Let $\kappa>0$ be a cardinal, and let $[\kappa]^{<\kappa}$ denote the collection of subsets of $\kappa$ having cardinality strictly less than $\kappa$. Is it consistent that $$|[\kappa]^{<\kappa}| > \kappa$$ for all cardinals $\kappa>\aleph\_0$?



Is it consistent that $|[\kappa]^{<\kappa}| > \kappa$?
Yes. First, let's just agree that $|[\kappa]^{<\kappa}|=\kappa^{<\kappa}$. One direction is immediate, in the other direction note that every function in $\kappa^{<\kappa}$ is an element of $[\kappa\times\kappa]^{<\kappa}$.


If $2^\kappa=\kappa^{++}$ for all successor cardinals, and there are no inaccessible cardinals, which is consistent by Easton's theorem then we can compute:


Either $\kappa$ is $\mu^+$, in which case $\kappa^{<\kappa}=\mu^\mu=\mu^{++}=\kappa^+$, or $\kappa$ is a limit cardinal in which case it is singular and $\kappa^{<\kappa}>\kappa$ anyway by König's lemma.


