- Infinite set
In
set theory , an infinite set is a set that is not afinite set . Infinite sets may be countable or uncountable. Some examples are:
* the set of allinteger s, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and
* the set of allreal number s is an uncountably infinite set.Properties
The set of
natural numbers (whose existence is assured by theaxiom of infinity ) is infinite. It is the only set which is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in ZFC only by showing that it follows from the existence of the natural numbers.A set is infinite
if and only if for every natural number the set has asubset whosecardinality is that natural number.If the
axiom of choice holds, then a set is infiniteif and only if it includes a countable infinite subset.If a set of sets is infinite or contains an infinite element, then its union is infinite. The powerset of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite; if the axiom of choice holds, then it is infinite.
If an infinite set is well-ordered, then it must have a nonempty subset which has no greatest element.
In ZF, a set is infinite if and only if the powerset of its powerset is a
Dedekind-infinite set , having a proper subsetequinumerous to itself. If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.If an infinite set is well-orderable, then it has many well-orderings which are non-isomorphic.
ee also
*
Infinity
*Aleph number
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