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Kunen's inconsistency theorem
Theorem in transfinite set theory
Kunen's inconsistency theorem
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen (1971), shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences of Kunen's theorem (or its proof) are: There is no non-trivial elementary embedding of the universe V into itself. In other words, there is no Reinhardt cardinal. If j is an elementary embedding of the universe V into an inner model M, and λ is the smallest fixed point of j above the critical point κ of j, then M does not contain the set j "λ (the image of j restricted to λ). There is no ω-huge cardinal. There is no non-trivial elementary embedding of Vλ+2 into itself. It is not known if Kunen's theorem still holds in ZF (ZFC without the axiom of choice), though Suzuki (1999) showed that there is no definable elementary embedding from V into V. That is there is no formula J in the language of set theory such that for some parameter p∈V for all sets x∈V and y∈V:...
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