Cardinality

A crude sense of cardinality, an awareness that some groups of things or events differ from other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago.[4] Human expression of cardinality is seen as early as 40000 years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells.[5] The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerian mathematics and the manipulation of numbers without reference to a specific group of concrete things or events.[6]

From the 6th century BCE, the writings of Greek philosphers show the first hints of the cardinality of infinite sets. While they considered the notion of infinity as an endless series of actions, such as adding 1 to a number repeatedly, they did not consider the size of an infinite set of numbers to be a thing in itself.[7] The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. In Euclid's Elements, commensurability was described as the ability to compare the length of two line segments, a and b, as a ratio, as long as there were a third segment, no matter how small, that could be laid end-to-end a whole number of times into both a and b. But with the discovery of irrational numbers, it was seen that even the infinite set of all rational numbers was not enough to describe the length of every possible line segment.[8] Still, the concept of infinite sets was not regarded as something that had cardinality.

To better understand the infinite, a notion of cardinality was formulated circa 1880 by Georg Cantor, the originator of set theory. He examined the process of equating two sets with bijection (a one-to-one correspondence between the elements of two sets based on a unique relationship, e.g. doubling where 1 ↦ 2, 2 ↦ 4, 3 ↦ 6, etc.) In 1891, with the publication of Cantor's diagonal argument, he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e. uncountable sets that contain more elements than there are in the infinite set of natural numbers. Research continues to study how the cardinalities of different infinite sets compare to each other.

While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite).

If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| (a fact known as Schröder–Bernstein theorem). The axiom of choice is equivalent to the statement that |A| ≤ |B| or |B| ≤ |A| for every A, B.[10][11]

In the above section, "cardinality" of a set was defined functionally. In other words, it was not defined as a specific object itself. However, such an object can be defined as follows.

The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A. There are two ways to define the "cardinality of a set":

Assuming the axiom of choice, the cardinalities of the infinite sets are denoted

If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions:

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is,

However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory, if ZFC is consistent.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper subsets and proper supersets of an infinite set S that have the same size as S, although S contains elements that do not belong to its subsets, and the supersets of S contain elements that are not included in it.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−½π, ½π) and R (see also Hilbert's paradox of the Grand Hotel).

The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain such a proof.

From this, one can show that in general, the cardinalities of unions and intersections are related by the following equation:[16]