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Uncountable set

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Infinite set that is not countable

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

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Inmathematics, anuncountable set, informally, is aninfinite set that contains too manyelements to becountable. The uncountability of a set is closely related to itscardinal number: a set is uncountable if its cardinal number is larger thanaleph-null, the cardinality of thenatural numbers.

Examples of uncountable sets include the set⁠ $\mathbb {R}$ ⁠ of allreal numbers and set of all subsets of the natural numbers.

Characterizations

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There are many equivalent characterizations of uncountability. A setX is uncountable if and only if any of the following conditions hold:

There is noinjective function (hence nobijection) fromX to the set of natural numbers.
X is nonempty and for every ω-sequence of elements ofX, there exists at least one element of X not included in it. That is,X is nonempty and there is nosurjective function from the natural numbers toX.
Thecardinality ofX is neither finite nor equal to $\aleph _{0}$ (aleph-null).
The setX has cardinality strictly greater than $\aleph _{0}$ .

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

Properties

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If an uncountable setX is a subset of setY, thenY is uncountable.

Examples

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The best known example of an uncountable set is the set⁠ $\mathbb {R}$ ⁠ of allreal 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 infinitesequences ofnatural numbers⁠ $\mathbb {N}$ ⁠ (see: (sequenceA102288 in theOEIS)), and theset of all subsets of the set of natural numbers. The cardinality of⁠ $\mathbb {R}$ ⁠ is often called thecardinality of the continuum, and denoted by ${\mathfrak {c}}$ , or $2^{\aleph _{0}}$ , or $\beth _{1}$ (beth-one).

TheCantor set is an uncountablesubset of⁠ $\mathbb {R}$ ⁠. The Cantor set is afractal and hasHausdorff dimension greater than zero but less than one (⁠ $\mathbb {R}$ ⁠ has dimension one). This is an example of the following fact: any subset of⁠ $\mathbb {R}$ ⁠ of Hausdorff dimension strictly greater than zero must be uncountable.

Another example of an uncountable set is the set of allfunctions from⁠ $\mathbb {R}$ ⁠ to⁠ $\mathbb {R}$ ⁠. This set is even "more uncountable" than⁠ $\mathbb {R}$ ⁠ 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 countableordinal numbers, denoted by Ω or ω₁.^[1] The cardinality of Ω is denoted $\aleph _{1}$ (aleph-one). It can be shown, using theaxiom of choice, that $\aleph _{1}$ is thesmallest 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 his23 problems. The statement that $\aleph _{1}=\beth _{1}$ is now called thecontinuum hypothesis, and is known to be independent of theZermelo–Fraenkel axioms forset theory (including theaxiom of choice).

Without the axiom of choice

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Main article:Dedekind-infinite set

Without theaxiom of choice, there might exist cardinalitiesincomparable to $\aleph _{0}$ (namely, the cardinalities ofDedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. Since 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 the leastinitial 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

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^Weisstein, Eric W."Uncountably Infinite".mathworld.wolfram.com. Retrieved2020-09-05.

Bibliography

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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

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Proof thatR is uncountable

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