 Ultrapower embedding

For an ultrafilter $U$ on $I$ and a first-order structure $M$, the diagonal mapsends an element to the equivalence class of its constant function. The Łoś theorem makes it an elementary embedding. The target is an ultrapower; it need not be well-founded. A Mostowski collapse is available only after well-foundedness and extensionality have been established.

For an uncountable cardinal $\kappa$, a $\kappa$-complete nonprincipal ultrafilter on $\kappa$ gives a well-founded set-theoretic ultrapower of the universe, whose transitive collapse has critical point of an elementary embedding $\kappa$. Completeness makes every function from $\kappa$ to an ordinal below $\kappa$ constant on an ultrafilter-large set. Nonprincipality and completeness make the identity function's class lie above every constant ordinal below $\kappa$, forcing $j(\kappa )>\kappa$.

The uncountability qualification is essential. A nonprincipal ultrafilter on $\omega$ is only finitely complete and gives an externally ill-founded ultrapower of the natural numbers; it cannot produce a transitive elementary embedding with critical point $\omega$.