Is there a set-theoretic construction of tetration and even higher-order Ackermann functions?

Question

Cardinal addition, multiplication, and exponentiation have set-theoretic constructions, namely disjoint union, cartesian product, and the set of all functions from $S$ to $T$, respectively. Are there set-theoretic constructions of tetration, pentation, etc? Let me narrow down the question somewhat, by restricting ourselves to the case of finite sets. Given two finite sets $S$ and $T$, is there a "natural", "set-theoretic" construction of a set $U$ whose cardinality is the tetration of the cardinals of $S$ and $T$ in that order? (And similarly for pentation, etc.) Now, this is where the question becomes a bit vague and fuzzy. I put the words "natural" and "set-theoretic" in quotes because I do not have a rigorous definition of them, only a "I know it when I see it" meaning. I want to know whether there is a set-theoretic construction, if there is one. And if, as I suspect, there is no such set-theoretic construction, I want a formal and rigorous definition of a "natural set-theoretic construction", along with the proof that there is no such construction for tetration and higher-order operators.

Answer 1

I cannot fully answer to your question, but I suspect that we could solve the problem for given integer values of the base, such as $^n{2}$.

Let us consider Von Neumann's cumulative hierarchy, which gives us a set of the sets induced by the ordinal numbers set itself. So we have a sequence defined through $V_n$ as $n$ runs over the nonnegative integers, where $V_n$ indicates the set of all sets whose rank is $< n$.

Now, the cardinality of $V_n$ defines a one-to-one correspondence between $^n{2}$ and the number of elements of the set $V_n$, for any given value of $n$.