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.

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 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 $\kappa$.

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 $\hat{\kappa }$.