В математике функции Веблена — иерархия нормальных функций, строго возрастающих от ординала к ординалу, предложенная Освальдом Вебленом в 1908 году. Если
φ
0
{\displaystyle \varphi _{0}}
α
{\displaystyle \alpha }
φ
α
{\displaystyle \varphi _{\alpha }}
неподвижные точки всех
φ
β
{\displaystyle \varphi _{\beta }}
β
<
α
.
{\displaystyle \beta <\alpha .}
В частном случае, когда
φ
0
(
α
)
=
ω
α
{\displaystyle \varphi _{0}(\alpha )=\omega ^{\alpha }}
иерархией Веблена ;
φ
1
(
α
)
=
ε
α
,
φ
2
(
α
)
=
ζ
α
,
φ
3
(
α
)
=
η
α
.
{\displaystyle \varphi _{1}(\alpha )=\varepsilon _{\alpha },\,\varphi _{2}(\alpha )=\zeta _{\alpha },\,\varphi _{3}(\alpha )=\eta _{\alpha }.}
α
{\displaystyle \alpha }
α
=
φ
β
1
(
γ
1
)
+
φ
β
2
(
γ
2
)
+
⋯
+
φ
β
k
(
γ
k
)
,
{\displaystyle \alpha =\varphi _{\beta _{1}}(\gamma _{1})+\varphi _{\beta _{2}}(\gamma _{2})+\cdots +\varphi _{\beta _{k}}(\gamma _{k}),}
k
>
0
{\displaystyle k>0}
натуральное число ,
φ
β
m
(
γ
m
)
≥
φ
β
m
+
1
(
γ
m
+
1
)
{\displaystyle \varphi _{\beta _{m}}(\gamma _{m})\geq \varphi _{\beta _{m+1}}(\gamma _{m+1})}
γ
m
<
φ
β
m
(
γ
m
)
.
{\displaystyle \gamma _{m}<\varphi _{\beta _{m}}(\gamma _{m}).}
α
{\displaystyle \alpha }
α
[
n
]
=
φ
β
1
(
γ
1
)
+
⋯
+
φ
β
k
−
1
(
γ
k
−
1
)
+
φ
β
k
(
γ
k
)
[
n
]
{\displaystyle \alpha [n]=\varphi _{\beta _{1}}(\gamma _{1})+\cdots +\varphi _{\beta _{k-1}}(\gamma _{k-1})+\varphi _{\beta _{k}}(\gamma _{k})[n]}
Если
β
=
0
,
{\displaystyle \beta =0,}
φ
0
(
γ
+
1
)
[
n
]
=
ω
γ
⋅
n
,
{\displaystyle \varphi _{0}(\gamma +1)[n]=\omega ^{\gamma }\cdot n,}
φ
0
(
0
)
=
1
{\displaystyle \varphi _{0}(0)=1}
φ
0
(
γ
)
=
ω
γ
.
{\displaystyle \varphi _{0}(\gamma )=\omega ^{\gamma }.}
Если
γ
=
0
,
{\displaystyle \gamma =0,}
φ
β
+
1
(
0
)
[
0
]
=
0
{\displaystyle \varphi _{\beta +1}(0)[0]=0}
φ
β
+
1
(
0
)
[
n
+
1
]
=
φ
β
(
φ
β
+
1
(
0
)
[
n
]
)
,
{\displaystyle \varphi _{\beta +1}(0)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(0)[n]),}
φ
β
+
1
(
0
)
[
n
]
=
φ
β
n
(
0
)
.
{\displaystyle \varphi _{\beta +1}(0)[n]=\varphi _{\beta }^{n}(0).}
Если
γ
{\displaystyle \gamma }
предельный ординал , тогда
φ
β
(
γ
)
[
n
]
=
φ
β
(
γ
[
n
]
)
.
