- A quantity is called infinite if it is not finite.
With the help of the definition of the finite set, this can be reformulated as follows:
- A set is infinite if there is no natural number such that the set is of equal power (for this is the empty set ),
with the von Neumann model of natural numbers even more compact than
- a set is infinite if it is not equal to a natural number (according to its von Neumann representation).
To Richard Dedekind the following definition is the infinity of a lot of back:
- A set is considered to be infinite if it is equal to a real 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 infinity defined at the beginning, however, requires the axiom of choice . It is clear that Dedekind infinite sets are infinite, since a finite set cannot be equal to a real subset.
Conversely, if there is an infinite set, recursively choose elements using the axiom of choice
Since is infinite, there can never be, which is why the choice of a new one is always possible. The image
if for a , otherwise
is well-defined because it is unambiguous with . It shows that the real subset is equal and therefore Dedekind infinite.
Without an at least weak version of the axiom of choice (usually the countable axiom of choice ) one cannot show that infinite sets are also Dedekind-infinite.
Existence of infinite sets
In the Zermelo-Fraenkel set theory , that is, in the usual mathematical foundations accepted by most mathematicians, the existence of infinite sets is required by an axiom, the so-called infinity axiom . Indeed, one cannot infer the existence of infinite sets from the remaining axioms. This axiom of infinity is criticized by some mathematicians, so-called constructivists , because the existence of infinite sets cannot be proven from logical axioms. Therefore, infinite sets are also suspected in the Zermelo-Fraenkel set theory of possibly leading to contradictions, although Russell's antinomy is not possible there. In fact, the consistency of set theory and thus of mathematics cannot be proven according to Kurt Gödel 's theorem of incompleteness . For a more detailed discussion, see Potential and Actual Infinity .
Different thicknesses of infinite sets
The powers of finite sets are the natural numbers; The idea of extending the concept of power to infinite sets is more difficult and interesting.
The set- theoretical concept of infinity becomes even more interesting, since there are different sets that have an infinite number of elements, but which cannot be mapped bijectively to one another. These different thicknesses are denoted by the symbol ( Aleph , the first letter of the Hebrew alphabet) and an index (initially an integer), the indices run through the ordinal numbers .
The power of the natural numbers (the smallest infinity) is in this notation . Although the natural numbers are a true subset of the rational numbers , they both have sets and the same cardinality . (→ Cantor's first diagonal argument )
The real numbers form an infinite set that is more powerful than the set of natural and rational numbers; it is uncountable . One also speaks of the cardinality of the uncountable sets of the first order. (→ Cantor's second diagonal argument )
The continuum hypothesis is the assertion that the cardinality of the real numbers is the same , i.e. the cardinality after the next greater one. It can neither be proven nor refuted with the usual axioms of set theory ( ZFC ) alone .
For every infinite set, further infinities can be constructed by forming the power set (set of all subsets). The set of Cantor indicates that the cardinality of a power amount is larger than the cardinality of the set. A classic problem of set theory (the generalized continuum hypothesis ) is whether a set with a power set results in a set of the next greater power or a few orders of magnitude are skipped over . This process can (formally) be continued so that there are infinitely many infinities .
- Harro Heuser : Textbook of Analysis. Part 1 , Vieweg + Teubner, ISBN 978-3-8348-0777-9 , pages 137 ff.
- Oliver Deiser: Introduction to set theory , Springer Berlin 2004, ISBN 978-3-540-20401-5 , pages 91-108.
- David Foster Wallace : Georg Cantor: The mathematician of the century and the discovery of the infinite , Piper Verlag 2007, hardcover, ISBN 3-492-04826-9