Generalizing a concept in mathematics is always a problematic situation. In most cases there are several ways to generalize a notion and it is not easy to decide if a particular generalization is more natural, useful or better than the other alternatives.
The same situation happens in set theory when we are trying to generalize the arithmetic operators to infinities. The case of ordinal arithmetic is less controversial because it has a straightforward recursive definition applicable for all usual and hyperoperators. But in the case of cardinal arithmetic we have defined all addition, multiplication and exponentiation cases separately as follows:
$\kappa+\lambda$ is the size of disjoint union of two sets with $\kappa$ and $\lambda$ elements.
$\kappa.\lambda$ is the size of Cartesian product of two sets with $\kappa$ and $\lambda$ elements.
$\kappa^\lambda$ is the size of set of all functions from a set with $\lambda$ elements to a set with $\kappa$ elements.
In each case we isolated a combinatorial property and defined our arithmetic operator as the size of a set definable from parameters $\kappa$ and $\lambda$. As it is clear this method of generalizing operators could be done in many different ways and some of them could be non-equivalent. For example in finite combinatorics $m+n$ could be size of many different definable sets rather than disjoint union and these definable notions could be different on infinite cardinals.
Note that according to the usual definition of our cardinal arithmetic which remained unchanged since the beginning of set theory, the first two operators (addition and multiplication) became counter-intuitively equal and as trivial as maximum operator. Also they are completely determined within ZFC. On the other hand the third operator (exponentiation) became highly non-trivial and completely undetermined even in the simplest case. i.e. ZFC cannot decide about the value of $\aleph_0^{\aleph_0}$.
Why such a large gap exists between cardinal exponentiation and cardinal addition and multiplication? Why addition and multiplication are counter-intuitively equal? Are there better arithmetics on cardinal numbers which have a more natural behavior, a rich theory and a deep connection with each other?
Let us explore the above questions more precisely by considering all possible definitions for arithmetic operators over cardinals.
Definition 1: Let $*:\omega\times\omega\rightarrow\omega$ be an arithmetic operator (i.e. ordinary addition, multiplication, exponentiation, tetration, … over natural numbers), the first order formula $\phi (x,y,z)$ in the language of set theory is called a $*$ – notion over cardinals (e.g. addition notion over cardinals, multiplication notion over cardinals, etc.) if and only if
$(a)~~ZFC\vdash \forall \kappa,\lambda\in Card~~~~~\{x~|~\phi (x,\kappa,\lambda)\}$ is a set.
$(b)~~ZFC\vdash \forall m,n\in\omega~~~~~| \{x~|~\phi (x,m,n)\} |=m*n$
Associated with any $*$ – notion $\phi (x,y,z)$ over cardinals one can define an operator $*_{\phi}:Card\times Card\rightarrow Card$ as follows:
$\forall\kappa,\lambda\in Card~~~~~\kappa *_{\phi}\lambda :=|\{x~|~\phi (x,\kappa,\lambda)\}|$
Note that by property $(a)$, the operator $*_{\phi}$ is well-defined. Also by property $(b)$ it is a generalization of arithmetic operator $*$ to infinite cardinals.
Example: If $+:\omega\times\omega\rightarrow\omega$ is the ordinary addition of natural numbers then the first order formula $\phi (x,y,z):\exists s\in y~\exists t\in z~~~x=\langle s,0 \rangle \vee x=\langle t,1\rangle$ asserting that “$x$ is a member of disjoint union $y$ and $z$” is a $+$ – notion over cardinals and the corresponding operator $+_{\phi}:Card\times Card\rightarrow Card$ is the usual addition of cardinals.
Definition 2: If $*:\omega\times\omega\rightarrow\omega$ is an arithmetic operator and $\phi (x,y,z), \psi (x,y,z)$ are two $*$ – notions over cardinals, we call $\phi, \psi$ equivalent, $\phi \sim \psi$, if and only if their corresponding cardinal arithmetics are same. i.e.
