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BMS分析Part1：0~BO：修订间差异

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第1行： 第1行：
本条目展示[[Bashicu矩阵|BMS]]强度分析的第一部分
本条目展示[[Bashicu矩阵|BMS]]强度分析的第一部分
<math>\varnothing=0</math>
<math>(0)=1</math>
<math>(0)(0)=2</math>
<math>(0)(0)(0)=3</math>
<math>(0)(1)=(0)(0)(0)(0)(0)...=\omega</math>
<math>(0)(1)(0)=\omega+1</math>
<math>(0)(1)(0)(0)=\omega+2</math>
<math>(0)(1)(0)(1)=\omega\times2</math>
<math>(0)(1)(0)(1)(0)(1)=\omega\times3</math>
<math>(0)(1)(1)=(0)(1)(0)(1)(0)(1)...=\omega^2</math>
<math>(0)(1)(1)(0)=\omega^2+1</math>
<math>(0)(1)(1)(0)(1)=\omega^2+\omega</math>
<math>(0)(1)(1)(0)(1)(0)(1)=\omega^2+\omega\times2</math>
<math>(0)(1)(1)(0)(1)(1)=\omega^2\times2</math>
<math>(0)(1)(1)(0)(1)(1)(0)(1)=\omega^2\times2+\omega</math>
<math>(0)(1)(1)(0)(1)(1)(0)(1)(1)=\omega^2\times3</math>
<math>(0)(1)(1)(1)=\omega^3</math>
<math>(0)(1)(1)(1)(1)=\omega^4</math>
<math>(0)(1)(2)=(0)(1)(1)(1)(1)...=\omega^\omega</math>
<math>(0)(1)(2)(0)(1)(2)=\omega^\omega\times2</math>
<math>(0)(1)(2)(1)=\omega^{\omega+1}</math>
<math>(0)(1)(2)(1)(1)=\omega^{\omega+2}</math>
<math>(0)(1)(2)(1)(1)(1)=\omega^{\omega+3}</math>
<math>(0)(1)(2)(1)(2)=\omega^{\omega\times2}</math>
<math>(0)(1)(2)(1)(2)(1)(2)=\omega^{\omega\times3}</math>
<math>(0)(1)(2)(2)=\omega^{\omega^2}</math>
<math>(0)(1)(2)(2)(1)=\omega^{\omega^2+1}</math>
<math>(0)(1)(2)(2)(1)(2)=\omega^{\omega^2+\omega}</math>
<math>(0)(1)(2)(2)(1)(2)(2)=\omega^{\omega^2\times2}</math>
<math>(0)(1)(2)(2)(1)(2)(2)(1)(2)(2)=\omega^{\omega^2\times3}</math>
<math>(0)(1)(2)(2)(2)=\omega^{\omega^3}</math>
<math>(0)(1)(2)(2)(2)(2)=\omega^{\omega^4}</math>
<math>(0)(1)(2)(3)=(0)(1)(2)(2)(2)(2)...=\omega^{\omega^\omega}</math>
<math>(0)(1)(2)(3)(4)=\omega^{\omega^{\omega^\omega}}</math>
<math>(0)(1)(2)(3)(4)(5)=\omega^{\omega^{\omega^{\omega^\omega}}}</math>
<math>(0)(1,1)=(0)(1)(2)(3)(4)(5)(6)...=\varepsilon_0</math>
{| class="wikitable"
{| class="wikitable"
!BMS
|BMS
!Veblen
|Standard([[序数坍缩函数#BOCF|BOCF]])
!MOCF
|-
|-
|<math>(0)(1,1)</math>
|(0)
|<math>\varepsilon_0</math>
|1
|<math>\psi(0)</math>
|-
|-
|<math>(0)(1,1)(0,0)</math>
|(0)(0)
|<math>\varepsilon_0+1</math>
|2
|-
|-
|<math>(0)(1,1)(0,0)(1,0)</math>
|(0)(0)(0)
|<math>\varepsilon_0+\omega</math>
|3
|-
|-
|<math>(0)(1,1)(0,0)(1,1)</math>
|(0)(1)
|<math>\varepsilon_0\times2</math>
|ω = First Transfinte Ordinal
|-
|-
|<math>(0)(1,1)(0,0)(1,1)(0,0)(1,1)</math>
|(0)(1)(0)
|<math>\varepsilon_0\times3</math>
|ω+1
|-
|-
|<math>(0)(1,1)(1,0)</math>
|(0)(1)(0)(0)
|<math>\varepsilon_0\times\omega</math>
|ω+2
|-
|-
|<math>(0)(1,1)(1,0)(1,0)</math>
|(0)(1)(0)(1)
|<math>\varepsilon_0\times\omega^2</math>
|ω2
|-
|-
|<math>(0)(1,1)(1,0)(2,0)</math>
|(0)(1)(0)(1)(0)
