# Regular cardinal
A **regular cardinal** is an infinite cardinal number \( \kappa \) whose [cofinality](https://www.edgepedia.org/cofinality) equals itself, written \( \operatorname{cf}(\kappa) = \kappa \); a cardinal that is not regular is called **singular**[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf). The cofinality \( \operatorname{cf}(\kappa) \) is the smallest size of a cofinal subset of \( \kappa \), that is, a subset unbounded in \( \kappa \)[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf). Equivalently, \( \kappa \) is regular when it is not the union of fewer than \( \kappa \) many sets, each of size less than \( \kappa \)[2](https://iep.utm.edu/set-theo/). The regular/singular division, introduced by [Felix Hausdorff](https://www.edgepedia.org/felix-hausdorff) in his 1906 paper on order-types, splits infinite cardinals into two classes with very different arithmetic behavior[3](https://www.sciencedirect.com/topics/mathematics/hausdorff) • [1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf).
| Key fact | Detail |
|---|---|
| Definition | \( \kappa \) is regular if \( \operatorname{cf}(\kappa) = \kappa \); otherwise singular[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf) |
| Union form | \( \kappa \) is regular iff it is not a union of fewer than \( \kappa \) sets of size less than \( \kappa \)[2](https://iep.utm.edu/set-theo/) |
| Successor cardinals | Every infinite successor cardinal \( \kappa^+ \) is regular[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf) |
| First singular cardinal | \( \aleph_\omega \), with \( \operatorname{cf}(\aleph_\omega) = \omega \)[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf) |
| König's corollary | \( \kappa < \kappa^{\operatorname{cf}(\kappa)} \); in particular \( \operatorname{cf}(2^{\aleph_0}) > \aleph_0 \)[4](https://math.colorado.edu/~kearnes/Teaching/Courses/F25/setsl.html) |
| Inaccessibility | An uncountable regular limit cardinal is weakly inaccessible; adding the strong limit property gives a strongly inaccessible cardinal, for which \( V_\kappa \) is a model of ZFC[5](https://plato.stanford.edu/entries/set-theory/) |
| Category-theoretic role | Regular cardinals index \( \lambda \)-filtered colimits, \( \lambda \)-presentable objects, and locally \( \lambda \)-presentable categories[6](https://ncatlab.org/nlab/show/regular%2Bcardinal) |
## Cofinality and the definition
For an infinite [limit ordinal](https://www.edgepedia.org/limit-ordinal) \( \alpha \), the cofinality \( \operatorname{cf}(\alpha) \) is the least length of an increasing sequence cofinal in \( \alpha \), equivalently the smallest cardinality of a cofinal subset[7](https://sites.math.washington.edu/~palmieri/Courses/2004/Math583/set-theory.html) • [1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf). Two basic facts follow immediately: \( \operatorname{cf}(\operatorname{cf}(\alpha)) = \operatorname{cf}(\alpha) \), and \( \operatorname{cf}(\alpha) \) is always a cardinal[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf). In particular \( \operatorname{cf}(\kappa) \) is a regular cardinal for every infinite cardinal \( \kappa \)[8](https://spot.colorado.edu/~szendrei/ST_S21/hw.html).
The union formulation makes the meaning concrete.
## Which cardinals are regular
**Successor cardinals are always regular.** In ZFC every infinite successor cardinal \( \kappa^+ \) satisfies \( \operatorname{cf}(\kappa^+) = \kappa^+ \)[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf). Thus \( \aleph_1, \aleph_2, \aleph_3, \ldots \) are all regular, and \( \omega = \aleph_0 \) is regular as well[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf).