{\displaystyle \varphi _{\beta }(\gamma )[n]=\varphi _{\beta }(\gamma [n]).}
Если
β
{\displaystyle \beta }
предельный ординал , тогда
φ
β
(
0
)
[
n
]
=
φ
β
[
n
]
(
0
)
{\displaystyle \varphi _{\beta }(0)[n]=\varphi _{\beta [n]}(0)}
φ
β
(
γ
+
1
)
[
n
]
=
φ
β
[
n
]
(
φ
β
(
γ
)
+
1
)
.
{\displaystyle \varphi _{\beta }(\gamma +1)[n]=\varphi _{\beta [n]}(\varphi _{\beta }(\gamma )+1).}
Иначе
φ
β
+
1
(
γ
+
1
)
[
0
]
=
φ
β
+
1
(
γ
)
+
1
{\displaystyle \varphi _{\beta +1}(\gamma +1)[0]=\varphi _{\beta +1}(\gamma )+1}
φ
β
+
1
(
γ
+
1
)
[
n
+
1
]
=
φ
β
(
φ
β
+
1
(
γ
+
1
)
[
n
]
)
,
{\displaystyle \varphi _{\beta +1}(\gamma +1)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(\gamma +1)[n]),}
φ
β
+
1
(
γ
+
1
)
[
n
]
=
φ
β
n
(
φ
β
+
1
(
γ
)
+
1
)
.
{\displaystyle \varphi _{\beta +1}(\gamma +1)[n]=\varphi _{\beta }^{n}(\varphi _{\beta +1}(\gamma )+1).}
применение правила 2
применение правила 5
φ
1
(
0
)
[
0
]
=
ε
0
[
0
]
=
0
{\displaystyle \varphi _{1}(0)[0]=\varepsilon _{0}[0]=0}
φ
1
(
1
)
[
0
]
=
φ
1
(
0
)
+
1
=
ε
0
+
1
=
ε
1
[
0
]
{\displaystyle \varphi _{1}(1)[0]=\varphi _{1}(0)+1=\varepsilon _{0}+1=\varepsilon _{1}[0]}
φ
1
(
0
)
[
1
]
=
φ
0
(
φ
1
(
0
)
[
0
]
)
=
φ
0
(
0
)
=
ω
0
=
1
{\displaystyle \varphi _{1}(0)[1]=\varphi _{0}(\varphi _{1}(0)[0])=\varphi _{0}(0)=\omega ^{0}=1}
φ
1
(
1
)
[
1
]
=
φ
0
(
φ
1
(
1
)
[
0
]
)
=
ε
1
[
1
]
=
ω
ε
0
+
1
{\displaystyle \varphi _{1}(1)[1]=\varphi _{0}(\varphi _{1}(1)[0])=\varepsilon _{1}[1]=\omega ^{\varepsilon _{0}+1}}
φ
1
(
0
)
[
2
]
=
φ
0
(
φ
1
(
0
)
[
1
]
)
=
ω
1
=
ω
{\displaystyle \varphi _{1}(0)[2]=\varphi _{0}(\varphi _{1}(0)[1])=\omega ^{1}=\omega }
φ
1
(
1
)
[
2
]
=
φ
0
(
φ
1
(
1
)
[
1
]
)
=
ε
1
[
2
]
=
ω
ω
ε
0
+
1
{\displaystyle \varphi _{1}(1)[2]=\varphi _{0}(\varphi _{1}(1)[1])=\varepsilon _{1}[2]=\omega ^{\omega ^{\varepsilon _{0}+1}}}
φ
1
(
0
)
[
3
]
=
φ
0
(
φ
1
(
0
)
[
2
]
)
=
ω
ω
{\displaystyle \varphi _{1}(0)[3]=\varphi _{0}(\varphi _{1}(0)[2])=\omega ^{\omega }}
φ
1
(
1
)
[
3
]
=
φ
0
(
φ
1
(
1
)
[
2
]
)
=
ε
1
[
3
]
=
ω
ω
ω
ε
0
+
1
