= Kunen lemma
{c}
Let $j:V\to M$ be a nonidentity <elementary embedding> into a transitive class model of <ZFC>, with its set restrictions and <Kunen critical sequence> available. Let $\lambda=\sup_{n<\omega}j^n(\kappa)$ for $\kappa=\operatorname{crit}(j)$. Then $j(\lambda)=\lambda$ and
$$
j\mathbin{``}\lambda\notin M.
$$
These domain and closure assumptions are essential; the assertion is not a statement about an arbitrary elementary map between arbitrary structures.<580 - Cardinal arithmetic (10) by Andrés E. Caicedo>[Theorem14, image of the first fixed point above the critical point]{ref}
For the combinatorial argument, choose an <omega-Jónsson function> $f:[\lambda]^\omega\to\lambda$: every subset of $\lambda$ of cardinality $\lambda$ has all $\lambda$ possible values on its countable subsets. If $X=j\mathbin{``}\lambda$ belonged to $M$, its order type would be $\lambda$, so $M$ would see it as a set of cardinality $\lambda$. Elementarity would make $j(f)$ take every value below $\lambda$ on its internally countable subsets. But every such subset $Y\subseteq X$ is $j(Z)$ for a countable $Z\subseteq\lambda$, since $j$ fixes countable enumerations coordinatewise. Hence
$$
j(f)(Y)=j(f)(j(Z))=j(f(Z))\in j\mathbin{``}\lambda.
$$
The value $\kappa$ is omitted by that image, a contradiction. The existence of omega-Jónsson functions is the combinatorial input; the rest tracks precisely what the target model could contain.
Back to article page
