Uncountable set
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
: a set is uncountable if its cardinal number is larger than aleph-null, the cardinality of the
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s. Examples of uncountable sets include the set of all real numbers and set of all subsets of the natural numbers.


Characterizations

There are many equivalent characterizations of uncountability. A set ''X'' is uncountable if and only if any of the following conditions hold: * There is no injective function (hence no bijection) from ''X'' to the set of natural numbers. * ''X'' is nonempty and for every ω- sequence of elements of ''X'', there exists 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 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 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 number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s (see: ), and the set of all subsets of the set of natural numbers. The cardinality of is often called the cardinality of the continuum, and denoted by \mathfrak , or 2^, or \beth_1 ( beth-one). The Cantor set is an uncountable subset of . The Cantor set is a
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
and has Hausdorff dimension greater than zero but less than one ( has dimension one). This is an example of the following fact: any subset of of Hausdorff dimension strictly greater than zero must be uncountable. Another example of an uncountable set is the set of all functions from to . This set is even "more uncountable" than 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 number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s, 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 Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
(including the axiom of choice).


Without the axiom of choice

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. 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 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


References


Bibliography

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


External links


Proof that R is uncountable
{{Set theory Basic concepts in infinite set theory Infinity Cardinal numbers