Elementary embedding ($j:M\to N$)

An elementary embedding preserves every first-order formula with parameters from its domain. Its critical point $\mathrm{crit}(j)$ is the least ordinal moved by a nonidentity embedding.

Preservation of all first-order formulas with parameters by an embedding. For an inclusion of structures this is the condition for an elementary substructure.

In ZFC there is no nonidentity elementary embedding $j:V\to V$, understood as an ordinary class embedding whose set restrictions and critical sequence are available. Let $\lambda$ be the supremum of the Kunen critical sequence, so $j(\lambda )=\lambda$ and $j({\lambda }^{+})={\lambda }^{+}$. Partition the ordinals of cofinality $\omega$ below ${\lambda }^{+}$ into stationary sets indexed by the critical point. After applying $j$, the extra stationary cell at that critical point meets the club of ordinals closed under $j$. Such an ordinal of countable cofinality is fixed by $j$ and hence belongs both to that extra cell and an old-index cell, a contradiction.

If $\kappa =\mathrm{crit}(j)$, then $j$ fixes every member of $V_{\kappa }$ and every ordinal below $\kappa$, while $j(\kappa )>\kappa$.

An ultrafilter $U$ on $I$ gives an elementary map into the well-founded collapse of an ultrapower by sending $x$ to the class of the constant function with value $x$. For a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, its critical point is $\kappa$.

For inaccessible $\lambda$, a map $j:V_{\lambda }\to M$ with critical point $\kappa$ is $\beta$-strong when $M$ is transitive and $V_{\kappa +\beta }\subseteq M$.

A cardinal $\kappa <\lambda$ is 1-strong when it is the critical point of an elementary embedding $j:V_{\lambda }\to M$ into a transitive model satisfying $V_{\kappa +1}\subseteq M$.

A formula $\Phi (x,\kappa )$ defines a $\beta$-stable property of $\kappa$ when it is absolute between the universe and every transitive $M\supseteq V_{\kappa +\beta }$.

If a $\beta$-strong embedding has critical point $\kappa$ and the $\beta$-stable property $\Phi (\kappa )$ holds, then $\{\mu <\kappa :\Phi (\mu )\}$ is unbounded in $\kappa$. For each $\gamma <\kappa$, the target sees $\kappa$ as a witness between $\gamma$ and $j(\kappa )$; elementarity reflects a witness between $\gamma$ and $\kappa$.

A property $P$ reflects below a cardinal $\kappa$ when $\{\mu <\kappa :P(\mu )\}$ is unbounded in $\kappa$. For an elementary embedding with critical point $\kappa$, one often proves this by using $\kappa$ as a witness below $j(\kappa )$ in the target and then invoking elementarity.

The critical sequence of $j$ starts with its critical point $\kappa$ and iterates ${\kappa }_{n+1}=j({\kappa }_n)$. Its supremum $\hat{\kappa }={\sup }_{n<\omega }{j}^n(\kappa )$ is the least fixed point of $j$ above $\kappa$ whenever the relevant iterates lie in the domain.

For an elementary embedding with critical-sequence supremum $\hat{\kappa }$, the setdoes not belong to the transitive target model. The proof uses an omega-Jonsson function on $\hat{\kappa }$.