Regular cardinal

<h2 id="examples">Examples</h2>
The ordinals less than 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 are finite. A finite sequence of finite ordinals always has a finite maximum, so 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 cannot be the limit of any sequence of type less than 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 whose elements are ordinals less than 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
, and is therefore a regular ordinal. 
 
 
 
 
 ℵ
 
 0
 
 
 
 
 {\displaystyle \aleph _{0}}
 
 (<a href="/facts/Aleph-null/kwe9I4v5">aleph-null</a>) is a regular cardinal because its initial ordinal, 
 
 
 
 ω
 
 
 {\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.

 
 
 
 ω
 +
 1
 
 
 {\displaystyle \omega +1}
 
 is the <a href="/facts/Successor_ordinal/k2acCuUG">next ordinal number</a> greater than 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
. It is singular, since it is not a limit ordinal. 
 
 
 
 ω
 +
 ω
 
 
 {\displaystyle \omega +\omega }
 
 is the next limit ordinal after 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
. It can be written as the limit of the sequence 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
, 
 
 
 
 ω
 +
 1
 
 
 {\displaystyle \omega +1}
 
, 
 
 
 
 ω
 +
 2
 
 
 {\displaystyle \omega +2}
 
, 
 
 
 
 ω
 +
 3
 
 
 {\displaystyle \omega +3}
 
, and so on. This sequence has order type 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
, so 
 
 
 
 ω
 +
 ω
 
 
 {\displaystyle \omega +\omega }
 
 is the limit of a sequence of type less than 
 
 
 
 ω
 +
 ω
 
 
 {\displaystyle \omega +\omega }
 
 whose elements are ordinals less than 
 
 
 
 ω
 +
 ω
 
 
 {\displaystyle \omega +\omega }
 
; therefore it is singular.

 
 
 
 
 ℵ
 
 1
 
 
 
 
 {\displaystyle \aleph _{1}}
 
 is the <a href="/facts/Successor_cardinal/NmowSfwc">next cardinal number</a> greater than 
 
 
 
 
 ℵ
 
 0
 
 
 
 
 {\displaystyle \aleph _{0}}
 
, so the cardinals less than 
 
 
 
 
 ℵ
 
 1
 
 
 
 
 {\displaystyle \aleph _{1}}
 
 are <a href="/facts/Countable_set/Wj4DmWPv">countable</a> (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So 
 
 
 
 
 ℵ
 
 1
 
 
 
 
 {\displaystyle \aleph _{1}}
 
 cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.

 
 
 
 
 ℵ
 
 ω
 
 
 
 
 {\displaystyle \aleph _{\omega }}
 
 is the next cardinal number after the sequence 
 
 
 
 
 ℵ
 
 0
 
 
 
 
 {\displaystyle \aleph _{0}}
 
, 
 
 
 
 
 ℵ
 
 1
 
 
 
 
 {\displaystyle \aleph _{1}}
 
, 
 
 
 
 
 ℵ
 
 2
 
 
 
 
 {\displaystyle \aleph _{2}}
 
, 
 
 
 
 
 ℵ
 
 3
 
 
 
 
 {\displaystyle \aleph _{3}}
 
, and so on. Its initial ordinal 
 
 
 
 
 ω
 
 ω
 
 
 
 
 {\displaystyle \omega _{\omega }}
 
 is the limit of the sequence 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
, 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
, 
 
 
 
 
 ω
 
 2
 
 
 
 
 {\displaystyle \omega _{2}}
 
, 
 
 
 
 
 ω
 
 3
 
 
 
 
 {\displaystyle \omega _{3}}
 
, and so on, which has order type 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
, so 
 
 
 
 
 ω
 
 ω
 
 
 
 
 {\displaystyle \omega _{\omega }}
 
 is singular, and so is 
 
 
 
 
 ℵ
 
 ω
 
 
 
 
 {\displaystyle \aleph _{\omega }}
 
. Assuming the axiom of choice, 
 
 
 
 
 ℵ
 
 ω
 
 
 
 
 {\displaystyle \aleph _{\omega }}
 
 is the first infinite cardinal that is singular (the first infinite ordinal that is singular is 
 
 
 
 ω
 +
 1
 
 
 {\displaystyle \omega +1}
 
, and the first infinite limit ordinal that is singular is 
 
 
 
 ω
 +
 ω
 
 
 {\displaystyle \omega +\omega }
 
). Proving the existence of singular cardinals requires the <a href="/facts/Axiom_schema_of_replacement/dmJn0BaS">axiom of replacement</a>, and in fact the inability to prove the existence of 
 
 
 
 
 ℵ
 
 ω
 
 
 
 
 {\displaystyle \aleph _{\omega }}
 
 in <a href="/facts/Zermelo_set_theory/o3fzd9nl">Zermelo set theory</a> is what led <a href="/facts/Adolf_Abraham_Halevi_Fraenkel/Hc8IBfmY">Fraenkel</a> to postulate this axiom.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
Uncountable (weak) <a href="/facts/Limit_cardinal/c7JXNqoe">limit cardinals</a> that are also regular are known as (weakly) <a href="/facts/Inaccessible_cardinals/EUydsllM">inaccessible cardinals</a>. 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 <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a> of the <a href="/facts/Aleph_number/kwe9I4v5">aleph function</a>, though not all fixed points are regular. For instance, the first fixed point is the limit of the 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
-sequence 
 
 
 
 
 ℵ
 
 0
 
 
 ,
 
 ℵ
 
 ω
 
 
 ,
 
 ℵ
 
 
 ω
 
 ω
 
 
 
 
 ,
 .
 .
 .
 
