Cardinal number

Cardinal numbers (Latin cardo, " hinge ", " focal point " ) in mathematics is a generalization of the natural numbers to describe the cardinality ( " cardinality " ) of sets.

The cardinality 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 the set theory to infinite sets and as you can expect infinite cardinal numbers.

Infinite sets can have different widths. These are designated with the symbol ℵ ( aleph, the first letter of the Hebrew alphabet ), and a ( first integer ) index. The cardinality of the natural numbers (the " smallest" infinity ) in this notation ℵ 0

A natural number can be used for two purposes: first, to describe the number of elements of a ( finite ) number, and, second, to specify the position of an element in an ordered set. While these two concepts coincide for finite sets, one needs to distinguish them for infinite sets. The description of the position in an ordered set leads to the concept of ordinal, while the size specification leads to cardinal numbers, which are described here.

Definition

Two sets X and Y are called the same powerful when there is a bijection from X to Y; to write then | X | = | Y |. The DC cardinality of sets is an equivalence relation on the class of all sets.

1 Definition: cardinal numbers as real classes

The problem with this definition is that the cardinal numbers are now even not sets, but real classes. ( Except for ).

This problem can be bypassed by using | not referred to the entire equivalence class, but an element selects therefrom, one chooses so to speak, a system of representatives of | X. To do this formally correct, one uses the theory of ordinal numbers, which must be defined according to previously in this approach:

2 Definition: cardinal numbers as a special ordinal

This set-theoretic handle the cardinality of a set itself is a lot again. It immediately follows the Vergleichbarkeitssatz: The cardinal numbers are totally ordered ( for they are a subset of the ordinals even well-ordered ). This can not be proved without the axiom of choice.

Motivation

Clearly serve cardinal numbers to compare the "size" of sets, without having to refer to the appearance of their elements. For finite sets this is easy: you simply counts the number of elements. In order to compare infinite sets, one needs a bit more work to characterize their thickness.

In the following, the terms most equal powerful and less powerful needs:

These terms are explained in the article cardinality closer.

For example, valid for finite sets that proper subsets are less powerful than the full amount, however, is in the article Hilbert's Hotel illustrated by an example that infinite sets have proper subsets that are equally powerful to them.

In examining this "large" quantities, the question arises whether the same powerful minor amounts necessary " matching" have orders. It turns out that this is not so for infinite sets, eg, the usual order of natural numbers of the ordered quantity is different. The set A is equal to powerful ( fancy 1 to 2, 2 to 3, ..., 1 ' to 1 from ), but in A there is in contrast to a greatest element. Taking into account the order of quantities, we arrive at ordinal. ( The ordinal of ω means and the means of A ω 1. )

Properties

In the article thickness is shown that the cardinal numbers are totally ordered.

A set M is called finite if there is a natural number (including 0) n are such that M has exactly n elements. So that means that M is either empty (which is the case n = 0), or that there is a bijection from M to the set { 1, ..., n}. A set M is infinite if there is no such natural number.

One can show that the infinite sets are exactly those sets that are equally powerful to a proper subset (see Dedekind - infinite).

It is true also that the cardinal number (read aleph zero, see Hebrew Alphabet ) the amount the smallest infinite cardinal number. The next larger cardinal number is (assuming the continuum hypothesis is, however, also applies without the continuum hypothesis certainly ). For each ordinal a there is an a-th infinite cardinal, and every infinite cardinal number is reached. Since the ordinals form a proper class, the class of cardinal numbers is real.

Note that without the axiom of choice, there are quantities that can not be well-ordered, and the specified section definition equation of cardinal numbers does not work with certain ordinals. One can then still define cardinal numbers as equivalence classes equal amounts of powerful, but these are then only partially ordered, as various cardinal numbers no longer need to be comparable ( this requirement is equivalent to the axiom of choice ). But you can also examine the cardinality of sets, cardinal numbers without using at all.

Arithmetic operations

If X and Y are disjoint sets, then we define

• .

In this case, a Cartesian product and the set of all functions from Y after X.

It can be shown that these links are natural numbers correspond with the usual processing operations. In addition, for any sets X, Y, Z:

• Addition and multiplication are associative and commutative.
• Addition and multiplication satisfy the distributive law.
• Apply power laws and.

The addition and multiplication of infinite cardinal numbers is ( assuming the axiom of choice ) easily: If X or Y is infinite and both sets is not empty, then applies

No cardinal number except 0 has an additive inverse ( an inverse element with respect to addition ), so are the cardinal numbers with the addition of no group, and certainly no ring.

It is 2 | X | is the cardinality of the power set P ( X) of X and Cantor's diagonal proof shows that 2 | X |> | X | for each X amount. It follows that there is no greatest cardinal number.

Continuum hypothesis

The " generalized continuum hypothesis " ( generalized continuum hypothesis, GCH ) states that for every infinite set X between the cardinal numbers | X | 2 and | X | are no other cardinal numbers. The continuum hypothesis ( continuum hypothesis, CH ) makes this claim for the case. It is independent of the Zermelo -Fraenkel set theory with the axiom of choice ( ZFC ).