the usual interpretation of "Betweenness"