Uncountable set

Template:Short description Template:Redirect Template:Pp-sock Template:Multiple In mathematics, an uncountable set, informally, 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 aleph-null, the cardinality of the natural numbers.

Examples of uncountable sets include the set Template:Tmath of all real numbers and the set of all subsets of the natural numbers.

Characterizations

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

• There is no injective function (hence no bijection) from Template:Mvar to the set of natural numbers.
• Template:Mvar is nonempty and for every ω-sequence of elements of Template:Mvar, there exists at least one element of X not included in it. That is, Template:Mvar is nonempty and there is no surjective function from the natural numbers to Template:Mvar.
• The cardinality of Template:Mvar is neither finite nor equal to ${\mathrm{\aleph }}_0$ (aleph-null).
• The set Template:Mvar has cardinality strictly greater than ${\mathrm{\aleph }}_0$.

The first three of these characterizations can be proved 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 Template:Mvar is a subset of set Template:Mvar, then Template:Mvar is uncountable.

Examples

The best known example of an uncountable set is the set Template:Tmath 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 Template:Tmath (see: Template:OEIS), and the set of all subsets of the set of natural numbers. The cardinality of Template:Tmath is often called the cardinality of the continuum, and denoted by $𝔠$, or $2^{{\mathrm{\aleph }}_0}$, or ${\beth }_1$ (beth-one).

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

Another example of an uncountable set is the set of all functions from Template:Tmath to Template:Tmath. This set is even "more uncountable" than Template:Tmath 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 ${\mathrm{\aleph }}_1$ (aleph-one). It can be shown, using the axiom of choice, that ${\mathrm{\aleph }}_1$ is the smallest uncountable cardinal number. Thus either ${\beth }_1$, the cardinality of the reals, is equal to ${\mathrm{\aleph }}_1$ or it is strictly larger. Georg Cantor was the first to propose the question of whether ${\beth }_1$ is equal to ${\mathrm{\aleph }}_1$. In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that ${\mathrm{\aleph }}_1={\beth }_1$ 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

Template:Main Without the axiom of choice, there might exist cardinalities incomparable to ${\mathrm{\aleph }}_0$ (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. 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:

• $\kappa \nleqq {\mathrm{\aleph }}_0;$
• $\kappa >{\mathrm{\aleph }}_0;$ and
• $\kappa \ge {\mathrm{\aleph }}_1$, where ${\mathrm{\aleph }}_1=|{\omega }_1|$ and ${\omega }_1$ is the 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.

See also

• Aleph number
• Beth number
• First uncountable ordinal
• Injective function

Bibliography

• Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. Template:Isbn (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. Template:Isbn (Paperback edition).
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