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PacletInstall["Wolfram/Lazy"]
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<<Wolfram`Lazy`
Represent ordinals as a lazy list with the conventional ordinal corresponding to its last element
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ω=LazyRange[]
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2ω=ω+ω
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Join[ω,ω]
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2
ω
=ω×ω
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Map[ω&,ω]
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Catenate@Map[ω&,ω]
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ω×=Catenate@Map[ω&,]
ω
ω
LazyNest[Catenate@Map[ω&,#]&,ω,Infinity]
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…
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Fold[Catenate@Map[ω&,#]&,ω,ω]
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…

ω
=LazyNest[Catenate@Map[ω&,#]&,ω,]=Fold[Catenate@Map[ω&,#]&,ω,]
ε
0
=
⋰
ω
ω
ω
=
ε
0
ω
LazyNest[LazyNest[Catenate@Map[ω&,#]&,#,Infinity]&,ω,Infinity]
In[]:=
ε0=Fold[Fold[Catenate@Map[ω&,#]&,ω,#]&,ω,ω]
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ε
1
=
ε
1
ε
0
ε1=Fold[Fold[Catenate@Map[ε0&,#]&,ε0,#]&,ε0,ω]
ε
ω
εω=Fold[ε|->Fold[Fold[Catenate@Map[ε&,#]&,ε,#]&,ε,ω],ω,ω]