{\displaystyle \varphi _{1}(1)[3]=\varphi _{0}(\varphi _{1}(1)[2])=\varepsilon _{1}[3]=\omega ^{\omega ^{\omega ^{\varepsilon _{0}+1}}}}
φ
1
(
0
)
[
4
]
=
φ
0
(
φ
1
(
0
)
[
3
]
)
=
ω
ω
ω
{\displaystyle \varphi _{1}(0)[4]=\varphi _{0}(\varphi _{1}(0)[3])=\omega ^{\omega ^{\omega }}}
φ
1
(
1
)
[
4
]
=
φ
0
(
φ
1
(
1
)
[
3
]
)
=
ε
1
[
4
]
=
ω
ω
ω
ω
ε
0
+
1
{\displaystyle \varphi _{1}(1)[4]=\varphi _{0}(\varphi _{1}(1)[3])=\varepsilon _{1}[4]=\omega ^{\omega ^{\omega ^{\omega ^{\varepsilon _{0}+1}}}}}
φ
2
(
0
)
[
4
]
=
φ
1
(
φ
1
(
φ
1
(
φ
1
(
0
)
)
)
)
=
ε
ε
ε
ε
0
=
ζ
0
[
4
]
{\displaystyle \varphi _{2}(0)[4]=\varphi _{1}(\varphi _{1}(\varphi _{1}(\varphi _{1}(0))))=\varepsilon _{\varepsilon _{\varepsilon _{\varepsilon _{0}}}}=\zeta _{0}[4]}
φ
2
(
1
)
[
4
]
=
φ
1
(
φ
1
(
φ
1
(
φ
1
(
φ
2
(
0
)
+
1
)
)
)
)
=
ε
ε
ε
ε
ζ
0
+
1
=
ζ
1
[
4
]
{\displaystyle \varphi _{2}(1)[4]=\varphi _{1}(\varphi _{1}(\varphi _{1}(\varphi _{1}(\varphi _{2}(0)+1))))=\varepsilon _{\varepsilon _{\varepsilon _{\varepsilon _{\zeta _{0}+1}}}}=\zeta _{1}[4]}
φ
3
(
0
)
[
4
]
=
φ
2
(
φ
2
(
φ
2
(
φ
2
(
0
)
)
)
)
=
ζ
ζ
ζ
ζ
0
=
η
0
[
4
]
{\displaystyle \varphi _{3}(0)[4]=\varphi _{2}(\varphi _{2}(\varphi _{2}(\varphi _{2}(0))))=\zeta _{\zeta _{\zeta _{\zeta _{0}}}}=\eta _{0}[4]}
φ
3
(
1
)
[
4
]
=
φ
2
(
φ
2
(
φ
2
(
φ
2
(
φ
3
(
0
)
+
1
)
)
)
)
=
ζ
ζ
ζ
ζ
η
0
+
1
=
η
1
[
4
]
{\displaystyle \varphi _{3}(1)[4]=\varphi _{2}(\varphi _{2}(\varphi _{2}(\varphi _{2}(\varphi _{3}(0)+1))))=\zeta _{\zeta _{\zeta _{\zeta _{\eta _{0}+1}}}}=\eta _{1}[4]}
φ
0
(
3
)
[
n
]
=
ω
2
⋅
n
{\displaystyle \varphi _{0}(3)[n]=\omega ^{2}\cdot n}
φ
0
(
φ
0
(
1
)
)
[
n
]
=
φ
0
(
φ
0
(
1
)
[
n
]
)
=
ω
ω
[
n
]
=
ω
n
{\displaystyle \varphi _{0}(\varphi _{0}(1))[n]=\varphi _{0}(\varphi _{0}(1)[n])=\omega ^{\omega [n]}=\omega ^{n}}
φ
1
(
ω
)
[
n
]
=
φ
1
(
ω
[
n
]
)
=
φ
1
(
n
)
=
ε
n
{\displaystyle \varphi _{1}(\omega )[n]=\varphi _{1}(\omega [n])=\varphi _{1}(n)=\varepsilon _{n}}
φ
1
(
φ
1
(
0
)
)
[
n
]
=
φ
1
(
φ
1
(
0
)
[
n
]
)
=
φ
1
(
ε
0
[
n
]
)
=
ε
ω
↑↑
(
n
−
1
)
