Sperner spaces and first-order logic

51A15, 03B30, 03C75, 03E75We study the class of Sperner spaces, a generalized version of affine spaces, as defined in the language of point-line incidence and line parallelity. We show that, although the class of Sperner spaces is a pseudo-elementaryclass, it is not elementary nor even-axiomatizable. We also axiomatize the first-order theory of this class.