|<math>\varepsilon_0\times\omega^\omega</math>
|ω2+1
|-
|-
|<math>(0)(1,1)(1,0)(2,0)(3,0)</math>
|(0)(1)(0)(1)(0)(1)
|<math>\varepsilon_0\times\omega^{\omega^\omega}</math>
|ω3
|-
|-
|<math>(0)(1,1)(1,0)(2,1)</math>
|(0)(1)(1)
|<math>\varepsilon_0^2</math>
|ω^2
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(1,0)(2,0)</math>
|(0)(1)(1)(0)
|<math>\varepsilon_0^2\times\omega</math>
|ω^2+1
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(1,0)(2,0)(3,0)</math>
|(0)(1)(1)(0)(1)
|<math>\varepsilon_0^2\times\omega^\omega</math>
|ω^2+ω
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(1,0)(2,1)</math>
|(0)(1)(1)(0)(1)(0)
|<math>\varepsilon_0^3</math>
|ω^2+ω+1
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(1,0)(2,1)(1,0)(2,1)</math>
|(0)(1)(1)(0)(1)(0)(1)
|<math>\varepsilon_0^4</math>
|ω^2+ω2
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)</math>
|(0)(1)(1)(0)(1)(1)
|<math>\varepsilon_0^\omega</math>
|ω^2*2
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)</math>
|(0)(1)(1)(0)(1)(1)(0)(1)(1)
|<math>\varepsilon_0^{\omega+1}</math>
|ω^2*3
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)(1,0)(2,1)</math>
|(0)(1)(1)(1)
|<math>\varepsilon_0^{\omega+2}</math>
|ω^3
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)(2,0)</math>
|(0)(1)(1)(1)(1)
|<math>\varepsilon_0^{\omega\times2}</math>
|ω^4
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)(2,0)(1,0)(2,1)(2,0)</math>
|(0)(1)(2)
|<math>\varepsilon_0^{\omega\times3}</math>
|ω^ω = Linear Array Ordinal
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(2,0)</math>
|(0)(1)(2)(0)
|<math>\varepsilon_0^{\omega^2}</math>
|ω^ω+1
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(3,0)</math>
|(0)(1)(2)(0)(1)(2)
|<math>\varepsilon_0^{\omega^\omega}</math>
|ω^ω*2
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(3,0)(4,0)</math>
|(0)(1)(2)(1)
|<math>\varepsilon_0^{\omega^{\omega^\omega}}</math>
|ω^(ω+1)
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(3,1)</math>
|(0)(1)(2)(1)(1)
|<math>\varepsilon_0^{\varepsilon_0}</math>
|ω^(ω+2)
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)</math>
|(0)(1)(2)(1)(2)
|<math>\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}</math>
|ω^(ω2)
|-
|-
|<math>(0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)</math>
|(0)(1)(2)(1)(2)(1)(2)
|<math>\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}</math>
|ω^(ω3)
|-
|-
|<math>(0)(1,1)(1,1)</math>
|(0)(1)(2)(2)
|<math>\varepsilon_1</math>
|ω^ω^2
|<math>\psi(1)</math>
|-
|-
|<math>(0)(1,1)(1,1)(0,0)(1,1)(1,1)</math>
|(0)(1)(2)(2)(2)
|<math>\varepsilon_1\times2</math>
|ω^ω^3
|-
|-
|<math>(0)(1,1)(1,1)(1,0)</math>
|(0)(1)(2)(3)
|<math>\varepsilon_1\times\omega</math>
|ω^ω^ω
|-
|-
|<math>(0)(1,1)(1,1)(1,0)(2,1)</math>
|(0)(1)(2)(3)(1)
