Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 23 / 1 / e / Solution

First use the fact that a small set is absent from a complete nonprincipal ultrafilter. Indeed, if $A\subseteq \kappa$ and $|A|<\kappa$, then every $\kappa \setminus \{\alpha \}$, $\alpha \in A$, belongs to $U$ by nonprincipality. Kappa-completeness gives $\kappa \setminus A\in U$, so $A\notin U$. In particular every final segment $\{i:i>\alpha \}$ belongs to $U$.

Suppose for contradiction that the ultrapower has at most $\kappa$ elements. List representatives $f_{\alpha }:\kappa \to M$ for all its classes, indexed by $\alpha <\kappa$; repetitions are allowed. At coordinate $i<\kappa$, fewer than $\kappa$ values occur among $f_{\alpha }(i)$ with $\alpha <i$. Since $|M|\ge \kappa$, chooseThis diagonal argument for ultrapower cardinality uses the axiom of choice. For each fixed $\alpha$, the functions $g$ and $f_{\alpha }$ disagree throughout the final segment $i>\alpha$, which belongs to $U$. Their equality set therefore cannot belong to $U$, and $[g]\ne [f_{\alpha }]$.

This contradicts the assumed enumeration of the ultrapower. HenceThe proof uses only that each ordinal $i<\kappa$ has cardinality below $\kappa$; it does not require a separate assumption of regularity.