{\displaystyle \varphi _{1}(\varphi _{1}(0))[n]=\varphi _{1}(\varphi _{1}(0)[n])=\varphi _{1}(\varepsilon _{0}[n])=\varepsilon _{\omega \uparrow \uparrow (n-1)}}
φ
φ
0
(
1
)
(
0
)
[
n
]
=
φ
ω
(
0
)
[
n
]
=
φ
ω
[
n
]
(
0
)
=
φ
n
(
0
)
{\displaystyle \varphi _{\varphi _{0}(1)}(0)[n]=\varphi _{\omega }(0)[n]=\varphi _{\omega [n]}(0)=\varphi _{n}(0)}
φ
φ
1
(
0
)
(
0
)
[
3
]
=
φ
ε
0
(
0
)
[
3
]
=
φ
ε
0
[
3
]
(
0
)
=
φ
ω
ω
(
0
)
{\displaystyle \varphi _{\varphi _{1}(0)}(0)[3]=\varphi _{\varepsilon _{0}}(0)[3]=\varphi _{\varepsilon _{0}[3]}(0)=\varphi _{\omega ^{\omega }}(0)}
Соответствующие примеры для быстрорастущей иерархии :
f
φ
1
(
0
)
(
4
)
=
f
φ
1
(
0
)
[
4
]
(
4
)
=
f
ω
ω
ω
(
4
)
{\displaystyle f_{\varphi _{1}(0)}(4)=f_{\varphi _{1}(0)[4]}(4)=f_{\omega ^{\omega ^{\omega }}}(4)}
f
φ
1
(
1
)
(
3
)
=
f
φ
1
(
1
)
[
3
]
(
3
)
=
f
ω
ω
ω
ε
0
+
1
(
3
)
{\displaystyle f_{\varphi _{1}(1)}(3)=f_{\varphi _{1}(1)[3]}(3)=f_{\omega ^{\omega ^{\omega ^{\varepsilon _{0}+1}}}}(3)}
Функция Γ перечисляет ординалы
α
,
{\displaystyle \alpha ,}
φ
α
(
0
)
=
α
.
{\displaystyle \varphi _{\alpha }(0)=\alpha .}
α
,
{\displaystyle \alpha ,}
ординалом Фефермана [англ.]
Γ
0
.
{\displaystyle \Gamma _{0}.}
Γ
0
[
0
]
=
0
{\displaystyle \Gamma _{0}[0]=0\,}
Γ
0
[
n
+
1
]
=
φ
Γ
0
[
n
]
(
0
)
.
{\displaystyle \Gamma _{0}[n+1]=\varphi _{\Gamma _{0}[n]}(0).}
Для
Γ
β
+
1
{\displaystyle \Gamma _{\beta +1}}
Γ
β
+
1
[
0
]
=
Γ
β
+
1
{\displaystyle \Gamma _{\beta +1}[0]=\Gamma _{\beta }+1}
Γ
β
+
1
[
n
+
1
]
=
φ
Γ
β
+
1
[
n
]
(
0
)
.
{\displaystyle \Gamma _{\beta +1}[n+1]=\varphi _{\Gamma _{\beta +1}[n]}(0).}
Если
β
{\displaystyle \beta }
β
<
Γ
β
,
{\displaystyle \beta <\Gamma _{\beta },}
Γ
β
[
n
]
=
Γ
β
[
n
]
.
{\displaystyle \Gamma _{\beta }[n]=\Gamma _{\beta [n]}.}
Функция Веблена
φ
α
(
β
)
{\displaystyle \varphi _{\alpha }(\beta )}
φ
(
α
,
β
)
{\displaystyle \varphi (\alpha ,\beta )}
φ
(
α
n
,
α
n
−
1
,
.
.
.
,
α
0
)
{\displaystyle \varphi (\alpha _{n},\alpha _{n-1},...,\alpha _{0})}
φ
(
α
)
=
ω
α
{\displaystyle \varphi (\alpha )=\omega ^{\alpha }}
φ
(
0
,
α
n
−
1
,
.
.
.
,
α
0
)
=
φ
(
α
n
−
1
,
.
