Kunen’s inconsistency theorem is a theorem in set theory proven by Kenneth Kunen, which says that a Reinhardt cardinal is inconsistent with having both the axiom of choice and the axiom of replacement together in the set theory. As a result, any other large cardinal axiom implying Reinhardt cardinals are also affected by Kunen’s inconsistency theorem, which include Berkeley cardinals, club Berkeley cardinals, and limit club Berkeley cardinals.
However, Kunen’s proof relies on both the axiom of choice and the axiom of replacement. Thus, if one wants these large cardinal axioms in their set theory, the solution is usually to either remove the axiom of choice and move to ZF, or to remove the axiom of replacement and move to ZC.
Kenneth Kunen: Elementary embeddings and infinitary combinatorics, Journal of Symbolic Logic. 1971;36(3):407-413. [doi:10.2307/2269948]
Wikipedia, Kunen’s inconsistency theorem
Rohan Srivastava: The Landscape of Large Cardinals [arXiv:2205.01787]
Last revised on September 22, 2026 at 02:52:17. See the history of this page for a list of all contributions to it.