Sperner spaces and first‐order logic

We study the class of Sperner spaces, a generalized version of affine spaces, as defined in the language of pointline incidence and line parallelity. We show that, although the class of Sperner spaces is a pseudo‐elementary class, it is not elementary nor even ??_∞ω‐axiomatizable. We also axiomatize the first‐order theory of this class.