In number theory there are several operators like addition, multiplication and exponentiation defined from $\omega\times\omega$ to $\omega$. Each of them is defined as an iteration of the other. The sequence of building such iterated operators can go further to define faster and faster [hyperoperators][1]. The first of them is [tetration][2] which is defined as iterated exponentiation. Let $m\uparrow n$ denote the tetration of $m$ and $n$ that is $\underbrace{m^{m^{m^{.^{.^{.}}}}}}_{n - times}$. This operator appears in several interesting occasions in logic, computations and combiantorics, for example see these Wikipedia articles on [Graham's number][3], [Ackermann's function][4], [busy Beaver function][5] and [Chaitin's incompleteness theorem][6].
Now consider the infinitary case. In set theory addition, multiplication and exponentiation are defined for cardinal numbers.
>**Question 1.** What about $\kappa\uparrow\lambda$? How should we define this?
Intuitively, we expect to define $\aleph_0\uparrow\aleph_0$ to be $\aleph_0^{\aleph_0^{\aleph_0^{.^{.^{.}}}}}.$
But this intuitive definition of tetration has some counter-intuitive properties, as then we expect to have $\aleph_0^{(\aleph_0\uparrow\aleph_0)}=\aleph_0\uparrow\aleph_0$ which is impossible by Cantor's theorem which says $\forall \kappa\geq\aleph_0\;\;\;\aleph_0^{\kappa}>\kappa$.
Note that for the cases of addition, multiplication and exponentiation, we have quite natural operations $f_+, f_\times$ and $f_e$ such that given cardinals $\kappa, \lambda$, we have $f_+(\kappa,\lambda)=\kappa+\lambda, f_\times(\kappa, \lambda)=\kappa\times \lambda$ and $f_e(\kappa,\lambda)=\kappa^\lambda.$
>**Question 2.** Is there a natural operation $f_t$ defined so that for all natural numbers $m,n$ we have $f_t(m,n)= m\uparrow n$, and so that its definition is so natural that it also works for infinite cardinal numbers?
[1]: http://en.wikipedia.org/wiki/Hyperoperation
[2]: http://en.wikipedia.org/wiki/Tetration
[3]: http://en.wikipedia.org/wiki/Graham%27s_number
[4]: http://en.wikipedia.org/wiki/Ackermann_function
[5]: http://en.wikipedia.org/wiki/Graham%27s_number
[6]: http://en.wikipedia.org/wiki/Kolmogorov_complexity#Chaitin.27s_incompleteness_theorem