**Limit cardinals are typically singular.** The first infinite singular cardinal is \( \aleph_\omega \): since \( \omega \) is regular and each \( \aleph_n \) for \( n > 0 \) is a successor cardinal, the first possible failure occurs at the limit index \( \omega \), and indeed \( \operatorname{cf}(\aleph_\omega) = \omega \)[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf). More generally \( \aleph_{\alpha+\omega} \) is singular for any ordinal \( \alpha \), since \( \operatorname{cf}(\aleph_{\alpha+\omega}) = \aleph_0 \)[9](https://math.colorado.edu/~kearnes/Teaching/Courses/setsl.html). The usual limit cardinals such as \( \aleph_\omega \), \( \aleph_{\omega+\omega} \), or \( \aleph_{\omega_1} \) are not regular[10](https://www.math.uni-hamburg.de/home/loewe/Lent2022/). Another singular example is the first fixed point of \( \alpha \mapsto \aleph_\alpha \), whose cofinality is \( \omega \)[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf).
Whether a regular limit cardinal exists at all is beyond ZFC: an uncountable regular limit cardinal is called **weakly inaccessible**, and ZFC cannot prove that any exists[1](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf) • [11](https://wwwpub.zih.tu-dresden.de/~bodirsky/Model-theory.pdf).
## König's theorem and exponentiation
König's theorem states that if \( \lambda_i < \kappa_i \) for all \( i \) in an index set \( I \), then \( \sum_{i \in I} \lambda_i < \prod_{i \in I} \kappa_i \)[4](https://math.colorado.edu/~kearnes/Teaching/Courses/F25/setsl.html). Its corollary for a single cardinal is \( \kappa < \kappa^{\operatorname{cf}(\kappa)} \), and further \( \aleph_\alpha < \operatorname{cf}(2^{\aleph_\alpha}) \)[4](https://math.colorado.edu/~kearnes/Teaching/Courses/F25/setsl.html). The second statement implies that the cardinality of the continuum, whatever it is, must have uncountable cofinality, so \( |\mathbb{R}| \neq \aleph_\omega \)[12](https://seop.lovelace.illc.uva.nl/entries/set-theory/index.html) • [9](https://math.colorado.edu/~kearnes/Teaching/Courses/setsl.html).
## Cardinal arithmetic at regular versus singular cardinals
The two classes behave very differently under the continuum function \( \kappa \mapsto 2^\kappa \). In 1963 Easton showed that for infinite regular cardinals \( \kappa \) the only constraints on \( 2^\kappa \) provable in ZFC are monotonicity, \( 2^\kappa > \kappa \), and \( \operatorname{cf}(2^\kappa) > \kappa \); any definable class function \( F \) on the infinite regular cardinals satisfying these three conditions yields a consistent model of ZFC in which \( 2^\kappa = F(\kappa) \) for all infinite regular \( \kappa \)[13](https://plato.stanford.edu/entries/continuum-hypothesis/). Set theorists had thereby pushed the cardinal arithmetic of regular cardinals as far as ZFC allows[13](https://plato.stanford.edu/entries/continuum-hypothesis/).
At singular cardinals the situation is more constrained. Silver's theorem (1974) shows that if \( \aleph_\delta \) is a singular cardinal of uncountable cofinality and the GCH holds below \( \aleph_\delta \), then it holds at \( \aleph_\delta \) itself; in particular GCH cannot first fail at \( \aleph_{\omega_1} \)[13](https://plato.stanford.edu/entries/continuum-hypothesis/). Shelah's pcf (possible cofinalities) theory gives ZFC bounds such as \( 2^{\aleph_\omega} < \aleph_{\omega_4} \) whenever \( \aleph_\omega \) is a strong limit cardinal[13](https://plato.stanford.edu/entries/continuum-hypothesis/). In summary, the continuum function at regular cardinals is relatively unconstrained in ZFC, while at singular cardinals it is provably constrained by the values on smaller cardinals[13](https://plato.stanford.edu/entries/continuum-hypothesis/).
