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.** 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 $\forall \kappa\geq\aleph_0\;\;\;\aleph_0^{\kappa}>\kappa$.
[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