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. The first of them is tetration 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, Ackermann's function, busy Beaver function and Chaitin's incompleteness theorem.
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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 ‎$‎‎‎\forall ‎‎\kappa‎\geq\aleph_0\;\;\;\aleph_0^{‎\kappa‎}>‎\kappa‎$.