Ordinal number in the context of Enumeration

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Konstantin Gutberlet [de; it] was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications).

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Georg Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name.

In set theory and its applications throughout mathematics, a class is a collection of mathematical objects (often sets) that can be unambiguously defined by a property that all its members share. Classes act as a way to have set-like collections while differing from sets so as to avoid paradoxes, especially Russell's paradox (see § Paradoxes). The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.

A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.

In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or ${\displaystyle \Delta _{2}^{1}}$ functions of the theory.

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.

A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them. This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the set-theoretic paradoxes, and for stating the axiom of global choice, which is stronger than ZFC's axiom of choice.

The knuckles are the joints of the fingers. The word is cognate to similar words in other Germanic languages, such as the Dutch "knokkel" (knuckle) or German "Knöchel" (ankle), i.e., Knöchlein, the diminutive of the German word for bone (Knochen). Anatomically, it is said that the knuckles consist of the metacarpophalangeal (MCP) and interphalangeal (IP) joints of the finger. The knuckles at the base of the fingers may be referred to as the 1st or major knuckles while the knuckles at the midfinger are known as the 2nd and 3rd, or minor, knuckles. However, the ordinal terms are used inconsistently and may refer to any of the knuckles.

In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term transfinite was coined in 1895 by Georg Cantor, who wished to avoid some of the implications of the word infinite. In particular he believed that "truly infinite" is a perfect and thus divine quality and so refused to attribute this term to mathematical constructs comprehensible by humans. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers. Nevertheless, the term transfinite also remains in use.

Notable work on transfinite numbers was done by Wacław Sierpiński: Leçons sur les nombres transfinis (1928 book) much expanded into Cardinal and Ordinal Numbers (1958, 2nd ed. 1965).

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

The smallest cardinality of an infinite set is that of the natural numbers, denoted by ${\displaystyle \aleph _{0}}$ (read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-ordered set is ${\displaystyle \aleph _{1},}$ then ${\displaystyle \aleph _{2},}$ then ${\displaystyle \aleph _{3},}$ and so on. Continuing in this manner, it is possible to define an infinite cardinal number ${\displaystyle \aleph _{\alpha }}$ for every ordinal number ${\displaystyle \alpha ,}$ as described below.

In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.

The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations (see ordinal analysis). However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available.

Since there are only countably many notations, all ordinals with notations are exhausted well below the first uncountable ordinal ω₁; their supremum is called Church–Kleene ω₁ or ω
₁ (not to be confused with the first uncountable ordinal, ω₁), described below. Ordinal numbers below ω
₁ are the recursive ordinals (see below). Countable ordinals larger than this may still be defined, but do not have notations.

Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Konstantin Gutberlet (de; it) was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications).

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.

The following are tables of the oldest people in the world in ordinal ranks. To avoid including false or unconfirmed claims of old age, names here are restricted to those people whose ages have been validated by an international body dealing in longevity research, such as the Gerontology Research Group, LongeviQuest, or Guinness World Records, and others who have otherwise been reliably sourced.

The longest documented and verified human lifespan is that of Jeanne Calment of France (1875–1997), a woman who lived to the age of 122 years and 164 days. As women live longer than men on average, women predominate in combined records. The longest lifespan for a man is that of Jiroemon Kimura of Japan (1897–2013), who lived to the age of 116 years and 54 days.

The symbol # is known as the number sign, hash, (in North America) the pound sign, and has a variety of other names. The symbol has historically been used for a wide range of purposes including the designation of an ordinal number and as a ligatured abbreviation for pounds avoirdupois – having been derived from the now-rare ℔.

Since 2007, widespread usage of the symbol to introduce metadata tags on social media platforms has led to such tags being known as "hashtags", and from that, the symbol itself is sometimes incorrectly called a hashtag.

In formal language theory within theoretical computer science, an infinite word is an infinite-length sequence (specifically, an ω-length sequence) of symbols, and an ω-language is a set of infinite words. Here, ω refers to the first infinite ordinal number, modeling a set of natural numbers.

One millionth is equal to 0.000 001, or 1 x 10 in scientific notation. It is the reciprocal of a million, and can be also written as 1⁄1,000,000. Units using this fraction can be indicated using the prefix "micro-" from Greek, meaning "small". Numbers of this quantity are expressed in terms of μ (the Greek letter mu).

"Millionth" can also mean the ordinal number that comes after the nine hundred, ninety-nine thousand, nine hundred, ninety-ninth and before the million and first.