Infinite set

Infinite set is a term used in set theory, a branch of mathematics. Even the use of the negating prefix un suggests the following definition:

• A set is infinite if it is not finite.

Using the definition of the finite set can be rephrase that as follows:

• A set is infinite if there is no natural number so that the amount is equally powerful ( for that is the empty set).

Examples of infinite sets are the set of natural numbers or the set of real numbers.

Dedekind - infinity

On Richard Dedekind, the following definition of an infinite set goes back:

• A set is infinite if it is equally powerful to a proper subset.

More precisely, one speaks in this case of Dedekind infinity. The advantage of this definition is that it does not refer to the natural numbers. The equivalence to the above-defined infinity, however, requires the axiom of choice. That Dedekind - infinite sets are infinite, it is clear there is no finite set to a proper subset can be equally powerful. Conversely, if an infinite amount, so you choose with the help of the axiom of choice recursive elements

Since is infinite, can never be, which is why the choice of is always possible. The figure

Then shows that the proper subset is equally powerful and therefore Dedekind - infinite.

It is even true that you can not show without at least a weak version of the axiom of choice that infinite sets are Dedekind - infinite.

Existence of infinite sets

In the Zermelo -Fraenkel set theory, that is, in the usual, accepted by most mathematicians basis of mathematics, the existence of infinite sets is required by an axiom, the so-called axiom of infinity. In fact, one can not rule out the other axioms the existence of infinite sets. This axiom of infinity is by some mathematicians, so-called constructivists criticized, since the existence of infinite sets of logical axioms is not provable. Therefore, infinite sets are also suspected in the Zermelo -Fraenkel set theory, may lead to contradictions, although Russell's antinomy is not possible there. In fact can not be proved the consistency of set theory and thus mathematics after going back to Kurt Godel 's incompleteness. For a further discussion, see Potential and actual infinity.

Different thicknesses of infinite sets

The widths of finite sets are the natural numbers; more difficult and more interesting is the idea of ​​extending the notion of cardinality on infinite sets.

The set-theoretic concept of infinity is even more interesting because there are different amounts that have infinitely many elements, but which can not be mapped bijectively to each other. These different widths are denoted by the symbol ( Aleph, the first letter of the Hebrew alphabet ), and a ( first integer ) Index, the indices run through the ordinals.

The cardinality of the natural numbers (the " smallest" infinity ) in this notation. Although the natural numbers are a proper subset of the rational numbers, have both volumes and the same thickness. (→ Cantor's first diagonal argument)

The real numbers form an infinite amount which is more powerful than the set of natural and rational numbers; it is uncountable. This is also called the cardinality of uncountable sets first stage. (→ Cantor's second diagonal argument)

The continuum hypothesis is the claim that the cardinality of the real numbers, so after next larger cardinality is equal. She is alone with the usual axioms of set theory ( ZFC ) neither provable nor refutable.

For every infinite set is more infinities by forming the power set ( set of all subsets ) can be constructed. The set of Cantor states that the cardinality of a power set is greater than the cardinality of the set. Whether with thickness caused by power set formation from a lot a lot of the next larger cardinality or several orders of magnitude are skipped, a classical problem of the theory (the generalized continuum hypothesis). This process can ( formally) be always continued so that there are infinitely many infinities.

There are several in set theory " number systems " containing infinitely large numbers. The best known are ordinal numbers, cardinal numbers, hyper- real numbers and the surreal numbers.