Cardinal number

Cardinal numbers (lat. cardo "hinge", "pivot") are in mathematics a generalization of the natural numbers to describe the power, also called cardinality, of sets.

The power of a finite set is a natural number - the number of elements in the set. The mathematician Georg Cantor described how to generalize this concept within set theory to infinite sets and how to calculate with infinite cardinal numbers.

Infinite sets can have different powers. These are denoted by the symbol $\aleph$ (Aleph, the first letter of the Hebrew alphabet), and an index (initially integer). The power of the natural numbers $\mathbb {N}$ , the smallest infinity, is in this notation $\aleph _{0}$ .

A natural number can be used for two purposes: first, to describe the number of elements in a finite set, and second, to indicate the position of an element in a finitely ordered set. While these two concepts are consistent for finite sets, they must be distinguished for infinite sets. The description of the position in an ordered set leads to the notion of ordinal numbers, while the indication of size leads to cardinal numbers, which are described here.

Definition

Two sets and called equipotent if there is a bijection from to ; one then writes $\left\vert X\right\vert = \left\vert Y\right\vert$ or $X\sim Y$ . Equivalence $\sim$ is an equivalence relation on the class of all sets.

Cardinal numbers as real classes

The equivalence class of the set with respect to the relation of equality is called the cardinal number $\left\vert X\right\vert$ .

The problem with this definition is that the cardinals are then themselves not sets, but real classes. (With the exception of $\left\vert\emptyset\right\vert$ ).

This problem can be avoided by using $\left\vert X\right\vert$ not to denote the whole equivalence class, but to select an element from it, one selects a representative system, so to speak. In order to do this formally correct, one uses the theory of ordinal numbers, which must be defined accordingly beforehand in this approach:

Cardinal numbers as special ordinal numbers

Every set is equi-efficient to a well-ordered set (provided that one assumes the well-ordering theorem equivalent to the axiom of choice). To belongs an ordinal number. can be chosen such that this ordinal number becomes smallest possible, since ordinal numbers are themselves well-ordered; then is an initial number. One can equate the cardinal number $\left\vert A\right\vert$ with this smallest ordinal number.

By this set-theoretic handle, the cardinality of a set is itself again a set. It follows immediately the comparability theorem that the cardinals are totally ordered, because they are even well-ordered as a subset of the ordinals. This cannot be proved without the axiom of choice.

Motivation

Descriptively, cardinal numbers serve to compare the size of sets without having to refer to the appearance of their elements. For finite sets, this is easy. You simply count the number of elements. To compare the power of infinite sets, you need a little more work.

In the following, the terms at most equally powerful and less powerful are needed:

If there is a bijection from to a subset of then is called at most equal to . One then writes $\left\vert A\right\vert \leq \left\vert B\right\vert$ .

If there is a bijection from to a subset of but there isno bijection from to then said to be less powerful than and said to be more powerful than . One then writes $\left\vert A\right\vert < \left\vert B\right\vert$ .

These terms are explained in more detail in the article Thickness.

For example, for finite sets it holds that real subsets are less powerful than the whole set, whereas in Hilbert's Hotel article it is illustrated by an example that infinite sets have real subsets which are equally powerful to them.

In the study of these large sets, the question arises whether equally-powerful ordered sets necessarily have matching orders. It turns out that this is not so for infinite sets, e.g. the ordinary order of the natural numbers is $\N = \{0 < 1 < 2 < 3 < \dotsb\}$ different from the ordered set $A := \{0 < 1 < 2 < 3 < \dotsb < 0^\prime\}$ . The set is equi-empowered to $\mathbb {N}$ . Thus $f\colon 0\mapsto 1, 1\mapsto 2, 2\mapsto 3, \dots,0^\prime\mapsto 0$ a bijection, but in unlike $\mathbb {N}$ a largest element. Considering the order of sets, we arrive at ordinal numbers. The ordinal number of $\mathbb {N}$ is called ω $\omega$ and that of is ω $\omega+1$ .

Cardinal number

Definition

Motivation

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