|<math>\varepsilon_1\times\varepsilon_0</math>
|ω^(ω^ω+1)
|-
|-
|<math>(0)(1,1)(1,1)(1,0)(2,1)(2,0)(3,1)</math>
|(0)(1)(2)(3)(2)
|<math>\varepsilon_1\times\varepsilon_0^2</math>
|ω^ω^(ω+1)
|-
|-
|<math>(0)(1,1)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)</math>
|(0)(1)(2)(3)(3)
|<math>\varepsilon_1\times\varepsilon_0^{\varepsilon_0}</math>
|ω^ω^(ω^2)
|-
|-
|<math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)</math>
|(0)(1)(2)(3)(4)
|<math>\varepsilon_1^2</math>
|ω^ω^ω^ω
|-
|-
|<math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)</math>
|(0)(1)(2)(3)(4)(5)
|<math>\varepsilon_1^\omega</math>
|ω^^5
|-
|-
|<math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)</math>
|(0)(1,1)
|<math>\varepsilon_1^{\varepsilon_0}</math>
|ψ(Ω) = Small Cantor's Ordinal
|-
|-
|<math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)</math>
|(0)(1,1)(0)
|<math>\varepsilon_1^{\varepsilon_1}</math>
|ψ(Ω)+1
|-
|-
|<math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)(3,0)(4,1)(4,1)</math>
|(0)(1,1)(0)(1,1)
|<math>\varepsilon_1^{\varepsilon_1^{\varepsilon_1}}</math>
|ψ(Ω)*2
|-
|-
|<math>(0)(1,1)(1,1)(1,1)</math>
|(0)(1,1)(1)
|<math>\varepsilon_2</math>
|ψ(Ω+1)
|<math>\psi(2)</math>
|-
|-
|<math>(0)(1,1)(1,1)(1,1)(1,1)</math>
|(0)(1,1)(1)(2)
|<math>\varepsilon_3</math>
|ψ(Ω+ω)
|<math>\psi(3)</math>
|-
|-
|<math>(0)(1,1)(2,0)</math>
|(0)(1,1)(1)(2,1)
|<math>\varepsilon_\omega</math>
|ψ(Ω+ψ(Ω))
|<math>\psi(\omega)</math>
|-
|-
|<math>(0)(1,1)(2,0)(1,0)</math>
|(0)(1,1)(1)(2,1)(1)
|<math>\varepsilon_\omega\times\omega</math>
|ψ(Ω+ψ(Ω)+1)
|-
|-
|<math>(0)(1,1)(2,0)(1,0)(2,1)</math>
|(0)(1,1)(1)(2,1)(2)
|<math>\varepsilon_\omega\times\varepsilon_0</math>
|ψ(Ω+ψ(Ω+1))
|-
|-
|<math>(0)(1,1)(2,0)(1,0)(2,1)(2,1)</math>
|(0)(1,1)(1)(2,1)(2)(3,1)
|<math>\varepsilon_\omega\times\varepsilon_1</math>
|ψ(Ω+ψ(Ω+ψ(Ω)))
|-
|-
|<math>(0)(1,1)(2,0)(1,0)(2,1)(2,1)(2,1)</math>
|(0)(1,1)(1,1)
|<math>\varepsilon_\omega\times\varepsilon_2</math>
|ψ(Ω*2)
|-
|-
|<math>(0)(1,1)(2,0)(1,0)(2,1)(2,1)(2,1)(2,1)</math>
|(0)(1,1)(1,1)(1)
|<math>\varepsilon_\omega\times\varepsilon_3</math>
|ψ(Ω*2+1)
|-
|-
|<math>(0)(1,1)(2,0)(1,0)(2,1)(3,0)</math>
|(0)(1,1)(1,1)(1)(2)
|<math>\varepsilon_\omega^2</math>
|ψ(Ω*2+ω)
|-
|-
|<math>(0)(1,1)(2,0)(1,0)(2,1)(3,0)(2,0)(3,1)(4,0)</math>
|(0)(1,1)(1,1)(1)(2,1)
|<math>\varepsilon_\omega^{\varepsilon_\omega}</math>
|ψ(Ω*2+ψ(Ω))
|
|-
|-
|<math>(0)(1,1)(2,0)(1,1)</math>
|(0)(1,1)(1,1)(1)(2,1)(2,1)
|<math>\varepsilon_{\omega+1}</math>
|ψ(Ω*2+ψ(Ω*2))
|<math>\psi(\omega+1)</math>
|-
|-
|<math>(0)(1,1)(2,0)(1,1)(1,1)</math>
|(0)(1,1)(1,1)(1)(2,1)(2,1)(2)
|<math>\varepsilon_{\omega+2}</math>
|ψ(Ω*2+ψ(Ω*2+1))
|<math>\psi (\omega +2)</math>
|-
|-
|<math>(0)(1,1)(2,0)(1,1)(1,1)(1,1)</math>
