infinite number
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, transfinite numbers or infinite numbers are numbers that are " infinite" in the sense that they are larger than all
finite Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...
numbers. These include the transfinite cardinals, which are
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s used to quantify the size of infinite sets, and the transfinite ordinals, which are
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s used to provide an ordering of infinite sets. The term ''transfinite'' was coined in 1895 by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
, who wished to avoid some of the implications of the word ''infinite'' in connection with these objects, which were, nevertheless, not ''finite''. 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 Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions ...
: ''Leçons sur les nombres transfinis'' (1928 book) much expanded into '' Cardinal and Ordinal Numbers'' (1958, 2nd ed. 1965).


Definition

Any finite
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of marbles), whereas ordinal numbers specify the order of a member within an ordered set (e.g., "the man from the left" or "the day of January"). When extended to transfinite numbers, these two concepts are no longer in
one-to-one correspondence In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ...
. A transfinite cardinal number is used to describe the size of an infinitely large set, while a transfinite ordinal is used to describe the location within an infinitely large set that is ordered. The most notable ordinal and cardinal numbers are, respectively: *\omega (
Omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...
): the lowest transfinite ordinal number. It is also the
order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y su ...
of the
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s under their usual linear ordering. *\aleph_0 (
Aleph-null 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 ...
): the first transfinite cardinal number. It is also the
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
of the natural numbers. If the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
holds, the next higher cardinal number is
aleph-one 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 t ...
, \aleph_1. If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-null. Either way, there are no cardinals between aleph-null and aleph-one. The
continuum hypothesis In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
is the proposition that there are no intermediate cardinal numbers between \aleph_0 and the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \bold\mathfrak c (lowercase Fraktur "c") or \ ...
(the cardinality of the set of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s): or equivalently that \aleph_1 is the cardinality of the set of real numbers. In
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
, neither the continuum hypothesis nor its negation can be proved. Some authors, including P. Suppes and J. Rubin, use the term ''transfinite cardinal'' to refer to the cardinality of a
Dedekind-infinite set In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' onto s ...
in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the
axiom of countable choice The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain \mathbb ( ...
is not assumed or is not known to hold. Given this definition, the following are all equivalent: * \mathfrak is a transfinite cardinal. That is, there is a Dedekind infinite set A such that the cardinality of ''A'' is \mathfrak . * \mathfrak + 1 = \mathfrak. * \aleph_0 \leq \mathfrak. * There is a cardinal \mathfrak such that \aleph_0 + \mathfrak = \mathfrak. Although transfinite ordinals and cardinals both generalize only the natural numbers, other systems of numbers, including the
hyperreal number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer n
s and
surreal number In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value th ...
s, provide generalizations of the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s.


Examples

In Cantor's theory of ordinal numbers, every integer number must have a successor.
John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...
, (1976) ''
On Numbers and Games ''On Numbers and Games'' is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpr ...
''. Academic Press, ISBN 0-12-186350-6. ''(See Chapter 3.)''
The next integer after all the regular ones, that is the first infinite integer, is named \omega. In this context, \omega+1 is larger than \omega, and \omega\cdot2, \omega^ and \omega^ are larger still. Arithmetic expressions containing \omega specify an ordinal number, and can be thought of as the set of all integers up to that number. A given number generally has multiple expressions that represent it, however, there is a unique Cantor normal form that represents it, essentially a finite sequence of digits that give coefficients of descending powers of \omega. Not all infinite integers can be represented by a Cantor normal form however, and the first one that cannot is given by the limit \omega^ and is termed \varepsilon_. \varepsilon_ is the smallest solution to \omega^=\varepsilon, and the following solutions \varepsilon_, ...,\varepsilon_, ...,\varepsilon_, ... give larger ordinals still, and can be followed until one reaches the limit \varepsilon_, which is the first solution to \varepsilon_=\alpha. This means that in order to be able to specify all transfinite integers, one must think up an infinite sequence of names: because if one were to specify a single largest integer, one would then always be able to mention its larger successor. But as noted by Cantor, even this only allows one to reach the lowest class of transfinite numbers: those whose size of sets correspond to the cardinal number \aleph_.


See also

*
Actual infinity In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, involves infinite entities as given, actual and completed objects. The concept of actual infinity was introduced into mathematics near the en ...
*
Aleph number 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 t ...
*
Beth number In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0, \beth_1, \beth_2, \beth_3, \dots, where \beth is the Hebrew lett ...
*
Epsilon number In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chos ...
*
Infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
*
Transfinite induction 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. Induction by cases Let P(\alpha) be a property defined for a ...


References


Bibliography

*Levy, Azriel, 2002 (1978) ''Basic Set Theory''. Dover Publications. *O'Connor, J. J. and E. F. Robertson (1998)
Georg Ferdinand Ludwig Philipp Cantor
" MacTutor History of Mathematics archive. * Rubin, Jean E., 1967. "Set Theory for the Mathematician". San Francisco: Holden-Day. Grounded in
Morse–Kelley set theory In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
. *
Rudy Rucker Rudolf von Bitter Rucker (; born March 22, 1946) is an American mathematician, computer scientist, science fiction author, and one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known f ...
, 2005 (1982) ''Infinity and the Mind''. Princeton Univ. Press. Primarily an exploration of the philosophical implications of
Cantor's paradise ''Cantor's paradise'' is an expression used by in describing set theory and infinite cardinal numbers developed by Georg Cantor. The context of Hilbert's comment was his opposition to what he saw as L. E. J. Brouwer's reductive attempts to circu ...
. . *
Patrick Suppes Patrick Colonel Suppes (; March 17, 1922 – November 17, 2014) was an American philosopher who made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psycholog ...
, 1972 (1960)
Axiomatic Set Theory
. Dover. . Grounded in ZFC. {{Authority control Basic concepts in infinite set theory Cardinal numbers Ordinal numbers