AbstractM. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2]satisfy the anti‐foundation axiomX1introduced in [3]. So it is interesting to study the axiomAFA, which is equivalent toX1in ZF, introduced by P. Aczel in [1]. We show in this paper thatAFAis inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It includes all general hyperuniverses as defined in [5]. In order to achieve our aim, we need to define ordinals in GPK and to study some of their properties.Mathematics Subject Classification:03E70, 03E1