.
.
,
α
0
)
,
{\displaystyle \varphi (0,\alpha _{n-1},...,\alpha _{0})=\varphi (\alpha _{n-1},...,\alpha _{0}),}
для
α
>
0
,
γ
↦
φ
(
α
n
,
.
.
.
,
α
i
+
1
,
α
,
0
,
.
.
.
,
0
,
γ
)
{\displaystyle \alpha >0,\,\gamma \mapsto \varphi (\alpha _{n},...,\alpha _{i+1},\alpha ,0,...,0,\gamma )}
ξ
↦
φ
(
α
n
,
.
.
.
,
α
i
+
1
,
β
,
ξ
,
0
,
.
.
.
,
0
)
{\displaystyle \xi \mapsto \varphi (\alpha _{n},...,\alpha _{i+1},\beta ,\xi ,0,...,0)}
β
<
α
.
{\displaystyle \beta <\alpha .}
Например,
φ
(
1
,
0
,
γ
)
{\displaystyle \varphi (1,0,\gamma )}
γ
{\displaystyle \gamma }
ξ
↦
φ
(
0
,
ξ
,
0
)
=
φ
(
ξ
,
0
)
,
{\displaystyle \xi \mapsto \varphi (0,\xi ,0)=\varphi (\xi ,0),}
Γ
γ
.
{\displaystyle \Gamma _{\gamma }.}
φ
(
1
,
0
,
0
)
=
Γ
0
{\displaystyle \varphi (1,0,0)=\Gamma _{0}}
φ
(
1
,
0
,
0
,
0
)
{\displaystyle \varphi (1,0,0,0)}
Предел для
φ
(
1
,
0
,
.
.
.
,
0
,
0
)
{\displaystyle \varphi (1,0,...,0,0)}
Hilbert Levitz, Transfinite Ordinals and Their Notations: For The Uninitiated PostScript )
Pohlers, Wolfram (1989), Proof theory , Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag, ISBN 3-540-51842-8 MR 1026933 Schütte, Kurt (1977), Proof theory , Grundlehren der Mathematischen Wissenschaften, vol. 225, Berlin-New York: Springer-Verlag, pp. xii+299, ISBN 3-540-07911-4 MR 0505313 Takeuti, Gaisi (1987), Proof theory , Studies in Logic and the Foundations of Mathematics, vol. 81 (Second ed.), Amsterdam: North-Holland Publishing Co., ISBN 0-444-87943-9 MR 0882549 Smorynski, C. (1982), "The varieties of arboreal experience", Math. Intelligencer , 4 (4): 182—189, doi :10.1007/BF03023553 Veblen, Oswald (1908), "Continuous Increasing Functions of Finite and Transfinite Ordinals", Transactions of the American Mathematical Society , 9 (3): 280—292, doi :10.2307/1988605 , JSTOR 1988605 Miller, Larry W. (1976), "Normal Functions and Constructive Ordinal Notations", The Journal of Symbolic Logic , 41 (2): 439—459, doi :10.2307/2272243 , JSTOR 2272243
Числа Функции Нотации
![{\displaystyle \alpha [n]=\varphi _{\beta _{1}}(\gamma _{1})+\cdots +\varphi _{\beta _{k-1}}(\gamma _{k-1})+\varphi _{\beta _{k}}(\gamma _{k})[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4901c6f145105dc243dfe7025e5e0a1fef7dc558)
![{\displaystyle \varphi _{0}(\gamma +1)[n]=\omega ^{\gamma }\cdot n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0495d8993d20c32fc060b1b0f295d9fb91f853de)
![{\displaystyle \varphi _{\beta +1}(0)[0]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b561894d6f1918c97b606d33420ba70c16f4e56)
![{\displaystyle \varphi _{\beta +1}(0)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(0)[n]),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52e7efabb39236eb33023d0e2da549a16283618d)
![{\displaystyle \varphi _{\beta +1}(0)[n]=\varphi _{\beta }^{n}(0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6f53b022a3ff26cfd50c172ef35e25fa25ba78)
![{\displaystyle \varphi _{\beta }(\gamma )[n]=\varphi _{\beta }(\gamma [n]).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e9fc66ed912125d18f53bbf205952bafca0c95)
![{\displaystyle \varphi _{\beta }(0)[n]=\varphi _{\beta [n]}(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b13122287844cc99e7c17752070a5f28bd033bd)
