- Aleph number
In the branch of

mathematics known asset theory , the**aleph numbers**are a sequence of numbers used to represent thecardinality (or size) ofinfinite set s. They are named after the symbol used to denote them, the Hebrew letter aleph ($aleph$).The cardinality of the

natural number s is $aleph\_0$ (aleph-null, also aleph-naught or aleph-zero), the next larger cardinality is aleph-one $aleph\_1$, then $aleph\_2$ and so on. Continuing in this manner, it is possible to define acardinal number $aleph\_alpha$ for everyordinal number α, as 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 (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of theextended real number line .**Aleph-null**Aleph-null ($aleph\_0$) is by definition the cardinality of the set of all

natural number s, and (assuming, as usual, theaxiom of choice ) is the smallest of all infinite cardinalities. A set has cardinality $aleph\_0$ if and only if it iscountably infinite , which is the case if and only if it can be put into a directbijection , or "one-to-one correspondence", with the natural numbers. Such sets include the set of allprime number s, the set of allinteger s, the set of allrational number s, the set ofalgebraic number s, and the set of all finitesubset s of anycountably infinite set.**Aleph-one**$aleph\_1$ is the cardinality of the set of all countable

ordinal number s, called**ω**or_{1}**Ω**. Note that this**ω**is itself an ordinal number larger than all countable ones, so it is an_{1}uncountable set . Therefore $aleph\_1$ is distinct from $aleph\_0$. The definition of $aleph\_1$ implies (in ZF,Zermelo-Fraenkel set theory "without" the axiom of choice) that no cardinal number is between $aleph\_0$ and $aleph\_1$. If theaxiom of choice (AC) is used, it can be further proved that the class of cardinal numbers istotally 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**: any countable subset of**Ω**has an upper bound in**Ω**. (This follows from the fact that a countable union of countable sets is countable, one of the most common applications of AC.) This fact is analogous to the situation in $aleph\_0$: any finite set of natural numbers has a maximum which is also a natural number; that is, 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 (vector space s, 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, viatransfinite 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 ofreal number s (cardinality of the continuum ) 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 theaxiom of choice ) that the celebratedcontinuum 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.**Aleph-ω**Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number $aleph\_omega$ is the smallest upper bound of

:$left\{,aleph\_n\; :\; ninleft\{,0,1,2,dots,\; ight\},\; ight\}.$

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 number s; 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 withcofinality $aleph\_0$, meaning there is an unbounded function from $aleph\_0$ to it.**Aleph-α for general α**To define $aleph\_alpha$ for arbitrary ordinal number $alpha$, we must define the successor cardinal operation, which assigns to any cardinal number $ho$ the next bigger

well-order ed cardinal $ho^+$. (If theaxiom of choice holds, this is the next bigger cardinal.)We can then define the aleph numbers as follows

:$aleph\_\{0\}\; =\; omega$:$aleph\_\{alpha+1\}\; =\; aleph\_\{alpha\}^+$

and for λ, an infinite limit ordinal,

:$aleph\_\{lambda\}\; =\; igcup\_\{eta\; lambda\}\; aleph\_eta.$

The α-th infinite initial ordinal is written $omega\_alpha$. Its cardinality is written $aleph\_alpha$. See

initial ordinal .**Fixed points of aleph**For any ordinal α we have:$alphaleqaleph\_alpha.$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, because of the

fixed-point lemma for normal functions . The first such is the limit of the sequence:$aleph\_0,\; aleph\_\{aleph\_0\},\; aleph\_\{aleph\_\{aleph\_0,ldots$Anyinaccessible cardinal is a fixed point of the aleph function as well.**Aleph number in popular culture*** The theme of the infinite runs throughout the work of

Jorge Luis Borges , whose short story "The Aleph" (" _es. El Aleph") deals with a point in space that contains all other points, seen from all possible angles, at all possible times.* In the "

Futurama " episode "Raging Bender ", the movie theater's name isLoew's $aleph\_0$-plex.* The science fiction novel "White Light" by

Rudy Rucker uses an imaginary universe to elucidate theset theory concept of aleph numbers.* The science fiction novel "

The Infinitive of Go " by John Brunner concerns ateleportation device based on transfinite mathematics which gives access to amultiverse of parallel realities whose cardinality is "at least "aleph-four".*

Scarlett Thomas 's book "PopCo ", features both a discussion of aleph-null and several events of importance that involve the concept.* Aleph One is the name of the open-source project for

Bungie Studios ' Marathon series of computer games. The last game of the series is entitled "Marathon Infinity ", so Aleph was chosen as the name because it was "going beyond Infinity".**External links***

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**Aleph (disambiguation)**— * Aleph or Alef is the first letter of the Semitic abjads descended from Proto Canaanite, Arabic alphabet, Phoenician alphabet, Hebrew alphabet, Syriac alphabet.Named after the letterPeople* Aleph (musician), an Italo disco artist and alias of… … Wikipedia**Aleph One (disambiguation)**— Aleph One may be: * aleph 1, the second aleph number * Elias Levy, known as Aleph One as a Bugtraq moderator * Aleph One (computer game), an enhanced cross platform version of the Marathon game engine … Wikipedia**Aleph Zero Records**— Aleph Zero Records, named after the famous aleph number, is an Israeli record label cofounded by Yaniv Shulman and Shahar Bar Itzhak. Artists * Bluetech * Omnimotion * Shulman See also * List of record labelsExternal links* [http://aleph… … Wikipedia**Aleph (psychedelic)**— IUPAC name 2 (2,5 Dimethoxy 4 methylsulfanyl phenyl) 1 methylethylamine … Wikipedia**Aleph kernel**— Aleph was an operating system kernel developed at the University of Rochester as part of their RIG project in 1975. Aleph used inter process communications to move data between programs and the kernel, so applications could transparently access… … Wikipedia**aleph-null**— [ä′lifzir′ōä′lifnul′] n. Math. in the theory of sets, the smallest infinite cardinal number; the cardinal number of the set of all positive integers: symbol, (א0): also called aleph zero [ä′lifzir′ō] … English World dictionary**Aleph One (computer game)**— infobox software name = Aleph One caption = Aleph One screenshot released = 17 January 2000 latest release version = 0.20.3 latest release date = September 2008 operating system = Cross platform platform = SDL language = genre = first person… … Wikipedia**Aleph Zadik Aleph**— The International Order of the Aleph Zadik Aleph (AZA) is an international youth led fraternal organization for Jewish teenagers, founded in 1924 and currently existing as the male wing of the B nai B rith Youth Organization (BBYO, Inc.), an… … Wikipedia**aleph-null**— noun the smallest infinite integer • Syn: ↑aleph nought, ↑aleph zero • Hypernyms: ↑large integer * * * | ̷ ̷(ˌ) ̷ ̷| ̷ ̷ noun also aleph zero | ̷ ̷(ˌ) ̷ ̷ˈ ̷ ̷(ˌ) ̷ ̷ … Useful english dictionary**aleph-null**— noun The first of the transfinite cardinal numbers; corresponds to the number of positive integers, also called natural numbers. Georg Cantor showed that even all the rational numbers could be put in one to one correspondence with them, and are… … Wiktionary