Regular cardinal

 ( Cardinal Numbers ) 

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that $\backslash kappa$ is a regular cardinal if and only if every unbounded subset $C\; \backslash subseteq\; \backslash kappa$ has cardinality $\backslash kappa$. Infinite cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.

In the presence of the axiom of choice, any cardinal number can be well-ordered, and so the following are equivalent:

1. $\backslash kappa$ is a regular cardinal.
2. If $\backslash kappa\; =\; \backslash textstyle\backslash sum\_\{i\; \backslash in\; I\}\; \backslash lambda\_i$ and for all $i$, then $|I|\; \backslash ge\; \backslash kappa$.
3. If $S\; =\; \backslash textstyle\backslash bigcup\_\{i\; \backslash in\; I\}\; S\_i$, and if and for all $i$, then . That is, every union of fewer than $\backslash kappa$ sets smaller than $\backslash kappa$ is smaller than $\backslash kappa$.
4. The category of sets of cardinality less than $\backslash kappa$ and all functions between them is closed under of cardinality less than $\backslash kappa$.
5. $\backslash kappa$ 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 the axiom 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 infinite ordinal number $\backslash alpha$ is a regular ordinal if it is a limit ordinal that is not the limit of a set of smaller ordinals that as a set has order type less than $\backslash alpha$. A regular ordinal is always an initial ordinal, though some initial ordinals are not regular, e.g., $\backslash omega\_\backslash omega$ (see the example below).

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

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

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

$\backslash aleph\_\backslash omega$ is the next cardinal number after the sequence $\backslash aleph\_0$, $\backslash aleph\_1$, $\backslash aleph\_2$, $\backslash aleph\_3$, and so on. Its initial ordinal $\backslash omega\_\backslash omega$ is the limit of the sequence $\backslash omega$, $\backslash omega\_1$, $\backslash omega\_2$, $\backslash omega\_3$, and so on, which has order type $\backslash omega$, so $\backslash omega\_\backslash omega$ is singular, and so is $\backslash aleph\_\backslash omega$. Assuming the axiom of choice, $\backslash aleph\_\backslash omega$ is the first infinite cardinal that is singular (the first infinite ordinal that is singular is $\backslash omega+1$, and the first infinite limit ordinal that is singular is $\backslash omega+\backslash omega$). Proving the existence of singular cardinals requires the axiom of replacement, and in fact the inability to prove the existence of $\backslash aleph\_\backslash omega$ in Zermelo set theory is what led Fraenkel to postulate this axiom.. Maddy cites two papers by Mirimanoff, "Les antinomies de Russell et de Burali-Forti et le problème fundamental de la théorie des ensembles" and "Remarques sur la théorie des ensembles et les antinomies Cantorienne", both in L'Enseignement Mathématique (1917).

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

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Properties 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 cardinalities 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 (see Easton's theorem). The continuum hypothesis postulates that the cardinality of the continuum is equal to $\backslash aleph\_1$, which is regular assuming choice.

Without the axiom of choice: there would be cardinal numbers that were not well-orderable. Moreover, the cardinal sum of an arbitrary collection could not be defined. Therefore, only the could meaningfully be called regular or singular cardinals.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 with ZF that $\backslash omega\_1$ be the limit of a countable sequence of countable ordinals as well as the set of be a countable union of countable sets. Furthermore, it is consistent with ZF when not including AC that every aleph bigger than $\backslash aleph\_0$ is singular (a result proved by Moti Gitik).

If $\backslash kappa$ is a limit ordinal, $\backslash kappa$ is regular iff the set of that are critical points of $\backslash Sigma\_1$-elementary embeddings $j$ with $j(\backslash alpha)=\backslash kappa$ is Club set in $\backslash kappa$.T. Arai, "Bounds on provability in set theories" (2012, p.2). Accessed 4 August 2022.

For cardinals , say that an elementary embedding $j:M\backslash to\; H(\backslash theta)$ a small embedding if $M$ is transitive and $j(\backslash textrm\{crit\}(j))=\backslash kappa$. A cardinal $\backslash kappa$ is uncountable and regular iff there is an $\backslash alpha>\backslash kappa$ such that for every $\backslash theta>\backslash alpha$, there is a small embedding $j:M\backslash to\; H(\backslash theta)$.Holy, Lücke, Njegomir, " Small embedding characterizations for large cardinals". Annals of Pure and Applied Logic vol. 170, no. 2 (2019), pp.251--271.^{Corollary 2.2}

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See also

• Inaccessible cardinal

• , Elements of Set Theory,
• , Set Theory, An Introduction to Independence Proofs,

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