|(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)(3,1)
|<math>\varepsilon_{\omega+3}</math>
|ψ(Ω*2+ψ(Ω*2+ψ(Ω*2)))
|<math>\psi(\omega+3)</math>
|-
|-
|<math>(0)(1,1)(2,0)(1,1)(2,0)</math>
|(0)(1,1)(1,1)(1,1)
|<math>\varepsilon_{\omega\times2}</math>
|ψ(Ω*3)
|<math>\psi (\omega \times 2)</math>
|-
|-
|<math>(0)(1,1)(2,0)(1,1)(2,0)(1,1)(2,0)</math>
|(0)(1,1)(1,1)(1,1)(1)(2,1)(2,1)(2,1)
|<math>\varepsilon_{\omega\times3}</math>
|ψ(Ω*3+ψ(Ω*3))
|<math>\psi (\omega \times 3)</math>
|-
|-
|<math>(0)(1,1)(2,0)(2,0)</math>
|(0)(1,1)(1,1)(1,1)(1,1)
|<math>\varepsilon_{\omega^2}</math>
|ψ(Ω*4)
|<math>\psi (\omega ^2)</math>
|-
|-
|<math>(0)(1,1)(2,0)(2,0)(2,0)</math>
|(0)(1,1)(1,1)(1,1)(1,1)(1,1)
|<math>\varepsilon_{\omega^3}</math>
|ψ(Ω*5)
|<math>\psi (\omega ^3)</math>
|-
|-
|<math>(0)(1,1)(2,0)(3,0)</math>
|(0)(1,1)(2)
|<math>\varepsilon_{\omega^\omega}</math>
|ψ(Ω*ω)
|<math>\psi(\omega^\omega)</math>
|-
|-
|<math>(0)(1,1)(2,0)(3,0)(4,0)</math>
|(0)(1,1)(2)(1)
|<math>\varepsilon_{\omega^{\omega^\omega}}</math>
|ψ(Ω*ω+1)
|<math>\psi(\omega^{\omega^\omega})</math>
|-
|-
|<math>(0)(1,1)(2,0)(3,1)</math>
|(0)(1,1)(2)(1)(2)
|<math>\varepsilon_{\varepsilon_0}</math>
|ψ(Ω*ω+ω)
|<math>\psi(\psi(0))</math>
|-
|-
|<math>(0)(1,1)(2,0)(3,1)(1,1)</math>
|(0)(1,1)(2)(1)(2,1)
|<math>\varepsilon_{\varepsilon_0+1}</math>
|ψ(Ω*ω+ψ(Ω))
|<math>\psi(\psi(0)+1)</math>
|-
|-
|<math>(0)(1,1)(2,0)(3,1)(3,1)</math>
|(0)(1,1)(2)(1)(2,1)(2,1)
|<math>\varepsilon_{\varepsilon_1}</math>
|ψ(Ω*ω+ψ(Ω*2))
|<math>\psi(\psi(1))</math>
|-
|-
|<math>(0)(1,1)(2,0)(3,1)(4,0)</math>
|(0)(1,1)(2)(1)(2,1)(3)
|<math>\varepsilon_{\varepsilon_\omega}</math>
|ψ(Ω*ω+ψ(Ω*ω))
|<math>\psi(\psi(\omega))</math>
|-
|-
|<math>(0)(1,1)(2,0)(3,1)(4,0)(5,1)</math>
|(0)(1,1)(2)(1)(2,1)(3)(2)
|<math>\varepsilon_{\varepsilon_{\varepsilon_0}}</math>
|ψ(Ω*ω+ψ(Ω*ω+1))
|<math>\psi(\psi(\psi(0)))</math>
|-
|-
|<math>(0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)</math>
|(0)(1,1)(2)(1)(2,1)(3)(2)(3,1)
|<math>\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}</math>
|ψ(Ω*ω+ψ(Ω*ω+ψ(Ω)))
|<math>\psi(\psi(\psi(\psi(0))))</math>
|-
|-
|<math>(0)(1,1)(2,1)</math>
|(0)(1,1)(2)(1,1)
|<math>\zeta_0</math>
|ψ(Ω*(ω+1))
|<math>\psi(\Omega)</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,0)(2,1)(3,1)</math>
|(0)(1,1)(2)(1,1)(1)
|<math>\zeta_0^2</math>
|ψ(Ω*(ω+1)+1)
|-
|-
|<math>(0)(1,1)(2,1)(1,0)(2,1)(3,1)(2,0)(3,1)(4,1)</math>
|(0)(1,1)(2)(1,1)(1)(2,1)
|<math>\zeta_0^{\zeta_0}</math>
|ψ(Ω*(ω+1)+ψ(Ω))
|-
|-
|<math>(0)(1,1)(2,1)(1,1)</math>
|(0)(1,1)(2)(1,1)(1)(2,1)(3)(2,1)
|<math>\varepsilon_{\zeta_0+1}</math>
|ψ(Ω*(ω+1)+ψ(Ω*(ω+1)))
|<math>\psi(\Omega+1)</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(1,0)(2,1)(3,1)(2,1)</math>
|(0)(1,1)(2)(1,1)(1,1)