## Large cardinals built on regularity
Regularity is one of the two components of the smallest large cardinal notions. A cardinal \( \kappa \) is a strong limit cardinal if \( 2^\lambda < \kappa \) for every \( \lambda < \kappa \); \( \kappa \) is strongly inaccessible when it is uncountable, regular, and a strong limit[5](https://plato.stanford.edu/entries/set-theory/) • [2](https://iep.utm.edu/set-theo/). For a strongly inaccessible \( \kappa \), the rank initial segment \( V_\kappa \) is a model of ZFC, which is why the existence of such cardinals cannot be proved in ZFC itself[5](https://plato.stanford.edu/entries/set-theory/) • [2](https://iep.utm.edu/set-theo/). If the GCH holds, every weakly inaccessible cardinal is strongly inaccessible[5](https://plato.stanford.edu/entries/set-theory/). The first weakly inaccessible cardinal is the smallest of all large cardinals[5](https://plato.stanford.edu/entries/set-theory/).
Regularity also enters stronger hierarchies. A regular cardinal \( \kappa \) is **Mahlo** when the set of strongly inaccessible cardinals below \( \kappa \) is stationary, meaning it meets every closed unbounded subset of \( \kappa \); the first [Mahlo cardinal](https://www.edgepedia.org/mahlo-cardinal) is far above the first strongly inaccessible, with \( \kappa \)-many inaccessibles below it[5](https://plato.stanford.edu/entries/set-theory/).
## Use outside set theory
Regular cardinals serve as size parameters in category theory. They appear in the definitions of \( \lambda \)-filtered colimits, \( \lambda \)-presentable objects, \( \lambda \)-accessible categories, locally \( \lambda \)-presentable categories, and \( \lambda \)-ind-completions, along with derived notions such as \( \lambda \)-combinatorial model categories[6](https://ncatlab.org/nlab/show/regular%2Bcardinal). A category is locally presentable when it is locally \( \lambda \)-presentable for some regular cardinal \( \lambda \)[14](https://www.ub.edu/topologia/seminars/Locally_presentable.pdf). If \( \lambda \) is not regular, having \( \lambda \)-filtered colimits is equivalent to having \( \lambda^+ \)-filtered colimits, and \( \lambda^+ \) is always regular assuming the axiom of choice, so the regular cardinals are exactly the levels at which the theory needs to be stated[6](https://ncatlab.org/nlab/show/regular%2Bcardinal).
## References
1. [Set Theory notes (Mark, Berkeley), Definition 10.10 and surrounding material](https://math.berkeley.edu/~marks/notes/set_theory_notes_4.pdf)
2. [Set Theory, Internet Encyclopedia of Philosophy](https://iep.utm.edu/set-theo/)
3. [Hausdorff, ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/hausdorff)
4. [Lecture Topics for Set Theory Students (Kearnes, Colorado)](https://math.colorado.edu/~kearnes/Teaching/Courses/F25/setsl.html)
5. [Set Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/set-theory/)
6. [Regular cardinal, nLab](https://ncatlab.org/nlab/show/regular%2Bcardinal)
7. [Notes on set theory (Palmieri, Washington)](https://sites.math.washington.edu/~palmieri/Courses/2004/Math583/set-theory.html)
8. [Set Theory homework notes (Szendrei, Colorado)](https://spot.colorado.edu/~szendrei/ST_S21/hw.html)
9. [Lecture Topics for Set Theory Students (Kearnes, Colorado, 2021)](https://math.colorado.edu/~kearnes/Teaching/Courses/setsl.html)
10. [Large Cardinals (L22), Benedikt Löwe, Universität Hamburg](https://www.math.uni-hamburg.de/home/loewe/Lent2022/)
11. [Model theory notes (Bodirsky, TU Dresden), Appendix A.4](https://wwwpub.zih.tu-dresden.de/~bodirsky/Model-theory.pdf)
12. [Set Theory, Stanford Encyclopedia of Philosophy (mirror)](https://seop.lovelace.illc.uva.nl/entries/set-theory/index.html)
13. [The Continuum Hypothesis, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/continuum-hypothesis/)
14. [Locally presentable categories seminar notes (UB)](https://www.ub.edu/topologia/seminars/Locally_presentable.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Ordinal and cardinal numbers › Cardinal numbers › Cofinality, regular and singular cardinals*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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