The Kunen Inconsistency is an important milestone in the study of axiomatic set theory, placing a hard limit on how close the target model of a non-trivial elementary embedding can be to the full universe. In particular, it shows that the existence of a Reinhardt embedding, that is a non-trivial embedding of the full universe into itself, is inconsistent. It is well-known that all proofs we currently have rely extensively on the fact that we are working with the full power of ZFC, most notably the essential use of choice.

In this talk we shall discuss the notion of a Reinhardt embedding over several weakened base theories, primarily ZFC without Power Set, Zermelo and Power Kripke Platek. We shall see how to obtain some upper bounds, lower bounds and equiconsistency results in terms of the usual ZFC large cardinal hierarchy as well as many unexpected characteristics such embeddings can have. Moreover, we shall see that, under reasonable additional assumptions, it is possible to reobtain Kunen-type inconsistency results in both ZFC without Power Set and Power Kripke Platek plus Well-Ordering.