Inset theory, aregular cardinal is acardinal number that is equal to its owncofinality. More explicitly, this means that${\displaystyle \kappa }$ is a regular cardinal if and only if everyunbounded subset${\displaystyle C\subseteq \kappa }$ has cardinality${\displaystyle \kappa }$. Infinitewell-ordered cardinals that are not regular are calledsingular cardinals. Finite cardinal numbers are typically not called regular or singular.

In the presence of theaxiom of choice, any cardinal number can be well-ordered, and then the following are equivalent for a cardinal${\displaystyle \kappa }$:

1. ${\displaystyle \kappa }$ is a regular cardinal.
2. If${\displaystyle \kappa =\sum _{i\in I}{\lambda }_i}$ and for all${\displaystyle i}$, then${\displaystyle |I|\ge \kappa }$.
3. If${\displaystyle S=\bigcup _{i\in I}{S}_i}$, and if and for all${\displaystyle i}$, then. That is, every union of fewer than${\displaystyle \kappa }$ sets smaller than${\displaystyle \kappa }$ is smaller than${\displaystyle \kappa }$.
4. Thecategory of sets of cardinality less than${\displaystyle \kappa }$ and all functions between them is closed undercolimits of cardinality less than${\displaystyle \kappa }$.
5. ${\displaystyle \kappa }$ is a regular ordinal (see below).

Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts.

The situation is slightly more complicated in contexts where theaxiom of choice might fail, as in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above equivalence holds for well-orderable cardinals only.

An infiniteordinal${\displaystyle \alpha }$ is aregular ordinal if it is alimit ordinal that is not the limit of a set of smaller ordinals that as a set hasorder type less than${\displaystyle \alpha }$. A regular ordinal is always aninitial ordinal, though some initial ordinals are not regular, e.g.,${\displaystyle {\omega }_{\omega }}$ (see the example below).

The ordinals less than${\displaystyle \omega }$ are finite. A finite sequence of finite ordinals always has a finite maximum, so${\displaystyle \omega }$ cannot be the limit of any sequence of type less than${\displaystyle \omega }$ whose elements are ordinals less than${\displaystyle \omega }$, and is therefore a regular ordinal.${\displaystyle {\mathrm{\aleph }}_0}$ (aleph-null) is a regular cardinal because its initial ordinal,${\displaystyle \omega }$, 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.

${\displaystyle \omega +1}$ is thenext ordinal number greater than${\displaystyle \omega }$. It is singular, since it is not a limit ordinal.${\displaystyle \omega +\omega }$ is the next limit ordinal after${\displaystyle \omega }$. It can be written as the limit of the sequence${\displaystyle \omega }$,${\displaystyle \omega +1}$,${\displaystyle \omega +2}$,${\displaystyle \omega +3}$, and so on. This sequence has order type${\displaystyle \omega }$, so${\displaystyle \omega +\omega }$ is the limit of a sequence of type less than${\displaystyle \omega +\omega }$ whose elements are ordinals less than${\displaystyle \omega +\omega }$; therefore it is singular.

${\displaystyle {\mathrm{\aleph }}_1}$ is thenext cardinal number greater than${\displaystyle {\mathrm{\aleph }}_0}$, so the cardinals less than${\displaystyle {\mathrm{\aleph }}_1}$ arecountable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So${\displaystyle {\mathrm{\aleph }}_1}$ cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.

${\displaystyle {\mathrm{\aleph }}_{\omega }}$ is the next cardinal number after the sequence${\displaystyle {\mathrm{\aleph }}_0}$,${\displaystyle {\mathrm{\aleph }}_1}$,${\displaystyle {\mathrm{\aleph }}_2}$,${\displaystyle {\mathrm{\aleph }}_3}$, and so on. Its initial ordinal${\displaystyle {\omega }_{\omega }}$ is the limit of the sequence${\displaystyle \omega }$,${\displaystyle {\omega }_1}$,${\displaystyle {\omega }_2}$,${\displaystyle {\omega }_3}$, and so on, which has order type${\displaystyle \omega }$, so${\displaystyle {\omega }_{\omega }}$ is singular, and so is${\displaystyle {\mathrm{\aleph }}_{\omega }}$. Assuming the axiom of choice,${\displaystyle {\mathrm{\aleph }}_{\omega }}$ is the first infinite cardinal that is singular (the first infiniteordinal that is singular is${\displaystyle \omega +1}$, and the first infinitelimit ordinal that is singular is${\displaystyle \omega +\omega }$). Proving the existence of singular cardinals requires theaxiom of replacement, and in fact the inability to prove the existence of${\displaystyle {\mathrm{\aleph }}_{\omega }}$ inZermelo set theory is what ledFraenkel to postulate this axiom.^[1]

Uncountable (weak)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 necessarilyfixed points of thealeph function, though not all fixed points are regular. For instance, the first fixed point is the limit of the${\displaystyle \omega }$-sequence${\displaystyle {\mathrm{\aleph }}_0,{\mathrm{\aleph }}_{\omega },{\mathrm{\aleph }}_{{\omega }_{\omega }},...}$ and is therefore singular.

If theaxiom of choice holds, then everysuccessor 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 cardinalities cannot be proven to be equal to any particular aleph, for instance thecardinality of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (seeEaston's theorem). Thecontinuum hypothesis postulates that the cardinality of the continuum is equal to${\displaystyle {\mathrm{\aleph }}_1}$, which is regular assuming choice.

Without the axiom of choice: there would be cardinal numbers that were not well-orderable.^{[citation needed]} Moreover, the cardinal sum of an arbitrary collection could not be defined.^{[citation needed]} Therefore, only thealeph numbers could meaningfully be called regular or singular cardinals.^{[citation needed]}Furthermore, a successor aleph would need not be regular. For instance, the union of a countable set of countable sets would not necessarily be countable. It is consistent withZF that${\displaystyle {\omega }_1}$ be the limit of a countable sequence of countable ordinals as well as the set ofreal numbers be a countable union of countable sets.^{[citation needed]} Furthermore, it is consistent with ZF when not including AC that every aleph bigger than${\displaystyle {\mathrm{\aleph }}_0}$ is singular (a result proved byMoti Gitik).

If${\displaystyle \kappa }$ is a limit ordinal,${\displaystyle \kappa }$ is regular iff the set of that are critical points of${\displaystyle {\mathrm{\Sigma }}_1}$-elementary embeddings${\displaystyle j}$ with${\displaystyle j(\alpha )=\kappa }$ isclub in${\displaystyle \kappa }$.^[2]

For cardinals, say that an elementary embedding${\displaystyle j:M\to H(\theta )}$ asmall embedding if${\displaystyle M}$ is transitive and${\displaystyle j(\text{crit}(j))=\kappa }$. A cardinal${\displaystyle \kappa }$ is uncountable and regular iff there is an${\displaystyle \alpha >\kappa }$ such that for every${\displaystyle \theta >\alpha }$, there is a small embedding${\displaystyle j:M\to H(\theta )}$.^{[3]Corollary 2.2}

• Inaccessible cardinal

• Herbert B. Enderton,Elements of Set Theory,ISBN 0-12-238440-7
• Kenneth Kunen,Set Theory, An Introduction to Independence Proofs,ISBN 0-444-85401-0

• Cardinal numbers
• Ordinal numbers

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