Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 116 / 1 / a

A strongly inaccessible cardinal is an uncountable regular cardinal $\kappa$ that is also a strong limit cardinal:

A theory is $\kappa$-satisfiable when every subtheory of size below $\kappa$ has a model. An uncountable cardinal $\kappa$ is weakly compact when every $\kappa$-satisfiable theory in an infinitary language $L_{\kappa ,\kappa }$ with at most $\kappa$ nonlogical symbols is satisfiable.

Equivalently, $\kappa \to (\kappa {)}_2^2$: every coloring $c:[\kappa {]}^2\to 2$ has a homogeneous subset of cardinality $\kappa$.

A filter $U$ is $\kappa$-complete when intersections of fewer than $\kappa$ members remain in $U$. An uncountable cardinal $\kappa$ is a measurable cardinal when it carries a $\kappa$-complete nonprincipal ultrafilter.

An uncountable cardinal $\kappa$ is a strongly compact cardinal when every $\kappa$-satisfiable theory in any $L_{\kappa ,\kappa }$ language is satisfiable. Unlike weak compactness, there is no cardinality bound on the language.