![{\displaystyle \varphi _{\beta }(\gamma +1)[n]=\varphi _{\beta [n]}(\varphi _{\beta }(\gamma )+1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f263b4f7d0d085477d62ddfae1aa299bcd29bd4a)
![{\displaystyle \varphi _{\beta +1}(\gamma +1)[0]=\varphi _{\beta +1}(\gamma )+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df39d847f5d3ca5ae4e7f43f5dce29da0987fc77)
![{\displaystyle \varphi _{\beta +1}(\gamma +1)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(\gamma +1)[n]),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb444b1dbe0d54a851ddf315e68c89949b0c8cb4)
![{\displaystyle \varphi _{\beta +1}(\gamma +1)[n]=\varphi _{\beta }^{n}(\varphi _{\beta +1}(\gamma )+1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28c4a617f338163ddbdae09b4277d5b3ee178f33)
![{\displaystyle \varphi _{1}(0)[0]=\varepsilon _{0}[0]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a35d0d702a70a240c6c98a0bd092aa4816a13e3)
![{\displaystyle \varphi _{1}(1)[0]=\varphi _{1}(0)+1=\varepsilon _{0}+1=\varepsilon _{1}[0]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a936187c81e4db7234b8f6d7c10f081f8389ee49)
![{\displaystyle \varphi _{1}(0)[1]=\varphi _{0}(\varphi _{1}(0)[0])=\varphi _{0}(0)=\omega ^{0}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3498e3a0b5c9ca531cf7f6a371a449f6c67fff72)
![{\displaystyle \varphi _{1}(1)[1]=\varphi _{0}(\varphi _{1}(1)[0])=\varepsilon _{1}[1]=\omega ^{\varepsilon _{0}+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f8ca9b8a8566185de89090983b3bfdb1130fba)
![{\displaystyle \varphi _{1}(0)[2]=\varphi _{0}(\varphi _{1}(0)[1])=\omega ^{1}=\omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d92aef3dfe4613171a5f8ef381fc63512b49f00)
![{\displaystyle \varphi _{1}(1)[2]=\varphi _{0}(\varphi _{1}(1)[1])=\varepsilon _{1}[2]=\omega ^{\omega ^{\varepsilon _{0}+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec207f99d3f5b90c6d26067c7725a806ac84d4a)
![{\displaystyle \varphi _{1}(0)[3]=\varphi _{0}(\varphi _{1}(0)[2])=\omega ^{\omega }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e28c2b283a6bdca1f7916d84fd976573b100d23)
![{\displaystyle \varphi _{1}(1)[3]=\varphi _{0}(\varphi _{1}(1)[2])=\varepsilon _{1}[3]=\omega ^{\omega ^{\omega ^{\varepsilon _{0}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/573a35fd9465a05a77e6bf8c08006fa323a13449)
![{\displaystyle \varphi _{1}(0)[4]=\varphi _{0}(\varphi _{1}(0)[3])=\omega ^{\omega ^{\omega }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7577c97b7c3c89236e0acb951698255cff4df2f)
![{\displaystyle \varphi _{1}(1)[4]=\varphi _{0}(\varphi _{1}(1)[3])=\varepsilon _{1}[4]=\omega ^{\omega ^{\omega ^{\omega ^{\varepsilon _{0}+1}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3794a0edb28e7e8799fe502f586fd01ce7c3d678)
![{\displaystyle \varphi _{2}(0)[4]=\varphi _{1}(\varphi _{1}(\varphi _{1}(\varphi _{1}(0))))=\varepsilon _{\varepsilon _{\varepsilon _{\varepsilon _{0}}}}=\zeta _{0}[4]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4014e7150c5b4b16523056dae18a22462c88a552)
