Cardinal sum $\kappa \oplus \lambda$, product $\kappa \otimes \lambda$ and exponentiation $\kappa^{\lambda}$, and why they are written apart from the ordinal operations

Let $\kappa$ and $\lambda$ be cardinals (Cardinal (initial ordinal) and cardinality), and recall the notation of Disjoint union, cartesian product, function space and power set respect equinumerosity, and for ordinals $\alpha, \beta$ the sets $\alpha \sqcup \beta$ and $\alpha \times \beta$ carry explicit well-orders, so their cardinalities exist in ZF:

$\kappa \sqcup \lambda = (\{0\} \times \kappa) \cup (\{1\} \times \lambda), \qquad {}^{\lambda}\kappa = \{\, h : h \text{ is a function } \lambda \to \kappa \,\}.$

Sum and product.

$\kappa \oplus \lambda \;:=\; \lvert \kappa \sqcup \lambda \rvert, \qquad \kappa \otimes \lambda \;:=\; \lvert \kappa \times \lambda \rvert .$

Both values exist in ZF and are cardinals: claim (c) of Disjoint union, cartesian product, function space and power set respect equinumerosity, and for ordinals $\alpha, \beta$ the sets $\alpha \sqcup \beta$ and $\alpha \times \beta$ carry explicit well-orders, so their cardinalities exist in ZF well-orders each of the two sets explicitly, and A set equinumerous with some ordinal has a least such ordinal, that ordinal is a cardinal, and equinumerous sets get the same one; no choice principle is used then supplies the least equinumerous ordinal. No choice principle is used.

Exponentiation.

$\kappa^{\lambda} \;:=\; \lvert {}^{\lambda}\kappa \rvert ,$

the number of functions from a set of size $\lambda$ to a set of size $\kappa$. The right-hand side is defined exactly when ${}^{\lambda}\kappa$ is well-orderable. Assuming the Axiom of Choice (The Axiom of Choice) every set is well-orderable (The well-ordering theorem) and $\kappa^{\lambda}$ is defined for all cardinals; every statement on this page that writes $\kappa^{\lambda}$ for an infinite exponent says so in its own hypotheses.

Transport to arbitrary sets. If $A$ and $B$ are well-orderable with $\lvert A \rvert = \kappa$ and $\lvert B \rvert = \lambda$, then

$\lvert A \sqcup B \rvert = \kappa \oplus \lambda, \qquad \lvert A \times B \rvert = \kappa \otimes \lambda, \qquad \lvert {}^{B}A \rvert = \kappa^{\lambda}$

whenever the sets on the left have cardinalities at all, because $A \approx \kappa$ and $B \approx \lambda$ (Equinumerous sets, $A \approx B$ and $A \preceq B$) and the three constructions respect $\approx$ (claim (a) of Disjoint union, cartesian product, function space and power set respect equinumerosity, and for ordinals $\alpha, \beta$ the sets $\alpha \sqcup \beta$ and $\alpha \times \beta$ carry explicit well-orders, so their cardinalities exist in ZF, Injection, surjection, bijection). So the operations may be computed from any representatives.

Finite and infinite cardinals. A cardinal $\kappa$ is finite when $\kappa \in \omega$ and infinite when $\omega \subseteq \kappa$, that is $\omega \le \kappa$; by trichotomy (Trichotomy and well-ordering of the ordinals) and $\omega$ is the least limit ordinal every cardinal is exactly one of the two.

The symbols $\oplus$ and $\otimes$ are not decoration. Ordinal addition and ordinal multiplication (Ordinal addition $\alpha + \beta$, Ordinal multiplication $\alpha \cdot \beta$) are defined on the same objects — cardinals are ordinals — and give different values. With $\omega$ read as a cardinal, $\omega \oplus \omega = \omega$, whereas the ordinal sum $\omega + \omega$ is strictly larger than $\omega$; and $\omega \otimes \omega = \omega$, whereas the ordinal product $\omega \cdot \omega$ is larger still. Writing both operations with $+$ and $\cdot$ would make every equation on this page ambiguous, so the cardinal operations get their own symbols and the plain $+$ and $\cdot$ on this page always mean the ordinal ones.

Exponentiation keeps the symbol, under a hard rule. There is no comparably readable alternative to $\kappa^{\lambda}$, and Ordinal $\alpha^{\beta}$ and cardinal $\kappa^{\lambda}$ are different operations that share one notation already records that $\alpha^{\beta}$ is used for two different operations: as ordinals $2^{\omega} = \omega$, while the cardinal $2^{\omega}$ counts the functions $\omega \to \{0,1\}$ and is uncountable. The rule adopted here, and followed on this page and its companion, is:

In an exponential, the base and the exponent are always alephs, letters or expressions denoting cardinals — $\kappa$, $\lambda$, $\mu$, $\mathfrak{c}$, $\operatorname{cf}(\kappa)$, $\lvert A \rvert$ — or a natural number read as a cardinal; never $\omega$, never $\omega_1$, and never a letter denoting an ordinal, such as $\alpha, \beta, \gamma, \xi, \eta$.

So $2^{\aleph_0}$, $\kappa^{\lambda}$ and $\aleph_1^{\aleph_0}$ are cardinal exponentials, and an expression such as $\omega^{\omega}$ or $\alpha^{\beta}$ is never written here at all. Where a value has to be named in both readings, the two are given different letters.

What is being counted, in each case. $\kappa \oplus \lambda$ is the size of two disjoint blocks laid side by side; the tagging in $\sqcup$ is what makes "disjoint" true even though one of $\kappa$ and $\lambda$ is always a subset of the other, so their intersection is the smaller of the two. $\kappa \otimes \lambda$ is the size of a rectangle. $\kappa^{\lambda}$ is the number of ways to choose a value in $\kappa$ for each of $\lambda$ positions, independently. None of the three is sensitive to the order in which the elements are arranged, which is exactly what distinguishes them from the ordinal operations (Ordinal exponentiation $\alpha^{\beta}$, with the conventions $\alpha^{0} = 1$ and $0^{0} = 1$ included), whose values depend on the arrangement.

The zero and one cases are not special. $0 = \varnothing$ and $1 = \{0\}$ are cardinals (Ordinal (von Neumann)), and the definitions apply to them unchanged: $\kappa \sqcup 0 = \{0\} \times \kappa$, $\kappa \times 0 = \varnothing$, and ${}^{0}\kappa = \{\varnothing\}$ has exactly one element, the empty function. The resulting unit laws are proved rather than stipulated, in Commutativity, associativity, distributivity and monotonicity of $\oplus$ and $\otimes$, the unit laws, the two exponent laws, and $\kappa \le \lambda$ if and only if $\kappa$ injects into $\lambda$.