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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 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?

The next question is taken from Noah's answer, where an answer to it may help in defining the tetration for higher infinite.

Question 3. What is $m\uparrow n$ counting?

See also What combinatorial quantity the tetration of two natural numbers represents?. But note that the answers given in the above question are so that they are not suitable for treating infinite cardinals.

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.
‎

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?

The next question is taken from Noah's answer, where an answer to it may help in defining the tetration for higher infinite.

Question 3. What is $m\uparrow n$ counting?

See also What combinatorial quantity the tetration of two natural numbers represents?. But note that the answers given in the above question are so that they are not suitable for treating infinite cardinals.

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.
‎

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?

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.
‎

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?

The next question is taken from Noah's answer, where an answer to it may help in defining the tetration for higher infinite.

Question 3. What is $m\uparrow n$ counting?

See also What combinatorial quantity the tetration of two natural numbers represents?. But note that the answers given in the above question are so that they are not suitable for treating infinite cardinals.