Additively indecomposable ordinal

In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an additively indecomposable ordinal α is any <a href="/facts/Ordinal_number/GelFLjJt">ordinal number</a> that is not 0 such that for any 
 
 
 
 β
 ,
 γ
 <
 α
 
 
 {\displaystyle \beta ,\gamma <\alpha }
 
, we have 
 
 
 
 β
 +
 γ
 <
 α
 .
 
 
 {\displaystyle \beta +\gamma <\alpha .}
 
 Additively indecomposable ordinals were named the gamma numbers by Cantor,p.20 and are also called additive principal numbers. The <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of additively indecomposable ordinals may be denoted 
 
 
 
 
 H
 
 
 
 {\displaystyle \mathbb {H} }
 
, from the German "Hauptzahl". The additively indecomposable ordinals are precisely those ordinals of the form 
 
 
 
 
 ω
 
 β
 
 
 
 
 {\displaystyle \omega ^{\beta }}
 
 for some ordinal 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
.
From the continuity of addition in its right argument, we get that if 
 
 
 
 β
 <
 α
 
 
 {\displaystyle \beta <\alpha }
 
 and α is additively indecomposable, then 
 
 
 
 β
 +
 α
 =
 α
 .
 
 
 {\displaystyle \beta +\alpha =\alpha .}

Obviously 1 is additively indecomposable, since 
 
 
 
 0
 +
 0
 <
 1.
 
 
 {\displaystyle 0+0<1.}
 
 No <a href="/facts/Finite_set/za1newlP">finite</a> ordinal other than 
 
 
 
 1
 
 
 {\displaystyle 1}
 
 is additively indecomposable. Also, 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every <a href="/facts/Infinite_set/9rTN0xGH">infinite</a> <a href="/facts/Initial_ordinal/EHFlKG96">initial ordinal</a> (an ordinal corresponding to a <a href="/facts/Cardinal_number/ccoKwU4R">cardinal number</a>) is additively indecomposable.
The class of additively indecomposable numbers is <a href="/facts/Order_topology/EHISytz6">closed</a> and unbounded. Its enumerating function is <a href="/facts/Normal_function/GCXHAu3H">normal</a>, given by 
 
 
 
 
 ω
 
 α
 
 
 
 
 {\displaystyle \omega ^{\alpha }}
 
.
The derivative of 
 
 
 
 
 ω
 
 α
 
 
 
 
 {\displaystyle \omega ^{\alpha }}
 
 (which enumerates its fixed points) is written 
 
 
 
 
 ε
 
 α
 
 
 
 
 {\displaystyle \varepsilon _{\alpha }}
 
 Ordinals of this form (that is, <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a> of 
 
 
 
 
 ω
 
 α
 
 
 
 
 {\displaystyle \omega ^{\alpha }}
 
) are called <a href="/facts/Epsilon_numbers_(mathematics)/t7w8KgA9">epsilon numbers</a>. The number 
 
 
 
 
 ε
 
 0
 
 
 =
 
 ω
 
 
 ω
 
 
 ω
 
 ⋯
 
 
 
 
 
 
 
 
 {\displaystyle \varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdots }}}}
 
 is therefore the first fixed point of the <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
 
 
 
 ω
 ,
 
 ω
 
 ω
 
 
 
 ,
 
 ω
 
 
 ω
 
 ω
 
 
 
 
 
 
 ,
 …
 
 
 {\displaystyle \omega ,\omega ^{\omega }\!,\omega ^{\omega ^{\omega }}\!\!,\ldots }

Additively indecomposable ordinal open-in-new

Additively indecomposable ordinal