ⓘ Kunens inconsistency theorem. In set theory, a branch of mathematics, Kunens inconsistency theorem, proved by Kenneth Kunen, shows that several plausible large ..

                                     

ⓘ Kunens inconsistency theorem

In set theory, a branch of mathematics, Kunens inconsistency theorem, proved by Kenneth Kunen, shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.

Some consequences of Kunens theorem or its proof are:

  • 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 non-trivial elementary embedding of the universe V into itself. In other words, there is no Reinhardt cardinal.
  • There is no ω-huge cardinal.
  • There is no non-trivial elementary embedding of V λ+2 into itself.

It is not known if Kunens 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: j x = y ↔ J x, y, p. {\displaystyle jx=y\leftrightarrow Jx,y,p\.}

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 replacement holds for formulas involving j. 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.

It is possible to have models of set theory that have elementary embeddings into themselves, at least if one assumes some mild large cardinal axioms. For example, if 0# exists then there is an elementary embedding from the constructible universe L into itself. This does not contradict Kunens theorem because if 0# exists then L cannot be the whole universe of sets.

                                     
  • elementary embeddings are not classes of the model. Kunen 1971 proved Kunen s inconsistency theorem showing that the existence of such an embedding contradicts
  • Kunen s inconsistency theorem stating roughly that no such embedding exists. More specifically, as Samuel Gomes da Silva states, the inconsistency
  • jn κ for positive integers n. However Kunen s inconsistency theorem shows that such cardinals are inconsistent in ZFC, though it is still open whether
  • suspected to be inconsistent in ZFC as it was thought possible that Kunen s inconsistency theorem that Reinhardt cardinals are inconsistent with the axiom
  • attempted solutions by amateurs of famous problems such as Fermat s last theorem or the squaring of the circle. It also does not include unpublished preprints
  • proved within the theory itself, as shown by Godel s second incompleteness theorem Formally, ZFC is a one - sorted theory in first - order logic. The signature
  • finitely axiomatizable, while ZFC and MK are not. A key theorem of NBG is the class existence theorem which states that for every formula whose quantifiers
  • what is meant by resemble Weak forms of the reflection principle are theorems of ZF set theory due to Montague 1961 while stronger forms can be new
  • that it has an integer root if and only if ZFC is inconsistent A stronger version of Fubini s theorem for positive functions, where the function is no
  • methods formalizable within arithmetic itself. Godel s incompleteness theorems show that Hilbert s program cannot be realized: if a consistent recursively
  • may be any uncountable cardinal of uncountable cofinality see Easton s theorem The continuum hypothesis postulates that the cardinality of the continuum