$$ZFC\vdash \forall\kappa,\lambda\in Card~~~~~\kappa *_{\phi}\lambda = \kappa *_{\psi}\lambda$$
Question 1: Is there any addition notion for cardinals non-equivalent to the usual addition notion of cardinals? If yes, how many of such addition notions are there up to $\sim$ equivalence relation? In the other words, is there any formula $\phi (x,y,z)$ such that:
$ZFC\vdash \forall \kappa,\lambda\in Card~~~~~\{x~|~\phi (x,\kappa,\lambda)\}$ is a set
$ZFC\vdash \forall m,n\in\omega~~~~~| \{x~|~\phi (x,m,n)\} |=m+n$
$ZFC\nvdash \forall\kappa,\lambda\in Card~~~~~\kappa *_{\phi}\lambda = \kappa +\lambda$
What about multiplication and exponentiation? In particular, is there an exponentiation notion over cardinals that its value in the simplest case, $\aleph_0$, remains undetermined even if we have a full knowledge of values of the usual cardinal exponentiation by assuming GCH? Precisely, is there any formula $\phi (x,y,z)$ in the language of set theory such that:
$ZFC\vdash \forall \kappa,\lambda\in Card~~~~~\{x~|~\phi (x,\kappa,\lambda)\}$ is a set
$ZFC\vdash \forall m,n\in\omega~~~~~| \{x~|~\phi (x,m,n)\} |=m^n$
$ZFC+GCH\nvdash \aleph_0 *_{\phi} \aleph_0=\aleph_0^{\aleph_0}$ (equivalently $ZFC+GCH\nvdash | \{x~|~\phi (x,\aleph_0,\aleph_0)\} |=\aleph_1$)
Besides nice behavior of each particular arithmetic operator over cardinals, it is important to consider the relative situation of these operators with respect to each other. In fact we should search for a sequence of generalizations for addition, multiplication, exponentiation, … on cardinals that satisfies some nice properties.
Definition 3: Let $\{*^i\}_{i\in \omega}$ be the natural enumeration of all arithmetic operators on natural numbers (i.e. $*^0$, $*^1$, $*^2$, $*^3$ are addition, multiplication, exponentiation and tetration respectively.) and $\{\phi_{i}(x,y,z)\}_{i\in\omega}$ is a sequence of formulas such that $\forall i\in\omega~~~\phi_{i}(x,y,z)$ is a $*^i$ – notion. Now consider the sequence $\{*^{i}_{\phi_{i}}\}_{i\in\omega}$ of arithmetic notions over cardinals.
$$\forall i\in\omega~\forall\kappa,\lambda\in Card\setminus\{0,1\}~~~~~\kappa *^{i}_{\phi_{i}}\lambda >\kappa, \lambda$$
e.g. Each sequence of cardinal arithmetics which contains the usual addition (or multiplication) of infinite cardinals is not natural because for all infinite cardinals $\kappa, \lambda$ we have $\kappa +\lambda = \kappa$ or $\kappa +\lambda = \kappa$.
Question 2: How many natural sequences of cardinal arithmetics are there up to equivalence relation defined in definition 3?
Question 3: Which other properties can we add to “being natural” to get a unique sequence of cardinal arithmetics up to equivalence? Could such a unique sequence with nice properties, be our standard cardinal arithmetic?
As an answer to the question 1, a simple example for two non-equivalent exponentiation notions is the following:
$\phi(x,y,z):$ $x$ is a function from $z$ to $y$.
$\psi (x,y,z):$ $x$ is a function from $z$ to $y$ and $x$ is a finite set.
Then we have:
$ZFC\vdash \forall \kappa,\lambda\in Card~~~~~\{x~|~\phi (x,\kappa,\lambda)\}$ and $\{x~|~\psi (x,\kappa,\lambda)\}$ are sets.
$ZFC\vdash \forall m,n\in\omega~~~~~| \{x~|~\phi (x,m,n)\} |=| \{x~|~\psi (x,m,n)\} |=m^n$
So both $\phi, \psi$ are exponentiation notions over cardinals. In other words both definitions for exponentiation of two cardinals as “number of functions from one set to another” and “number of finite functions from one set to another” are valid generalizations for the notion of exponentiation of natural numbers to infinite cardinals.
But these two exponentiation notions are “not” equivalent over infinite cardinals because we have:
$ZFC\vdash \aleph_0 *_{\phi} \aleph_0=| \{x~|~\phi (x,\aleph_0,\aleph_0)\} |=2^{\aleph_0}$
$ZFC\vdash \aleph_0 *_{\psi} \aleph_0=| \{x~|~\psi (x,\aleph_0,\aleph_0)\} |=\aleph_0$
Thus $ZFC\vdash \aleph_0 *_{\phi} \aleph_0 \neq \aleph_0 *_{\psi} \aleph_0$
and so $ZFC\nvdash \forall \kappa, \lambda\in Card~~~~~\kappa *_{\phi} \lambda =\kappa *_{\psi} \lambda$
that means $\phi\nsim\psi$
Consider Hamkins’ and Yang’s paper “Satisfaction is not Absolute”.
There they show the following result:
$M_1$,$M_2$$\vDash$ZFC
$\mathbb N^{M_1}$=$\mathbb N^{M_2}$
$M_1$ believes $\mathbb N$$\vDash$$\theta$
$M_2$ believes $\mathbb N$$\vDash$$\lnot$$\theta$.
Without the assumption of the “higher-order ontological commitment, going strictly beyond the commitment to a definite nature for the underlying structure itself” (whatever that “ontological commitment” might be), why should we expect the situation to be better in the case of infinite cardinals? So the answer to the question “Are there non-equivalent cardinal arithmetics?” without the assumption of that “ontological commitment” is a definite yes.