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\begin{document}

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\section*{cardinal arithmetic}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{arithmetic}{}\paragraph*{{Arithmetic}}\label{arithmetic}

[[!include arithmetic geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Cardinal arithmetic is an [[arithmetic]] with [[cardinals]], which generalizes the ordinary [[arithmetic]] of [[natural numbers]] to non-[[finite set|finite numbers]].

The idea is that on [[finite sets]], whose [[cardinalities]] are [[natural numbers]], the usual operations of [[arithmetic]] -- [[addition]], [[multiplication]] and [[exponentiation]] -- are represented on [[finite sets]] by basic operations of [[set theory]], namely forming [[disjoint union]] of sets, forming [[Cartesian product]] of sets and forming [[function sets]] of sets.

\begin{tabular}{l|l}
$\phantom{A}$[[natural numbers]]$\phantom{A}$&$\phantom{A}$[[finite sets]]$\phantom{A}$\\
\hline 
$\phantom{A}$[[addition]]$\phantom{A}$&$\phantom{A}$[[disjoint union]]$\phantom{A}$\\
$\phantom{A}$[[multiplication]]$\phantom{A}$&$\phantom{A}$[[Cartesian product]]$\phantom{A}$\\
$\phantom{A}$[[exponentiation]]$\phantom{A}$&$\phantom{A}$[[function sets]]$\phantom{A}$\\
\end{tabular}

(Here passing from the left to the right column is an example of \emph{[[vertical categorification|categorification]]}, while passing from right to left is an example \emph{[[decategorification]]}.)

But the operations on sets on the right directly generalize from [[finite sets]] to general sets, hence from sets whose [[cardinality]] is a [[natural number]] to [[infinite cardinals]]. Therefore cardinal arithmetic is also called a \emph{[[transfinite arithmetic]]}.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

For $S$ a [[set]], write ${|S|}$ for its [[cardinality]]. Then the standard operations in the [[category]] [[Set]] induce [[arithmetic]] operations on [[cardinal numbers]]:

For $S_1$ and $S_2$ two sets, the \textbf{sum} of their cardinalities is the cardinality of their [[disjoint union]], the [[coproduct]] in $Set$:

\begin{displaymath}
{|S_1|} + {|S_2|} \coloneqq {|S_1 \amalg S_2|}
  \,.
\end{displaymath}
More generally, given any family $(S_i)_{i: I}$ of sets indexed by a set $I$, the \textbf{sum} of their cardinalities is the cardinality of their disjoint union:

\begin{displaymath}
\sum_{i: I} {|S_i|} \coloneqq {|\coprod_{i: I} S_i|}
  \,.
\end{displaymath}
Likewise, the \textbf{product} of their cardinalities is the cardinality of their [[cartesian product]], the [[product]] in $Set$:

\begin{displaymath}
{|S_1|} \, {|S_2|} \coloneqq {|S_1 \times S_2|}
  \,.
\end{displaymath}
More generally again, given any family $(S_i)_{i: I}$ of sets indexed by a set $I$, the \textbf{product} of their cardinalities is the cardinality of their cartesian product:

\begin{displaymath}
\prod_{i: I} {|S_i|} \coloneqq {|\prod_{i: I} S_i|}
  \,.
\end{displaymath}
Also, the \textbf{exponential} of one cardinality raised to the power of the other is the cardinality of their [[function set]], the [[exponential object]] in $Set$:

\begin{displaymath}
{|S_1|}^{|S_2|} \coloneqq {|Set(S_2,S_1)|}
  \,.
\end{displaymath}
In particular, we have $2^{|S|}$, which (assuming the law of [[excluded middle]]) is the cardinality of the [[power set]] $P(S)$. In [[constructive mathematics|constructive]] (but not [[predicative mathematics|predicative]]) mathematics, the cardinality of the power set is $\Omega^{|S|}$, where $\Omega$ is the cardinality of the set of [[truth values]].

The usual way to define an ordering on cardinal numbers is that ${|S_1|} \leq {|S_2|}$ if there exists an [[injection]] from $S_1$ to $S_2$:

\begin{displaymath}
({|S_1|} \leq {|S_2|})
  \;:\Leftrightarrow\;
  (\exists (S_1 \hookrightarrow S_2))
  \,.
\end{displaymath}
Classically, this is almost equivalent to the existence of a [[surjection]] $S_2 \to S_1$, except when $S_1$ is [[empty set|empty]]. Even restricting to [[inhabited sets]], these are not equivalent conditions in [[constructive mathematics]], so one may instead define that ${|S_1|} \leq {|S_2|}$ if there exists a [[subset]] $X$ of $S_2$ and a surjection $X \to S_1$. Another alternative is to require that $S_1$ (or $X$) be a [[decidable subset]] of $S_2$. All of these definitions are equivalent using [[excluded middle]].

This order relation is [[antisymmetric relation|antisymmetric]] (and therefore a [[partial order]]) by the [[Cantor–Schroeder–Bernstein theorem]] (proved by Cantor using the [[well-ordering theorem]], then proved by Schroeder and Bernstein without it). That is, if $S_1 \hookrightarrow S_2$ and $S_2 \hookrightarrow S_1$ exist, then a bijection $S_1 \cong S_2$ exists. This theorem is not constructively valid, however.

The well-ordered cardinals are [[well-order|well-ordered]] by the ordering $\lt$ on [[ordinal numbers]]. Assuming the [[axiom of choice]], this agrees with the previous order in the sense that $\kappa \leq \lambda$ iff $\kappa \lt \lambda$ or $\kappa = \lambda$. Another definition is to define that $\kappa \lt \lambda$ if $\kappa^+ \leq \lambda$, using the successor operation below.

