Uncountable set

"Uncountable" redirects here. For the linguistic concept, see Uncountable noun.

In mathematics, an uncountable set (or uncountably infinite set)^[1] is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

Characterizations

There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions holds:

There is no injective function from X to the set of natural numbers.
X is nonempty and for every ω-sequence of elements of X, there exist at least one element of X not included in it. That is, X is nonempty and there is no surjective function from the natural numbers to X.
The cardinality of X is neither finite nor equal to $\aleph _{0}$ (aleph-null, the cardinality of the natural numbers).
The set X has cardinality strictly greater than $\aleph _{0}$ .

The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

Properties

If an uncountable set X is a subset of set Y, then Y is uncountable.

Examples

The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of natural numbers. The cardinality of R is often called the cardinality of the continuum and denoted by c, or $2^{\aleph _{0}}$ , or $\beth _{1}$ (beth-one).

The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one (R has dimension one). This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable.

Another example of an uncountable set is the set of all functions from R to R. This set is even "more uncountable" than R in the sense that the cardinality of this set is $\beth _{2}$ (beth-two), which is larger than $\beth _{1}$ .

A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω₁. The cardinality of Ω is denoted $\aleph _{1}$ (aleph-one). It can be shown, using the axiom of choice, that $\aleph _{1}$ is the smallest uncountable cardinal number. Thus either $\beth _{1}$ , the cardinality of the reals, is equal to $\aleph _{1}$ or it is strictly larger. Georg Cantor was the first to propose the question of whether $\beth _{1}$ is equal to $\aleph _{1}$ . In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that $\aleph _{1}=\beth _{1}$ is now called the continuum hypothesis and is known to be independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice).

Without the axiom of choice

Main article: Dedekind-infinite set

Without the axiom of choice, there might exist cardinalities incomparable to $\aleph _{0}$ (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above but not the fourth characterization. Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.

If the axiom of choice holds, the following conditions on a cardinal $\kappa \!$ are equivalent:

$\kappa \nleq \aleph _{0};$
$\kappa >\aleph _{0};$ and
$\kappa \geq \aleph _{1}$ , where $\aleph _{1}=|\omega _{1}|$ and $\omega _{1}\,$ is least initial ordinal greater than $\omega .\!$

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.

References

↑ Uncountably Infinite — from Wolfram MathWorld

Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition).
Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN 3-540-44085-2

External links

Proof that R is uncountable

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 Many-valued logic

Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus

Naive set theory	Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image 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

Set theory

Axioms	Choice countable dependent Constructibility (V=L) Determinacy 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 · 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 Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine

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