Contents

Idea

The cardinal numbers (or just cardinals) constitute a generalisation of natural numbers to numbers of possibly infinite magnitudes. Specifically, cardinal numbers generalise the concept of ‘the number of …’. In particular, the number of natural numbers is the first infinite cardinal number.

Definition

Naïvely, a cardinal number should be an isomorphism class of sets, and the cardinality of a set $S$ would be its isomorphism class. That is:

1. every set has a unique cardinal number as its cardinality;
2. every cardinal number is the cardinality of some set;
3. two sets have the same cardinality if and only if they are isomorphic as sets.

Then a finite cardinal is the cardinality of a finite set, while an infinite cardinal or transfinite cardinal is the cardinality of an infinite set. (If you interpret both terms in the strictest sense, then there may be cardinals that are neither finite nor infinite, without some form of the axiom of choice).

Taking this definition literally in material set theory, each cardinal is then a proper class (so one could not make further sets using them as elements). For this reason, in axiomatic set theory one usually defines a cardinal number as a particular representative of this equivalence class. There are several ways to do this:

• The cardinality of a set $S$ is the smallest possible ordinal rank of any well-order on $S$. In other words, it is the smallest ordinal number (usually defined following von Neumann) which can be put in bijection with $S$. A cardinal number is then any cardinality, i.e. any ordinal which is not in bijection with any smaller ordinal.

  • On well-orderable sets, this cardinality function satisfies (1–3), but one needs the axiom of choice (precisely, the well-ordering theorem) to prove that every set is well-orderable. This approach is probably the most common one in the presence of the axiom of choice.

  • In the absence of excluded middle, when the “correct” definition of well-order is different from the usual one (and so “the least ordinal such that …” may not exist), a better definition of the cardinality of $S$ is as the set of all ordinal numbers less than the ordinal rank of every well-order on $S$.

• Alternatively, we can define the cardinality of a set $X$ to be the set of all well-founded pure sets that are isomorphic as sets to $X$ and such that no pure set of smaller hereditary rank (that is, which occurs earlier in the von Neumann hierarchy) is isomorphic to $X$.

  • On those sets that are isomorphic to some well-founded set, this cardinality function satisfies (1–3), but one needs some assumption to prove that every set is isomorphic to a well-founded set. (This will follow directly from the axiom of foundation; it will also follow from the axiom of choice, since then every set is isomorphic to a von Neumann ordinal.)

In the absence of the appropriate axioms, the definitions above can still be used to define well-ordered cardinals and well-founded cardinals, respectively.

From the perspective of structural set theory, it is evil to care about distinctions between isomorphic objects, and unnecessary to insist on a canonical choice of representatives for isomorphism classes. Therefore, from this point of view it is natural to simply say:

• A cardinal is a set (that is, an object of Set).

However, one still may need sets of cardinals, that is sets that serve as the target of a cardinality function satisfying (1–3) on any (small) collection of sets. One can construct this as a quotient set of that collection.

Lowercase Greek letters starting from $\kappa$ are often used for cardinal numbers.

In homotopy type theory

A cardinal in homotopy type theory is an element of the type of cardinals $\kappa \colon Card_\mathcal{U}$ relative to a universe $\mathcal{U}$. The type of cardinals is defined as the set-truncation of the type of sets $Set_\mathcal{U}$ relative to $\mathcal{U}$, $Card_\mathcal{U} \coloneqq \left| Set_\mathcal{U} \right|_0$ with $Set_\mathcal{U}$ defined as

Cardinal arithmetic

Definition

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

For $S_1$ and $S_2$ two sets, the sum of their cardinalities is the cardinality of their disjoint union, the coproduct in $Set$:

More generally, given any family $(S_i)_{i: I}$ of sets indexed by a set $I$, the sum of their cardinalities is the cardinality of their disjoint union:

Likewise, the product of their cardinalities is the cardinality of their cartesian product, the product in $Set$:

More generally again, given any family $(S_i)_{i: I}$ of sets indexed by a set $I$, the product of their cardinalities is the cardinality of their cartesian product:

Also, the exponential of one cardinality raised to the power of the other is the cardinality of their function set, the exponential object in $Set$:

In particular, we have $2^{|S|}$, which (assuming the law of excluded middle) is the cardinality of the power set $P(S)$. In constructive (but not 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 $S_2$:

Classically, this is almost equivalent to the existence of a surjection $S_2 \to S_1$, except when $S_1$ is 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 (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-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 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.

