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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, transfinite numbers are numbers that are "
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
" in the sense that they are larger than all
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * 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 marke ...
numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
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 least n ...
s used to provide an ordering of infinite sets. The term ''transfinite'' was coined by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
in 1895, 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.


Definition

Any finite
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
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 five marbles), whereas ordinal numbers specify the order of a member within an ordered set (e.g., "the third man from the left" or "the twenty-seventh day of January"). When extended to transfinite numbers, these two concepts become distinct. 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 (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/isopsephy (gematria), it has a value of 800. The wo ...
): 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 such ...
of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
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 that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named af ...
): the first transfinite cardinal number. It is also the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the natural numbers. If the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
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 that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named ...
, \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, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
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 \mathfrak c (lowercase fraktur "c") or , \mathb ...
(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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 such as ...
, neither the continuum hypothesis nor its negation can be proven. 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 ...
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 N (where ...
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, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
s and
surreal number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
, (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 unpre ...
''. 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 In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...
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 involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extend ...
*
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 second H ...
*
Epsilon numbers (mathematics) 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 quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...


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 The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathemati ...
. * Rubin, Jean E., 1967. "Set Theory for the Mathematician". San Francisco: Holden-Day. Grounded in Morse–Kelley set theory. *
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. . *
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, psychology ...
, 1972 (1960)
Axiomatic Set Theory
. Dover. . Grounded in ZFC. {{Authority control Basic concepts in infinite set theory Cardinal numbers Ordinal numbers