 
 {\displaystyle \aleph _{0},\aleph _{\omega },\aleph _{\omega _{\omega }},...}
 
 and is therefore singular.

<h2 id="properties">Properties</h2>
If the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a> holds, then every <a href="/facts/Successor_cardinal/NmowSfwc">successor cardinal</a> 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 <a href="/facts/Cardinality_of_the_continuum/ob578Q9l">cardinality of the continuum</a>, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see <a href="/facts/Easton%2527s_theorem/g6o8sRrU">Easton's theorem</a>). The <a href="/facts/Continuum_hypothesis/R0yGqHND">continuum hypothesis</a> postulates that the cardinality of the continuum is equal to 
 
 
 
 
 ℵ
 
 1
 
 
 
 
 {\displaystyle \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 <a href="/facts/Aleph_number/kwe9I4v5">aleph numbers</a> 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 <a href="/facts/Zermelo%25E2%2580%2593Fraenkel_set_theory/UYIsPbXw">ZF</a> that 
 
 
 
 
 ω
 
 1
 
 
 
 
 {\displaystyle \omega _{1}}
 
 be the limit of a countable sequence of countable ordinals as well as the set of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> be a countable union of countable sets. Furthermore, it is consistent with ZF when not including AC that every aleph bigger than 
 
 
 
 
 ℵ
 
 0
 
 
 
 
 {\displaystyle \aleph _{0}}
 
 is singular (a result proved by <a href="/facts/Moti_Gitik/FxanEoDd">Moti Gitik</a>).
If 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 is a limit ordinal, 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 is regular iff the set of 
 
 
 
 α
 <
 κ
 
 
 {\displaystyle \alpha <\kappa }
 
 that are critical points of 
 
 
 
 
 Σ
 
 1
 
 
 
 
 {\displaystyle \Sigma _{1}}
 
-<a href="/facts/Elementary_embedding/3C94AfFq">elementary embeddings</a> 
 
 
 
 j
 
 
 {\displaystyle j}
 
 with 
 
 
 
 j
 (
 α
 )
 =
 κ
 
 
 {\displaystyle j(\alpha )=\kappa }
 
 is <a href="/facts/Club_set/TJVtvQAT">club</a> in 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
For cardinals 
 
 
 
 κ
 <
 θ
 
 
 {\displaystyle \kappa <\theta }
 
, say that an elementary embedding 
 
 
 
 j
 :
 M
 →
 H
 (
 θ
 )
 
 
 {\displaystyle j:M\to H(\theta )}
 
 a small embedding if 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is transitive and 
 
 
 
 j
 (
 
 
 crit
 
 
 (
 j
 )
 )
 =
 κ
 
 
 {\displaystyle j({\textrm {crit}}(j))=\kappa }
 
. A cardinal 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 is uncountable and regular iff there is an 
 
 
 
 α
 >
 κ
 
 
 {\displaystyle \alpha >\kappa }
 
 such that for every 
 
 
 
 θ
 >
 α
 
 
 {\displaystyle \theta >\alpha }
 
, there is a small embedding 
 
 
 
 j
 :
 M
 →
 H
 (
 θ
 )
 
 
 {\displaystyle j:M\to H(\theta )}
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>Corollary 2.2

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Inaccessible_cardinal/EUydsllM">Inaccessible cardinal</a></li></ul>

<ul><li><a href="/facts/Herbert_Enderton/LVXAVgos">Herbert B. Enderton</a>, Elements of Set Theory, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-238440-7</li>
<li><a href="/facts/Kenneth_Kunen/jbJNPJKV">Kenneth Kunen</a>, Set Theory, An Introduction to Independence Proofs, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-85401-0</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Maddy, Penelope (1988), "Believing the axioms. I", Journal of Symbolic Logic, 53 (2): 481–511, doi:10.2307/2274520, JSTOR 2274520, MR 0947855, Early hints of the Axiom of Replacement can be found in Cantor's letter to Dedekind [1899] and in Mirimanoff [1917]. 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). <a href="/wiki/Penelope_Maddy" target="_blank">/wiki/Penelope_Maddy</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">T. Arai, "Bounds on provability in set theories" (2012, p.2). Accessed 4 August 2022. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Holy, Lücke, Njegomir, "Small embedding characterizations for large cardinals". Annals of Pure and Applied Logic vol. 170, no. 2 (2019), pp.251--271. <a href="https://www.sciencedirect.com/science/article/pii/S0168007218301167" target="_blank">https://www.sciencedirect.com/science/article/pii/S0168007218301167</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Regular cardinal open-in-new

Regular cardinal