Aleph number
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- This article is about the aleph numbers. For Aleph One as it pertains to the Marathon computer game, see that article.
In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. The name is that of the symbol used to denote these numbers, the Hebrew letter aleph (<math>\aleph<math>).
The cardinality of the natural numbers is aleph-null (<math>\aleph_0<math>) (also aleph-nought), the next larger cardinality is aleph-one <math>\aleph_1<math>, then comes <math>\aleph_2<math> and so on. Continuing in this manner, it is possible to define a cardinal number <math>\aleph_\kappa<math> for every ordinal number κ as will be described below.
The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.
It should be noted that the aleph numbers are unrelated to the ∞ commonly found in algebra and calculus. Alephs measure the sizes of sets. Infinity (∞), however, could roughly be defined as the extreme limit of the real number line. While some alephs are larger than others, ∞ is just ∞.
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Possible origins
Georg Cantor may have had Jewish roots. He may have been familiar with Kabbalah, Jewish mysticism. Kabbalah uses the letter א to denote God, who is referred to as אין סוף, Ayn Sof, the Infinite, in Kabbalah. Perhaps Cantor borrowed the abbreviation and used it to refer to mathematical infinity, rather than theological.
Aleph-null
Aleph-null (<math>\aleph_0<math>), also called aleph-nought, is by definition the cardinality of the set of all natural numbers, and (assuming, as usual, the axiom of choice), is the smallest of all infinite cardinalities. A set has cardinality <math>\aleph_0<math> if and only if it is countably infinite, which is the case if and only if it can be put into a direct one-to-one correspondence (see bijection) with the integers. Such sets include the set of all prime numbers and the set of all rational numbers.
Aleph-one
<math>\aleph_1<math> is the cardinality of the set of all countably infinite ordinal numbers, Ω. It can be demonstrated within the Zermelo-Fraenkel axioms (without the axiom of choice) that no cardinal number is between <math>\aleph_0<math> and <math>\aleph_1<math>. If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus <math>\aleph_1<math> is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set Ω (the standard example of a set of size <math>\aleph_1<math>): any countable subset of Ω has an upper bound (with respect to the standard well-ordering of ordinals) in Ω (the proof is easy: a countable union of countable sets is countable; this is one of the most common applications of AC). This fact is analogous to the situation in <math>\aleph_0<math>: any finite set of natural numbers (subset of ω) has a maximum which is also a natural number (has an upper bound in ω) — finite unions of finite sets are finite.
Ω is actually pretty useful, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations, e.g. trying to explicitly describe the sigma-algebra generated by an arbitrary collection of subsets. This is harder than most explicit descriptions of "generation" in algebra (for example vector spaces, groups, etc.) because in those cases we only have to close with respect to finite operations — sums, products, and the like. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of Ω.
The continuum hypothesis
The cardinality of the set of real numbers is <math>2^{\aleph_0}<math>. It is not clear where this number fits in the aleph number hierarchy. In Zermelo-Fraenkel set theory with the axiom of choice, the celebrated continuum hypothesis is equivalent to the identity
- <math>2^{\aleph_0}=\aleph_1.<math>
This proposition is independent of "ZFC", i.e., of Zermelo-Fraenkel set theory conjoined with the axiom of choice: it can be neither proved nor disproved within the context of that axiom system. That it is consistent with ZFC was demonstrated by Kurt Gödel in 1940; that it is independent of ZFC was demonstrated by Paul Cohen in 1963.
Aleph-ω
Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number <math>\aleph_\omega<math> is the smallest upper bound of
- <math>\left\{\,\aleph_n : n\in\left\{\,0,1,2,\dots\,\right\}\,\right\}.<math>
Aleph-ω is the first uncountable cardinal number that can be demonstrated within Zermelo-Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer n we can consistently assume that <math>2^{\aleph_0} = \aleph_n<math>, and moreover it is possible to assume <math>2^{\aleph_0}<math> is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality <math>\aleph_0<math>, meaning there is an unbounded function from <math>\aleph_0<math>.
Aleph-κ for general κ
To define aleph-κ for arbitrary ordinal number κ, we need the successor cardinal operation, which assigns to any cardinal number ρ the next bigger cardinal <math>\rho^+<math>.
We can then define the aleph numbers as follows
- <math>\aleph_0 = \omega<math>
- <math>\aleph_{\alpha+1} = \aleph_{\alpha}^+<math>
and for λ an infinite limit ordinal,
- <math>\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta.<math>