The \textbf{[[successor]]} of a well-ordered cardinal $\kappa$ is the smallest well-ordered cardinal larger than $\kappa$. Note that (except for finite cardinals), this is different from $\kappa$'s successor as an [[ordinal number]]. We can also take successors of arbitrary cardinals using the operation of [[Hartog's number]], although this won't quite have the properties that we want of a successor without the axiom of choice.

\hypertarget{remarks}{}\subsubsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item It is traditional to write [[ℵ]]${}_0$ for the first [[infinite cardinal]] (the cardinality of the [[natural numbers]]), $\aleph_1$ for the next (the first uncountable cardinality), and so on. In this way every cardinal (assuming choice) is labeled $\aleph_\mu$ for a unique [[ordinal number]] $\mu$, with $(\aleph_\mu))^+ = \aleph_{\mu^+}$.


\item For every cardinal $\pi$, we have $2^\pi \gt \pi$ (this is sometimes called [[Cantor's theorem]]). The question of whether $2^{\aleph_0} = \aleph_{1}$ (or more generally whether $2^{\aleph_\mu} = \aleph_{\mu^+}$) is called Cantor's continuum problem; the assertion that this is the case is called the (generalized) [[continuum hypothesis]]. It is known that the continuum hypothesis is undecidable in [[ZFC]].



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{theorem}
\label{}\hypertarget{}{}
For every [[infinite cardinal]] $\pi$ we have (using the axiom of choice) $\pi + \pi = \pi$ and $\pi \cdot \pi = \pi$, so addition and multiplication are [[idempotent]].

\end{theorem}
\begin{proof}
Since $\pi \leq \pi + \pi = 2 \cdot \pi \leq \pi \cdot \pi$, it suffices to prove $\pi \cdot \pi \leq \pi$. Establishing this is closely analogous to establishing one of the standard bijections $\mathbb{N} \cong \mathbb{N} \times \mathbb{N}$ -- not the one that enumerates along successive diagonals $x + y = k$ which are spheres in an $L^1$ [[p-norm|metric]] (and which involves additive structure), but one which enumerates along successive spheres in an $L^\infty$ metric (which involves only order structure).

With this hint in mind, regard $\pi$ as the least [[ordinal]] in its cardinality class (using the [[well-ordering theorem]]), and suppose $\pi$ is a minimal counterexample to the statement. Thus ${|\alpha|}^2 = {|\alpha|}$ for all ordinals $\alpha \lt \pi$. Consider $\pi^3 = \pi \times \pi \times \pi$ in [[lexicographic order]]. This is a well-ordered set, as is the subset

\begin{displaymath}
S = \{(x, y, z) \in \pi^3: x = \max(y, z)\}
\end{displaymath}
under the order inherited from $\pi^3$. Clearly $S \cong \pi^2$ as sets. Under the supposition ${|\pi|} \lt {|\pi|}^2$, the ordinal $\pi$ is isomorphic to one of the [[lower set|initial segments]]

\begin{displaymath}
S(a, b, c) \coloneqq \{(x, y, z) \in S: (x, y, z) \lt (a, b, c)\}
\end{displaymath}
of $S$, say $S(\alpha, \beta, \gamma)$. But then (regarding $\alpha$ as an ordinal less than $\pi$)

\begin{displaymath}
{|\pi|} = {|S(\alpha, \beta, \gamma)|} \leq {|S(\alpha, \alpha, \alpha)|} = {|\alpha|}^2 = {|\alpha|} \lt {|\pi|}
\end{displaymath}
which gives a contradiction.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Conversely, if (relative to ZF or any other standard set theory) we assume that any infinite set can be put in bijection with its cartesian square, then every set can be well-ordered, a result originally due to Tarski. Details may be found at [[Hartogs number]].

\end{remark}
\begin{cor}
\label{}\hypertarget{}{}
If one of two non-zero cardinals $\kappa, \lambda$ is infinite, then

\begin{displaymath}
\kappa + \lambda = \kappa \cdot \lambda = \max \{\kappa, \lambda\}.
\end{displaymath}
\end{cor}
\begin{proof}
Clearly, $\kappa \leq \kappa + \lambda$ and $\lambda \leq \kappa + \lambda$, so $\max \{\kappa, \lambda\} \leq \kappa + \lambda$, and similarly for multiplication instead of addition, assuming the cardinals are non-zero. Letting $\mu = \max \{\kappa, \lambda\}$, we also have

\begin{displaymath}
\kappa \cdot \lambda \leq \mu \cdot \mu = \mu
\end{displaymath}
and similarly for addition instead of multiplication.

\end{proof}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Traditional lecture notes include

\begin{itemize}%
\item Alexandru Baltag, \emph{Axiomatix set theory lecture 8: Cardinal arithmetic} (\href{http://www.vub.ac.be/CLWF/SS/Sets8.pdf}{pdf})


\item \emph{Cardinal arithmetic} (\href{https://www.math.ksu.edu/~nagy/real-an/ap-b-card.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Planetmath, \emph{\href{http://planetmath.org/cardinalarithmetic}{cardinal arithmetic}}

\end{itemize}
The point of view of [[categorification]] is amplified in

\begin{itemize}%
\item [[Emily Riehl]], \emph{Categorifying cardinal arithmetic} (\href{http://www.math.jhu.edu/~eriehl/arithmetic.pdf}{pdf})

\end{itemize}


\end{document}