Definitions

Uncountable set

Uncountable set

In mathematics, an uncountable set is an infinite set which is too big 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 natural numbers. The related term nondenumerable set is used by some authors as a synonym for "uncountable set" while other authors define a set to be nondenumerable if it is not an infinite countable set.

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 any ω-sequence of elements of X fails to include at least one element of X. 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 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 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 even the set of all infinite sequences consisting only of zeros and ones) 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 (omega) 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-Frankel axioms for set theory (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. 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.

See also

References

  • 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).
  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.

External links

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