Kunen's inconsistency theorem
Encyclopedia
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.
Some consequences of Kunen's theorem are:
It is not known if Kunen's theorem still holds in ZF (ZFC without the axiom of choice), though showed that there is no definable elementary embedding from V into V.
Notice that Kunen used Morse–Kelley set theory
in his proof, if the proof is re-written to use ZFC, then one must add the assumption that j is a definable class function, i.e. that there is a formula J in the language of set theory such that for some parameter p∈V for all sets x∈V and y∈V:
Otherwise one could not even show that j "λ exists as a set. The forbidden set j "λ is crucial to the proof. The proof first shows that it cannot be in M. The other parts of the theorem are derived from that.
Some consequences of Kunen's theorem are:
- There is no non-trivial elementary embedding of the universe V into itself. In other words, there is no Reinhardt cardinalReinhardt cardinalIn set theory, a branch of mathematics, a Reinhardt cardinal is a large cardinal κ, suggested by , that is the critical point of a non-trivial elementary embedding j of V into itself....
. - 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 pointCritical point (set theory)In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself....
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 showed that there is no definable elementary embedding from V into V.
Notice that Kunen used Morse–Kelley set theory
Morse–Kelley set theory
In the foundation of mathematics, Morse–Kelley set theory or Kelley–Morse set theory is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory...
in his proof, if the proof is re-written to use ZFC, then one must add the assumption that j is a definable class function, i.e. that there is a formula J in the language of set theory such that for some parameter p∈V for all sets x∈V and y∈V:
Otherwise one could not even show that j "λ exists as a set. The forbidden set j "λ is crucial to the proof. The proof first shows that it cannot be in M. The other parts of the theorem are derived from that.