Exponentiation of infinite cardinals: rules involving $\max\{\kappa_1, \kappa_2\}$?

Question

Th question is "simple". Let $\kappa_1, \kappa_2$ be two infinite cardinals. We have the simple rules for addition and multiplication: $$ \kappa_1+ \kappa_2=\kappa_1 \kappa_2=\max\{\kappa_1, \kappa_2\}. $$ Now do we have a similar rule for $$ \kappa_1^{\kappa_2}\space? $$ Could we share some reference for this?

Answer 1

No. We don't even have a rule for $2^\kappa.$ This is related to the independence of the continuum hypothesis: the value of $2^\kappa$ for regular $\kappa$ is (almost) completely independent of ZFC. For example, $2^{\aleph_0}$ (the cardinality of the reals) can consistently be any cardinal with uncountable cofinality.

See Jech Set Theory chapter 5 for information about what we do know, for instance, about how to reduce certain cardinal exponentiations to others. (e.g. look at theorem 5.20.) Under GCH, where we assume $2^\kappa=\kappa^+$ for all $\kappa,$ this leads to a general formula for $\kappa^\lambda$ (theorem 5.15). Under the SCH, $\kappa^\lambda$ is fully determined by the continuum function $\kappa\mapsto 2^\kappa$ on regular cardinals (theorem 5.22), but the continuum function on the regular cardinals can still consistently be almost anything. In ZFC alone, there are some nontrivial results on $\kappa^{\operatorname{cf}\kappa}$ and $2^\kappa$ for singular cardinals, though these take the form of bounds under assumptions on what the continuum function looks like for regular $\kappa,$ so this isn't anything approaching a "formula". These results are much harder, though, see for instance Jech chapter 24 or the book by Holz, Steffens, and Weltz.