Additively indecomposable ordinal

<h2 id="multiplicatively-indecomposable">Multiplicatively indecomposable</h2>
A similar notion can be defined for multiplication. If α is greater than the multiplicative identity, 1, and β < α and γ < α imply β·γ < α, then α is multiplicatively indecomposable. The finite ordinal 2 is multiplicatively indecomposable since 1·1 = 1 < 2. Besides 2, the multiplicatively indecomposable ordinals (named the delta numbers by Cantor<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>p.20) are those of the form 
 
 
 
 
 ω
 
 
 ω
 
 α
 
 
 
 
 
 
 
 {\displaystyle \omega ^{\omega ^{\alpha }}\,}
 
 for any ordinal α. Every <a href="/facts/Epsilon_numbers_(mathematics)/t7w8KgA9">epsilon number</a> is multiplicatively indecomposable; and every multiplicatively indecomposable ordinal (other than 2) is additively indecomposable. The delta numbers (other than 2) are the same as the <a href="/facts/Prime_ordinal/satc9FHy">prime ordinals</a> that are limits.

<h2 id="higher-indecomposables">Higher indecomposables</h2>
Exponentially indecomposable ordinals are equal to the epsilon numbers, tetrationally indecomposable ordinals are equal to the zeta numbers (fixed points of 
 
 
 
 
 ε
 
 α
 
 
 
 
 {\displaystyle \varepsilon _{\alpha }}
 
), and so on. Therefore, 
 
 
 
 
 φ
 
 ω
 
 
 (
 0
 )
 
 
 {\displaystyle \varphi _{\omega }(0)}
 
 is the first ordinal which is 
 
 
 
 
 ↑
 
 n
 
 
 
 
 {\displaystyle \uparrow ^{n}}
 
-indecomposable for all 
 
 
 
 n
 
 
 {\displaystyle n}
 
, where 
 
 
 
 ↑
 
 
 {\displaystyle \uparrow }
 
 denotes <a href="/facts/Knuth%27s_up-arrow_notation/F441JrlK">Knuth's up-arrow notation</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Ordinal_arithmetic/satc9FHy">Ordinal arithmetic</a></li></ul>

<ul><li><a href="/facts/Wac%C5%82aw_Sierpi%C5%84ski/ywT4mREz">Sierpiński, Wacław</a> (1958), <a href="/facts/Cardinal_and_Ordinal_Numbers/BlVTYVje">Cardinal and ordinal numbers</a>, Polska Akademia Nauk Monografie Matematyczne, vol. 34, Warsaw: Państwowe Wydawnictwo Naukowe, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0095787">0095787</a></li></ul>

This article incorporates material from <a href="https://planetmath.org/additivelyindecomposable">Additively indecomposable</a> on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.

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

<ol>
<li id="fn:1">A. Rhea, "The Ordinals as a Consummate Abstraction of Number Systems" (2017), preprint. <a href="https://arxiv.org/abs/1706.08908" target="_blank">https://arxiv.org/abs/1706.08908</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">W. Pohlers, "A short course in ordinal analysis", pp. 27–78. Appearing in Aczel, Simmons, Proof Theory: A selection of papers from the Leeds Proof Theory Programme 1990 (1992). Cambridge University Press, ISBN 978-0-521-41413-5 <a href="/wiki/Peter_Aczel" target="_blank">/wiki/Peter_Aczel</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">A. Rhea, "The Ordinals as a Consummate Abstraction of Number Systems" (2017), preprint. <a href="https://arxiv.org/abs/1706.08908" target="_blank">https://arxiv.org/abs/1706.08908</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

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Additively indecomposable ordinal