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 $\alpha \bigsqcup \beta$ and $\alpha \times \beta$ carry explicit well-orders, so their cardinalities exist in ZF:

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

Sum and product.

$\kappa \oplus \lambda :=\mid \kappa \bigsqcup \lambda \mid ,\kappa \otimes \lambda :=\mid \kappa \times \lambda \mid \mathrm{.}$

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 $\alpha \bigsqcup \beta$ 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.

${\kappa }^{\lambda }:=\mid {}^{\lambda }\kappa \mid ,$

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 $\mid A\mid =\kappa$ and $\mid B\mid =\lambda$, then

$\mid A\bigsqcup B\mid =\kappa \oplus \lambda ,\mid A\times B\mid =\kappa \otimes \lambda ,\mid {}^{B}A\mid ={\kappa }^{\lambda }$

whenever the sets on the left have cardinalities at all, because $A\approx \kappa$ and $B\approx \lambda$ (Equinumerous sets, $A\approx B$ and $A⪯B$) 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 $\alpha \bigsqcup \beta$ 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 — $\kappa$, $\lambda$, $\mu$, $\mathfrak{c}$, $\mathrm{cf}(\kappa )$, $\mid A\mid$ — 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 $2^{{\mathrm{\aleph }}_0}$, ${\kappa }^{\lambda }$ and ${\mathrm{\aleph }}_1^{{\mathrm{\aleph }}_0}$ 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 $\bigsqcup$ 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. $0=\mathrm{\varnothing }$ and $1=\{0\}$ are cardinals (Ordinal (von Neumann)), and the definitions apply to them unchanged: $\kappa \bigsqcup 0=\{0\}\times \kappa$, $\kappa \times 0=\mathrm{\varnothing }$, and ${}^0\kappa =\{\mathrm{\varnothing }\}$ 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 $\lambda$.