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 (). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In set theory, an infinite set is a set that is not a finite set. ... Note: This article contains special characters. ... is the reconstructed name of the first letter of the Proto-Canaanite alphabet, continued in descended Semitic alphabets as Phoenician , Syriac , Hebrew , , and Arabic . Aleph originally expressed the glottal stop (IPA ), usually transliterated as , a symbol based on the Greek spiritus lenis , for example in the transliteration of the letter...

The cardinality of the natural numbers is (aleph-null, also aleph-naught, aleph-zero or aleph-nought); the next larger cardinality is aleph-one , then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number α, as described below. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ...

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 ∞. The infinity symbol ∞ in several typefaces. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... In mathematics, the real line is simply the set of real numbers. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

Aleph-null

Aleph-null () 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 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, the set of all integers and the set of all rational numbers. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics the term countable set is used to describe the size of a set, e. ... A bijective function. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a prime number (or a prime) is a positive integer that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...

Aleph-one

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 and . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus 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 ): 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 : 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. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

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. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Group theory is that branch of mathematics concerned with the study of groups. ... Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ...

The continuum hypothesis

The cardinality of the set of real numbers is . 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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

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. [...]I dont believe in natural science. ... Paul Joseph Cohen (April 2, 1934 – March 23, 2007[1]) was an American mathematician. ...

Aleph-ω

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number is the smallest upper bound of

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 , and moreover it is possible to assume is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality , meaning there is an unbounded function from to it. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a ≤ b. ...

Aleph-α for general α

To define aleph-α for arbitrary ordinal number α, we must define the successor cardinal operation, which assigns to any cardinal number ρ the next bigger well-ordered cardinal ρ ⁺ . (If the axiom of choice holds, this is the next bigger cardinal.) In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

We can then define the aleph numbers as follows

and for λ, an infinite limit ordinal,

The α-th infinite initial ordinal is written ω_α. Its cardinality is written . See initial ordinal. The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ...

Fixed points of aleph

For any ordinal α we have

In many cases 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 In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...

Any inaccessible cardinal is a fixed point of the aleph function as well. In set theory, a cardinal number is called weakly inaccessible if it is an uncountable regular weak limit cardinal and strongly inaccessible, or just inaccessible, if it is an uncountable regular strong limit cardinal. ...

Aleph number in popular culture

• The theme of the infinite runs throughout the work of Jorge Luis Borges, whose short story "The Aleph" ("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 was ()PLEX. This is an obvious continuation of The Simpsons movie theater joke, based on googol, or more specifically googolplex—the Googolplex. 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.

• The science fiction novel White Light by Rudy Rucker uses an imaginary universe to elucidate the set theory concept of aleph numbers.

• The science fiction novel The Infinitive of Go by John Brunner concerns a teleportation device based on transfinite mathematics which gives access to a multiverse of parallel realities whose cardinality is "at least aleph-four".

• Faust Aleph-null is the alternative title of the fantasy novel Black Easter by James Blish.

• The novel "Zimmerman's Algorithm" uses aleph-null to stand for a group creating a godlike artificial intelligence.

Jorge Luis Borges (August 24, 1899 – June 14, 1986) was an Argentine writer who is considered one of the foremost literary figures of the 20th century. ... The Aleph is a short story by the Argentinian writer and poet Jorge Luis Borges. ... Futurama is an Emmy Award-winning animated sitcom created by Matt Groening (creator of The Simpsons) and David X. Cohen for the Fox network, and will resume airing in 2008 on Comedy Central. ... Raging Bender is episode 8 in season 2 of Futurama. ... Simpsons redirects here. ... A googol is the large number 10100, that is, the digit 1 followed by one hundred zeros (in decimal representation). ... Hilberts paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (1862 – 1943): In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. ... White Light is a work of science fiction by Rudy Rucker published in 1980 by Ace Books. ... Rudolf Rucker, Fall 2005. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... The Infinitive of Go is a science fiction novel by John Brunner. ... Notable people named John Brunner include: John Brunner (industrialist) John Brunner (novelist) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Teleportation is the movement of objects or elementary particles from one place to another, more or less instantaneously, without traveling through space. ... A multiverse is a set of many universes. ... Hugo Award nominated fantasy novel by James Blish in which an arms dealer hires a black magacian to unleash all the Demons of Hell on earth for a single day. ... James Benjamin Blish (East Orange, New Jersey, May 23, 1921 - Henley-on-Thames, July 29, 1975) was an American author of fantasy and science fiction. ...

See also

• Georg Cantor
• Cardinal number
• Uncountable set
• Continuum hypothesis
• Beth number

Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...

External links

• Eric W. Weisstein, Aleph-0 at MathWorld.
• aleph numbers at PlanetMath.