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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 (). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
Note: This article contains special characters. ...
ALEPH (Apparatus for LEP Physics at CERN) is one of the four detectors of the LEP collider Categories: Stub | Particle detectors ...
The cardinality of the natural numbers is aleph-null () (also aleph-nought), the next larger cardinality is aleph-one , then comes and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number κ as will be described below. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
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 Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ...
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 ∞. Infinity has discrete meanings in mathematics, philosophy, theology and everyday life. ...
In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...
In mathematics, the real line is simply the set of real numbers. ...
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. The word Jew ( Hebrew: יהודי) is used in a wide number of ways, but generally refers to a follower of the Jewish faith, a child of a Jewish mother, or someone of Jewish descent with a connection to Jewish culture or ethnicity and often a combination of these attributes. ...
The tree of life. ...
The term God is used to designate a Supreme Being, however, there are countless definitions of God. ...
In the Jewish Kabbalah tradition, Ayn Sof (Ain Sof, Hebrew boundlessness or without end), also known referred to as Divine Being, is the name for God as he is unknown, or the mysterious and ultimate source of all existence. ...
Aleph-null Aleph-null (), 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 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. In mathematics, the axiom of choice is an axiom of set theory. ...
In mathematics the term countable set is used to describe the size of a set, e. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...
In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself. ...
In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
Aleph-one 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 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. ...
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 axiom of choice is an axiom of set theory. ...
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. 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. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
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. In Zermelo-Fraenkel set theory with the axiom of choice, the celebrated continuum hypothesis is equivalent to the identity The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number. ...
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In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...
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. In mathematics, the axiom of choice is an axiom of set theory. ...
Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ...
Paul Joseph Cohen (born April 2, 1934) is 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 . 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. ...
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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 ρ + . In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. ...
We can then define the aleph numbers as follows
and for λ an infinite limit ordinal,
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