Infinite set


Infinite set

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:
* the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and
* the set of all real numbers is an uncountably infinite set.

Properties

The set of natural numbers (whose existence is assured by the axiom 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 a subset whose cardinality is that natural number.

If the axiom of choice holds, then a set is infinite if 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 subset equinumerous 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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • infinite set — noun (mathematics) A set that can be put into a one one correspondence with part of itself • • • Main Entry: ↑infinite …   Useful english dictionary

  • infinite set — (or collection or number) See denumerable/non denumerable …   Philosophy dictionary

  • Dedekind-infinite set — In mathematics, a set A is Dedekind infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind finite if it is not Dedekind… …   Wikipedia

  • set — /set/, v., set, setting, n., adj., interj. v.t. 1. to put (something or someone) in a particular place: to set a vase on a table. 2. to place in a particular position or posture: Set the baby on his feet. 3. to place in some relation to something …   Universalium

  • Set — /set/, n. Egyptian Relig. the brother and murderer of Osiris, represented as having the form of a donkey or other mammal and regarded as personifying the desert. Also, Seth. * * * I In mathematics and logic, any collection of objects (elements),… …   Universalium

  • infinite — [in′fə nit] adj. [ME < L infinitus: see IN 2 & FINITE] 1. lacking limits or bounds; extending beyond measure or comprehension: without beginning or end; endless 2. very great; vast; immense 3. a) Math. indefinitely large; greater than any …   English World dictionary

  • set theory — the branch of mathematics that deals with relations between sets. [1940 45] * * * Branch of mathematics that deals with the properties of sets. It is most valuable as applied to other areas of mathematics, which borrow from and adapt its… …   Universalium

  • Set theory — This article is about the branch of mathematics. For musical set theory, see Set theory (music). A Venn diagram illustrating the intersection of two sets. Set theory is the branch of mathematics that studies sets, which are collections of objects …   Wikipedia

  • infinite — I. adjective Etymology: Middle English infinit, from Anglo French or Latin; Anglo French, from Latin infinitus, from in + finitus finite Date: 14th century 1. extending indefinitely ; endless < infinite space > 2. immeasurably or inconceivably… …   New Collegiate Dictionary

  • Infinite Ryvius — 無限のリヴァイアス (Mugen no Ryvius) Genre Science fiction, Psychodrama …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.