In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, a limit ordinal is an
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 ...
that is neither zero nor a
successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal.
For example,
ω, the smallest ordinal greater than every
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 ...
is a limit ordinal because for any smaller ordinal (i.e., for any natural number) ''n'' we can find another natural number larger than it (e.g. ''n''+1), but still less than ω.
Using the
von Neumann definition of ordinals, every ordinal is the
well-ordered set
In mathematics, a well-order (or well-ordering or well-order relation) 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. The set ''S'' together with the well-ord ...
of all smaller ordinals. The union of a nonempty set of ordinals that has no
greatest element is then always a limit ordinal. Using
von Neumann cardinal assignment, every infinite
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. ...
is also a limit ordinal.
Alternative definitions
Various other ways to define limit ordinals are:
*It is equal to the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
of all the ordinals below it, but is not zero. (Compare with a successor ordinal: the set of ordinals below it has a maximum, so the supremum is this maximum, the previous ordinal.)
*It is not zero and has no maximum element.
*It can be written in the form ωα for α > 0. That is, in the
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 ...
there is no finite number as last term, and the ordinal is nonzero.
*It is a limit point of the class of ordinal numbers, with respect to the
order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If ''X'' is a totally ordered set, th ...
. (The other ordinals are
isolated points.)
Some contention exists on whether or not 0 should be classified as a limit ordinal, as it does not have an immediate predecessor;
some textbooks include 0 in the class of limit ordinals while others exclude it.
[for example, Kenneth Kunen, ''Set Theory. An introduction to independence proofs''. North-Holland.]
Examples
Because the
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently ...
of ordinal numbers is
well-order
In mathematics, a well-order (or well-ordering or well-order relation) 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. The set ''S'' together with the well-or ...
ed, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding ''limit''), as it is the
least upper bound of the
natural numbers
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 ...
. Hence ω represents 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 suc ...
of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·''n'' for any natural number ''n''. Taking the
union (the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
operation on any
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of ordinals) of all the ω·n, we get ω·ω = ω
2, which generalizes to ω
''n'' for any natural number ''n''. This process can be further iterated as follows to produce:
:
In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. All of the ordinals discussed so far are still
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
ordinals. However, there is no
recursively enumerable scheme for
systematically naming all ordinals less than the
Church–Kleene ordinal, which is a countable ordinal.
Beyond the countable, the
first uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. W ...
is usually denoted ω
1. It is also a limit ordinal.
Continuing, one can obtain the following (all of which are now increasing in cardinality):
:
In general, we always get a limit ordinal when taking the union of a nonempty set of ordinals that has no
maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
element.
The ordinals of the form ω²α, for α > 0, are limits of limits, etc.
Properties
The classes of successor ordinals and limit ordinals (of various
cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by
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 ...
or definitions by
transfinite recursion
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 ...
. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
in the order topology and this is usually desirable.
If we use the
von Neumann cardinal assignment, every infinite
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. ...
is also a limit ordinal (and this is a fitting observation, as ''cardinal'' derives from the Latin ''cardo'' meaning ''hinge'' or ''turning point''): the proof of this fact is done by simply showing that every infinite successor ordinal is
equinumerous to a limit ordinal via the
Hotel Infinity argument.
Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level).
Indecomposable ordinals
Additively indecomposable
A limit ordinal α is called additively indecomposable if it cannot be expressed as the sum of β < α ordinals less than α. These numbers are any ordinal of the form
for β an ordinal. The smallest is written
, the second is written
, etc.
Multiplicatively indecomposable
A limit ordinal α is called multiplicatively indecomposable if it cannot be expressed as the product of β < α ordinals less than α. These numbers are any ordinal of the form
for β an ordinal. The smallest is written
, the second is written
, etc.
Exponentially indecomposable and beyond
The term "exponentially indecomposable" does not refer to ordinals not expressible as the exponential product ''(?)'' of β < α ordinals less than α, but rather the
epsilon numbers, "tetrationally indecomposable" refers to the zeta numbers, "pentationally indecomposable" refers to the eta numbers, etc.
See also
*
Ordinal arithmetic
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 expl ...
*
Limit cardinal
*
Fundamental sequence (ordinals)
References
Further reading
*
Cantor, G., (1897), ''Beitrage zur Begrundung der transfiniten Mengenlehre. II'' (tr.: Contributions to the Founding of the Theory of Transfinite Numbers II), Mathematische Annalen 49, 207-24
English translation
*
Conway, J. H. and
Guy, R. K. "Cantor's Ordinal Numbers." In ''The Book of Numbers''. New York: Springer-Verlag, pp. 266–267 and 274, 1996.
* Sierpiński, W. (1965). ''
Cardinal and Ordinal Numbers'' (2nd ed.). Warszawa: Państwowe Wydawnictwo Naukowe. Also defines ordinal operations in terms of the Cantor Normal Form.
{{DEFAULTSORT:Limit Ordinal
Ordinal numbers