basic constructions:
strong axioms
further
The cardinal numbers (or just cardinals) constitute a generalisation of a natural numbers to 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.
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:
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 $X$.
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:
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.
Roger Witte First of all sorry if I am posting in the wrong place
While thinking about graphs, I wanted to define them as subobjects of naive cardinal 2 and this got me thinking about the behaviour of the full subcategories of Set defined by isomorphism classes. These categories turned out to be more interesting than I had expected.
If the background set theory is ZFC or similar, these are all large but locally small categories with all hom sets being isomorphic. They all contain the same number of objects (except 0, which contains one object and no non-identity morphisms) and are equinumerous with Set. Each hom Set contains $N^N$ arrows. In the finite case $N!$ of the morphisms in a particular hom set are isomorphisms. In particular, only 0 and 1 are groupoids. I haven’t worked out how this extends to infinite cardinalities, yet.
If the background theory is NF, then they are set and 1 is smaller than Set. I haven’t yet worked out how 2 compares to 1. I need to brush up on my NF to see how NF and category theory interact.
I am acutely aware that NF/NFU is regarded as career suicide by proffesional mathematicians, but, fortunately, I am a proffesional transport planner, not a mathematician.
Toby: Each of these categories is equivalent (but not isomorphic, except for 0) to a category with exactly one object, which may be thought of as a monoid. Given a cardinal $N$, if you pick a set $X$ with $N$ elements, then this is (up to equivalence, again) the monoid of functions from $X$ to itself. The invertible elements of this monoid form the symmetric group, with order $N!$ as you noticed. Even for infinite cardinalities, we can say $N^N$ and $N!$, where we define these numbers to be the cardinalities of the sets of functions (or invertible functions) from a set of cardinality $N$ to itself.
From a structural perspective, there's no essential difference between equivalent categories, so the fact that these categories (except for 0) are equinumerous with all of Set is irrelevant; what matters is not the number of objects but the number of isomorphism classes of objects (and similarly for morhpisms). That doesn't mean that your result that they are equinumerous with Set is meaningless, of course; it just means that it says more about how sets are represented in ZFC than about sets themselves. So it should be no surprise if it comes out differently in NF or NFU, but I'm afraid that I don't know enough about NF to say whether they do or not.
By the way, every time you edit this page, you wreck the links to external web pages (down towards the bottom in the last query box). It seems as if something in your editor is removing URLs.
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 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.
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.
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 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.
For a generalization of cardinality from sets to groupoids see groupoid cardinality.
The original article is
The book
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
there are also some references listed at
A good introduction to infinite cardinals is:
For a much deeper treatment, which assumes most of the material in Drake as a prerequisite, see:
This is a very readable and freely available historical introduction:
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 a critical discussion of the history of the meaning of Cantor’s “Kardinalen”, see
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).