Aleph number

From Wikipedia, the free encyclopedia

"Aleph One" redirects here. For hacker's nick, see Elias Levy, for the computer game, see Aleph One (computer game).

In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph ( $\aleph$ ).

The cardinality of the natural numbers is aleph-null ( $\aleph_0$ ) (also aleph-naught, aleph-nought); the next larger cardinality is aleph-one $\aleph_1$ , then $\aleph_2$ and so on. Continuing in this manner, it is possible to define a cardinal number $\aleph_\alpha$ for every ordinal number α, 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.

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line, or an extremal point of the extended real number line. While some alephs are larger than others, ∞ is just ∞.

1 Aleph-null
2 Aleph-one
3 The continuum hypothesis
4 Aleph-ω
5 Aleph-α for general α
6 Fixed points of aleph
7 Popular culture
8 See also
9 External links

[edit] Aleph-null

Aleph-null ( $\aleph_0$ ) 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 $\aleph_0$ if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Such sets include the set of all prime numbers and the set of all rational numbers.

[edit] Aleph-one

$\aleph_1$ is the cardinality of the set of all countable ordinal numbers, called ω₁ or Ω. Notice ω₁ is an uncountable set. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between $\aleph_0$ and $\aleph_1$ . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus $\aleph_1$ 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 $\aleph_1$ ): 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 $\aleph_0$ : 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 a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the σ-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.

[edit] The continuum hypothesis

The cardinality of the set of real numbers is $2^{\aleph_0}$ . It is not clear where this number fits in the aleph number hierarchy. It follows from ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that the celebrated continuum hypothesis, CH, is equivalent to the identity

$2^{\aleph_0}=\aleph_1.$

CH is independent of ZFC: it can be neither proven nor disproven 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.

[edit] Aleph-ω

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number $\aleph_\omega$ is the smallest upper bound of

$\left\{\,\aleph_n : n\in\left\{\,0,1,2,\dots\,\right\}\,\right\}.$

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 $2^{\aleph_0} = \aleph_n$ , and moreover it is possible to assume $2^{\aleph_0}$ is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality $\aleph_0$ , meaning there is an unbounded function from $\aleph_0$ to it.

[edit] Aleph-α for general α

To define aleph-α for arbitrary ordinal number α, we need the successor cardinal operation, which assigns to any cardinal number ρ the next bigger cardinal $ρ +$ .

We can then define the aleph numbers as follows

$\aleph_{0} = \omega$

$\aleph_{\alpha+1} = \aleph_{\alpha}^+$

and for λ, an infinite limit ordinal,

$\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta.$

[edit] Fixed points of aleph

For any ordinal α we have

$\alpha\leq\aleph_\alpha.$

In many cases $\aleph_{\alpha}$ is strictly greater than α. For example, for any successor ordinal α this holds. There are, however, some limit ordinals which are fixed points of the aleph function. The first such is the limit of the sequence

$\aleph_0, \aleph_{\aleph_0}, \aleph_{\aleph_{\aleph_0}},\ldots$

Any inaccessible cardinal is a fixed point of the aleph function as well.

[edit] Popular culture

In the Futurama episode Raging Bender, the movie theater's name was ( $\aleph_0$ )PLEX. This is an obvious continuation of The Simpsons movie theater joke, based on googol, or more specifically googolplex -- the Googleplex. This also seems to be a reference to the mathematical curiosity known as Hilbert's Hotel, or Hilbert's paradox of the Grand Hotel.

There is also an industrial band on Deathkon Medias label with the name Aleph.Null. The band is known for mixing the genres of IDM and Classical with heavily distorted and overamped drums.