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 postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (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 a well-ordered set, then it must have a nonempty subset that 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.^[1] If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.

If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.

References

↑ Lua error in package.lua at line 80: module 'strict' not found.. See in particular pp. 32–33.

External links

Weisstein, Eric W., "Infinite Set", MathWorld.

[1] Lua error in package.lua at line 80: module 'strict' not found.. See in particular pp. 32–33.

[1]

v t e Set theory
Axioms	Choice countable dependent Extensionality Infinity Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union
Concepts Methods	Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram
Set types	Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset<dot-separator>Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem
Set theorists	Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Lotfi A. Zadeh Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Willard Quine

v t e Mathematical logic
General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory
Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram
Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables
Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus
Naive set theory	Set Empty set Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Range Map Function Relation Ordered pair
Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory
Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity
Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax
Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

Infinite set

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