In set theory, a regular cardinal is an infinite well-orderable cardinal whose initial ordinal is regular, where a regular ordinal is an ordinal which is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... In mathematics, a well-order (or well-ordering) 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. ... 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). ... The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a ≤ b. ...

If the axiom of choice holds (so that any cardinal number can be well-ordered), an infinite cardinal κ is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than κ, the elements of which are cardinals less than κ. An infinite ordinal α is regular if and only if it is a limit ordinal which is not the limit of a set of smaller ordinals which set has order type less than α. A regular ordinal is always an initial ordinal, though some initial ordinals are not regular. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... It has been suggested that this article or section be merged with Logical biconditional. ... A limit ordinal is an ordinal number which is not a successor ordinal. ... In mathematics, especially in set theory, ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 0, the one after that 1, the next one 2, and so on) and to measure the length of the whole set by the least...

Infinite well-ordered cardinals which are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.

Examples

The ordinals less than ω are finite. A finite sequence of finite ordinals always has a finite maximum, so ω cannot be the limit of any sequence of type less than ω whose elements are ordinals less than ω, and is therefore a regular ordinal. (aleph-null) is a regular cardinal because its initial ordinal, ω, is regular. It can also be seen directly to be regular, as the cardinal sum of a finite number of finite cardinal numbers is itself finite. 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. ...

ω + 1 is the next ordinal number greater than ω. It is singular, since it is not a limit ordinal. ω + ω is the next limit ordinal after ω. It can be written as the limit of the sequence ω, ω + 1, ω + 2, ω + 3, and so on. This sequence has order type ω, so ω + ω is the limit of a sequence of type less than ω + ω whose elements are ordinals less than ω + ω, therefore it is singular. When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one. ...

is the next cardinal number greater than , so the cardinals less than are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So cannot be written as the sum of a countable set of countable cardinal numbers, and is regular. In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. ... In mathematics, a countable set is a set with the same cardinality (i. ...

is the next cardinal number after the sequence , , , , and so on. Its initial ordinal ω_ω is the limit of the sequence ω, ω₁, ω₂, ω₃, and so on, which has order type ω, so ω_ω is singular, and so is . Assuming the axiom of choice, is the first infinite cardinal which is singular (the first infinite ordinal which is singular is ω + 1). Proving the existence of singular cardinals requires the axiom of replacement, and in fact the inability to prove the existence of in Zermelo set theory is what led Fraenkel to postulate this axiom. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ... Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. ... Adolf Abraham Halevi Fraenkel (February 17, 1891 - October 15, 1965), known as Abraham Fraenkel, was a German / Israeli mathematician. ...

Properties

Uncountable limit cardinals that are also regular are known as weakly inaccessible cardinals. They cannot be proved to exist within ZFC, though their existence is not known to be inconsistent with ZFC. Their existence is sometimes taken as an additional axiom. Inaccessible cardinals are necessarily fixed points of the aleph function, though not all fixed points are regular. For instance, the first fixed point is the limit of the ω-sequence and is therefore singular. In set theory, a cardinal number is called weakly inaccessible if it is an uncountable regular weak limit cardinal and strongly inaccessible, or just inaccessible, if it is an uncountable regular strong limit cardinal. ... In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ... In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ...

If the axiom of choice holds, then every successor cardinal is regular. Thus the regularity or singularity of most aleph numbers can be checked depending on whether the cardinal is a successor cardinal or a limit cardinal. Some cardinal numbers cannot be proven to be equal to any particular aleph, for instance the cardinality of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality. The continuum hypothesis postulates that the cardinality of the continuum is equal to which is regular. In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

Without the axiom of choice, there would be cardinal numbers which were not well-orderable. Moreover, the cardinal sum of an arbitrary collection could not be defined. Therefore only the aleph numbers can meaningfully be called regular or singular cardinals. Furthermore, a successor aleph need not be regular. For instance, the union of a countable set of countable sets need not be countable. It is consistent with ZF that ω₁ be the limit of a countable sequence of countable ordinals. In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ... Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...

See also

• Inaccessible cardinal

In mathematics, a cardinal number k > (aleph-null) is called weakly inaccessible, or just inaccessible, if the following two conditions hold. ...

References