![{\displaystyle \varphi _{2}(1)[4]=\varphi _{1}(\varphi _{1}(\varphi _{1}(\varphi _{1}(\varphi _{2}(0)+1))))=\varepsilon _{\varepsilon _{\varepsilon _{\varepsilon _{\zeta _{0}+1}}}}=\zeta _{1}[4]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/484355ce96cc8391763b1d7dde3232b3e177b1de)
![{\displaystyle \varphi _{3}(0)[4]=\varphi _{2}(\varphi _{2}(\varphi _{2}(\varphi _{2}(0))))=\zeta _{\zeta _{\zeta _{\zeta _{0}}}}=\eta _{0}[4]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/871dae77fec31b1d262876e216369569713e3339)
![{\displaystyle \varphi _{3}(1)[4]=\varphi _{2}(\varphi _{2}(\varphi _{2}(\varphi _{2}(\varphi _{3}(0)+1))))=\zeta _{\zeta _{\zeta _{\zeta _{\eta _{0}+1}}}}=\eta _{1}[4]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b42d8acf9a15340bee0292b647d96070fce978ec)
![{\displaystyle \varphi _{0}(3)[n]=\omega ^{2}\cdot n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17e73c65c561c9b6001112d633af24659be4066c)
![{\displaystyle \varphi _{0}(\varphi _{0}(1))[n]=\varphi _{0}(\varphi _{0}(1)[n])=\omega ^{\omega [n]}=\omega ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5a19e09e6a64c480b0c46a63674c16acba9913)
![{\displaystyle \varphi _{1}(\omega )[n]=\varphi _{1}(\omega [n])=\varphi _{1}(n)=\varepsilon _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb65f4b9d37b5f9b4c24c592e1cc560f7ea950c)
![{\displaystyle \varphi _{1}(\varphi _{1}(0))[n]=\varphi _{1}(\varphi _{1}(0)[n])=\varphi _{1}(\varepsilon _{0}[n])=\varepsilon _{\omega \uparrow \uparrow (n-1)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a16b6b35b5959afcc430fd0b6da64c278353f7)
![{\displaystyle \varphi _{\varphi _{0}(1)}(0)[n]=\varphi _{\omega }(0)[n]=\varphi _{\omega [n]}(0)=\varphi _{n}(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c51c2b153d46c9ed5d89910ced82f83b9ec593)
![{\displaystyle \varphi _{\varphi _{1}(0)}(0)[3]=\varphi _{\varepsilon _{0}}(0)[3]=\varphi _{\varepsilon _{0}[3]}(0)=\varphi _{\omega ^{\omega }}(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bef065f3614b4acf6b222b392203884f079ecdc8)
![{\displaystyle f_{\varphi _{1}(0)}(4)=f_{\varphi _{1}(0)[4]}(4)=f_{\omega ^{\omega ^{\omega }}}(4)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37bb67855157a6901064d4f92151019fcdfdeb72)
![{\displaystyle f_{\varphi _{1}(1)}(3)=f_{\varphi _{1}(1)[3]}(3)=f_{\omega ^{\omega ^{\omega ^{\varepsilon _{0}+1}}}}(3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7374d5c7f223db2bb6704ae827626ae232e589d9)
![{\displaystyle \Gamma _{0}[0]=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28be40a1f97d402f8801a787cd2ef85c631d880c)
![{\displaystyle \Gamma _{0}[n+1]=\varphi _{\Gamma _{0}[n]}(0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83121ac5c0e5e474efa62260d9d56a165bb4e7c2)
![{\displaystyle \Gamma _{\beta +1}[0]=\Gamma _{\beta }+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4200ac4aad2f91e7d41afd5ff2b436a925d97273)
![{\displaystyle \Gamma _{\beta +1}[n+1]=\varphi _{\Gamma _{\beta +1}[n]}(0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/582f4f6e8acfd644efff0f0be2cfb35f0f168fd2)
![{\displaystyle \Gamma _{\beta }[n]=\Gamma _{\beta [n]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf9d84314dec4868cc4afbe8341311cadeafdb5)
