A <strongly inaccessible cardinal> is an uncountable <regular cardinal> $\kappa$ that is also a :
$$
\lambda<\kappa\quad\Longrightarrow\quad2^\lambda<\kappa.
$$

Solution

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

For a <regular cardinal> $\kappa$,
$$
\operatorname{Fn}_\kappa(I,J)=\{p:p:I\rightharpoonup J,\ |\operatorname{dom}p|<\kappa\}.
$$
Again $q\le p$ means $q\supseteq p$, and the maximal element is the empty function. The regularity of $\kappa$ ensures that the union of a descending sequence of fewer than $\kappa$ conditions still has domain of cardinality below $\kappa$ whenever the conditions form a compatible increasing chain of partial functions.

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 128 / 2 / a / ii / Solution

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An infinite cardinal $\kappa$ is regular when its cofinality is $\kappa$.


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