|<math>\varepsilon_{\zeta_0+1}^2</math>
|ψ(Ω*(ω+2))
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(1,1)</math>
|(0)(1,1)(2)(1,1)(1,1)(1,1)
|<math>\varepsilon_{\zeta_0+2}</math>
|ψ(Ω*(ω+3))
|<math>\psi(\Omega+2)</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,0)</math>
|(0)(1,1)(2)(1,1)(2)
|<math>\varepsilon_{\zeta_0+\omega}</math>
|ψ(Ω*(ω2))
|<math>\psi(\Omega+\omega)</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,0)(3,1)</math>
|(0)(1,1)(2)(1,1)(2)(1,1)
|<math>\varepsilon_{\zeta_0+\varepsilon_0}</math>
|ψ(Ω*(ω2+1))
|<math>\psi(\Omega+\psi(0))</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)</math>
|(0)(1,1)(2)(1,1)(2)(1,1)(2)
|<math>\varepsilon_{\zeta_0\times2}</math>
|ψ(Ω*(ω3))
|<math>\psi(\psi(\Omega)\times2)</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)</math>
|(0)(1,1)(2)(2)
|<math>\varepsilon_{\varepsilon_{\zeta_0+1}}</math>
|ψ(Ω*ω^2)
|<math>\psi(\psi(\Omega+1))</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(5,1)</math>
|(0)(1,1)(2)(2)(1,1)
|<math>\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}}</math>
|ψ(Ω*(ω^2+1))
|<math>\psi(\psi(\psi(\Omega+1)))</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,1)</math>
|(0)(1,1)(2)(2)(1,1)(2)(2)
|<math>\zeta_1</math>
|ψ(Ω*ω^2*2)
|<math>\psi(\Omega\times2)</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)</math>
|(0)(1,1)(2)(2)(2)
|<math>\varepsilon_{\zeta_1+1}</math>
|ψ(Ω*ω^3)
|<math>\psi(\psi(\Omega\times2+1))</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,0)</math>
|(0)(1,1)(2)(3)
|<math>\varepsilon_{\zeta_1+\omega}</math>
|ψ(Ω*ω^ω)
|<math>\psi(\psi(\Omega\times2+\omega))</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)</math>
|(0)(1,1)(2)(3)(4)
|<math>\zeta_2</math>
|ψ(Ω*ω^ω^ω)
|<math>\psi(\Omega\times3)</math>
|-
|-
|<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)</math>
|(0)(1,1)(2)(3,1)
|<math>\zeta_3</math>
|ψ(Ω*ψ(Ω))
|
|-
|-
|<math>(0)(1,1)(2,1)(2,0)</math>
|(0)(1,1)(2)(3,1)(3,1)
|<math>\zeta_\omega</math>
|ψ(Ω*ψ(Ω*2))
|-
|-
|<math>(0)(1,1)(2,1)(2,0)(2,0)</math>
|(0)(1,1)(2)(3,1)(4)
|<math>\zeta_{\omega^2}</math>
|ψ(Ω*ψ(Ω*ω))
|-
|-
|<math>(0)(1,1)(2,1)(2,0)(3,1)</math>
|(0)(1,1)(2)(3,1)(4)(5,1)
|<math>\zeta_{\varepsilon_0}</math>
|ψ(Ω*ψ(Ω*ψ(Ω)))
|
|-
|-
|<math>(0)(1,1)(2,1)(2,0)(3,1)(4,1)</math>
|(0)(1,1)(2,1)
|<math>\zeta_{\zeta_0}</math>
|ψ(Ω^2) = Cantor's Ordinal
|-
|-
|<math>(0)(1,1)(2,1)(2,0)(3,1)(4,1)(4,0)(5,1)(6,1)</math>
|(0)(1,1)(2,1)(1)
|<math>\zeta_{\zeta_{\zeta_0}}</math>
|ψ(Ω^2+1)
|-
|-
|<math>(0)(1,1)(2,1)(2,1)</math>
|(0)(1,1)(2,1)(1)(2,1)
|<math>\eta_0</math>
|ψ(Ω^2+ψ(Ω))
|-
|-
|<math>(0)(1,1)(2,1)(2,1)(1,1)</math>
|(0)(1,1)(2,1)(1)(2,1)(3,1)
|<math>\varepsilon_{\eta_0+1}</math>
|ψ(Ω^2+ψ(Ω^2))
|-
|-
|<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)</math>
