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 ℒ[sub]∞ω-axiomatizable. We also axiomatize the first-order theory of this class.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/61173/1/Sperner.pd