Properties

• 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 $(\aleph_\mu))^+ = \aleph_{\mu^+}$.

• 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.

• For every transfinite cardinal $\pi$ we have (using the axiom of choice) $\pi + \pi = \pi$ and $\pi \cdot \pi = \pi$, so addition and multiplication are idempotent.

Properties of cardinals

• A transfinite cardinal $\pi$ is a regular cardinal if no set of cardinality $\pi$ is the union of fewer than $\pi$ sets of cardinality less than $\pi$. Equivalently, $\pi$ is regular if given a function $P \to X$ (regarded as a family $\{P_x\}_{x\in X}$) such that ${|X|} \lt \pi$ and ${|P_x|} \lt \pi$ for all $x \in X$, then ${|P|} \lt \pi$. Again equivalently, $\pi$ is regular if the category $\Set_{\lt\pi}$ of sets of cardinality $\lt\pi$ has all colimits of size $\lt\pi$. The successor of any infinite cardinal, such as $\aleph_1$, is a regular cardinal.

• A cardinal is called singular if it is not regular. For instance, $\aleph_\omega = \bigcup_{n\in \mathbb{N}} \aleph_n$ is singular, more or less by definition, since $\aleph_n \lt \aleph_\omega$ and ${|\mathbb{N}|} = \aleph_0 \lt \aleph_\omega$.

• A limit cardinal is one which is not a successor of any other cardinal. Note that every infinite cardinal is a limit ordinal (in the picture where cardinals are identified with certain ordinals).

• A strong limit cardinal is a cardinal $\pi$ such that if $\lambda \lt \pi$, then $2^\lambda \lt \pi$, for any cardinal $\lambda$. Since $\lambda^+ \le 2^\lambda$, any strong limit is a limit. Conversely, assuming the continuum hypothesis, every limit is a strong limit. Since $2^\lambda$ is the cardinality of the power set $P(\lambda)$, a cardinal $\pi$ is a strong limit iff the category $\Set_{\lt\pi}$ is an elementary topos.

• An inaccessible cardinal is any (usually uncountable) regular strong limit cardinal. A weakly inaccessible cardinal is a regular limit cardinal.

• A cardinal $\kappa$ is a measurable cardinal if some (hence any) set of cardinality $\kappa$ admits a two-valued measure which is $\kappa$-additive, or equivalently an ultrafilter which is $\kappa$-complete.

Related concepts

• groupoid cardinality

• counting measure

• ordinal number

References

The original article is

• Georg Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, Math. Ann. 46 (1895) pp.481-512, reprinted from p. 282 on in Ernst Zermelo (ed.), Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Springer Berlin 1932 (online English translation)

The book

• Peter J Cameron, Sets, Logic and Categories (ISBN: 1-85233-056-2 )

contains a very readable account of ZFC and the definitions of both ordinal and cardinal numbers.

Any serious reference on set theory should cover cardinal numbers. The long-established respected tome is

• Thomas Jech Set Theory;

there are also some references listed at

• MathWorld and

• the English Wikipedia.

A good introduction to infinite cardinals is:

• Frank R. Drake, Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 76, Elsevier, 1974.

For a much deeper treatment, which assumes most of the material in Drake as a prerequisite, see:

• Akihiro Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Beginning, Springer, 2003.

This is a very readable and freely available historical introduction:

• Akihiro Kanamori and M. Magidor, The evolution of large cardinal axioms in set theory, in Higher Set Theory, Springer Lecture Notes in Mathematics 669, 1978. (pdf)

Standard approaches start with a material set theory, such as ZFC, whereas the approach here uses structural set theory as emphasized above. Since cardinality is isomorphism-invariant, it is easy to interpret the standard material structurally, although the basic definitions will be different. There does not seem to be a standard account of cardinality from within structural set theory.

For cardinal numbers in homotopy type theory, see chapter 10 of

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

For a critical discussion of the history of the meaning of Cantor’s “Kardinalen”, see

• William Lawvere, Cohesive Toposes and Cantor's "lauter Einsen",

which argues that Cantor’s original meaning of set was more like what today is cohesive set and that his Kardinalen refer to the underlying set (see at flat modality).