|(0)(1,1)(2,1)(1,1)
|<math>\zeta_{\eta_0+1}</math>
|ψ(Ω^2+Ω)
|-
|-
|<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,0)(3,1)(4,1)(4,1)(3,1)(4,1)</math>
|(0)(1,1)(2,1)(1,1)(1,1)
|<math>\zeta_{\zeta_{\eta_0+1}}</math>
|ψ(Ω^2+Ω*2)
|-
|-
|<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)</math>
|(0)(1,1)(2,1)(1,1)(2)
|<math>\eta_1</math>
|ψ(Ω^2+Ω*ω)
|-
|-
|<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)</math>
|(0)(1,1)(2,1)(1,1)(2)(3,1)
|<math>\eta_2</math>
|ψ(Ω^2+Ω*ψ(Ω))
|-
|-
|<math>(0)(1,1)(2,1)(2,1)(2,0)</math>
|(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)
|<math>\eta_\omega</math>
|ψ(Ω^2+Ω*ψ(Ω^2))
|-
|-
|<math>(0)(1,1)(2,1)(2,1)(2,1)</math>
|(0)(1,1)(2,1)(1,1)(2,1)
|<math>\varphi(4,0)</math>
|ψ(Ω^2*2)
|<math>\psi(\Omega^3)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)</math>
|(0)(1,1)(2,1)(1,1)(2,1)(1,1)
|<math>\varphi(\omega,0)</math>
|ψ(Ω^2*2+Ω)
|<math>\psi(\Omega^\omega)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(1,1)</math>
|(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2)
|<math>\varphi(1,\varphi(\omega,0)+1)</math>
|ψ(Ω^2*2+Ω*ω)
|<math>\psi(\Omega^\omega+1)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(1,1)(2,1)</math>
|(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)
|<math>\varphi(2,\varphi(\omega,0)+1)</math>
|ψ(Ω^2*3)
|<math>\psi(\Omega^\omega+\Omega)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(1,1)(2,1)(2,1)</math>
|(0)(1,1)(2,1)(2)
|<math>\varphi(3,\varphi(\omega,0)+1)</math>
|ψ(Ω^2*ω)
|<math>\psi(\Omega^\omega+\Omega^2)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)</math>
|(0)(1,1)(2,1)(2)(1,1)
|<math>\varphi(\omega,1)</math>
|ψ(Ω^2*ω+Ω)
|<math>\psi(\Omega^\omega\times2)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)</math>
|(0)(1,1)(2,1)(2)(1,1)(2,1)
|<math>\varphi(\omega,2)</math>
|ψ(Ω^2*(ω+1))
|<math>\psi(\Omega^\omega\times3)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(2,0)</math>
|(0)(1,1)(2,1)(2)(2)
|<math>\varphi(\omega,\omega)</math>
|ψ(Ω^2*ω2)
|<math>\psi(\Omega^\omega\times\omega)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(2,1)</math>
|(0)(1,1)(2,1)(2)(3)
|<math>\varphi(\omega+1,0)</math>
|ψ(Ω^2*ω^2)
|<math>\psi(\Omega^{\omega+1})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(2,1)(2,1)</math>
|(0)(1,1)(2,1)(2)(3)(4)
|<math>\varphi(\omega+2,0)</math>
|ψ(Ω^2*ω^ω)
|<math>\psi(\Omega^{\omega+2})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(2,1)(2,1)(2,1)</math>
|(0)(1,1)(2,1)(2)(3,1)
|<math>\varphi(\omega+3,0)</math>
|ψ(Ω^2*ψ(Ω))
|<math>\psi(\Omega^{\omega+3})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(2,1)(3,0)</math>
|(0)(1,1)(2,1)(2)(3,1)(3,1)
|<math>\varphi(\omega\times2,0)</math>
|ψ(Ω^2*ψ(Ω*2))
|<math>\psi(\Omega^{\omega\times2})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(2,1)(3,0)(2,1)(3,0)</math>
|(0)(1,1)(2,1)(2)(3,1)(4,1)
|<math>\varphi(\omega\times3,0)</math>
|ψ(Ω^2*ψ(Ω^2))
|<math>\psi(\Omega^{\omega\times3})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(3,0)</math>
|(0)(1,1)(2,1)(2)(3,1)(4,1)(4)
|<math>\varphi(\omega^2,0)</math>
|ψ(Ω^2*ψ(Ω^2*ω))
|<math>\psi(\Omega^{\omega^2})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(4,1)</math>
|(0)(1,1)(2,1)(2,1)
|<math>\varphi(\varphi(1,0),0)</math>
|ψ(Ω^3) = Large Cantor's Ordinal
|<math>\psi(\Omega^{\psi(0)})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(4,1)(5,1)</math>
|(0)(1,1)(2,1)(2,1)(1)
|<math>\varphi(\varphi(2,0),0)</math>
|ψ(Ω^3+1)
|<math>\psi(\Omega^{\psi(\Omega)})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)</math>
|(0)(1,1)(2,1)(2,1)(1)(2,1)
|<math>\varphi(\varphi(\omega,0),0)</math>
|ψ(Ω^3+ψ(Ω))
|<math>\psi(\Omega^{\psi(\Omega^\omega)})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)(9,0)</math>
|(0)(1,1)(2,1)(2,1)(1)(2,1)(3,1)(3,1)
|<math>\varphi(\varphi(\varphi(\omega,0),0),0)</math>
|ψ(Ω^3+ψ(Ω^3))
|<math>\psi(\Omega^{\psi(\Omega^{\psi(\Omega^\omega)})})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)</math>
|(0)(1,1)(2,1)(2,1)(1,1)
|<math>\varphi(1,0,0)=\Gamma_0</math>
|ψ(Ω^3+Ω)
|<math>\psi(\Omega^\Omega)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(1,1)
|<math>\varphi(1,0,1)=\Gamma_1</math>
|ψ(Ω^3+Ω*2)
|<math>\psi(\Omega^\Omega\times2)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2)
|<math>\varphi(1,0,2)=\Gamma_2</math>
|ψ(Ω^3+Ω*ω)
|<math>\psi(\Omega^\Omega\times3)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,0)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2)(3,1)
|<math>\varphi(1,0,\omega)=\Gamma_\omega</math>
|ψ(Ω^3+Ω*ψ(Ω))
|<math>\psi(\Omega^\Omega\times\omega)</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)(5,1)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2)(3,1)(4,1)(4,1)
|<math>\varphi(1,0,\varphi(1,0,0))=\Gamma_{\Gamma_0}</math>
|ψ(Ω^3+Ω*ψ(Ω^3))
|<math>\psi(\Omega^\Omega\times\psi(\Omega^\Omega))</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)(5,1)(4,0)(5,1)(6,1)(7,1)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2,1)
|<math>\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))=\Gamma_{\Gamma_{\Gamma_0}}</math>
|ψ(Ω^3+Ω^2)
|<math>\psi(\Omega^\Omega\times\psi(\Omega^\Omega\times\psi(\Omega^\Omega)))</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,1)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)
|<math>\varphi(1,1,0)</math>
|ψ(Ω^3*2)
|<math>\psi(\Omega^{\Omega+1})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,1)(2,1)</math>
|(0)(1,1)(2,1)(2,1)(2)
|<math>\varphi(1,2,0)</math>
|ψ(Ω^3*ω)
|<math>\psi(\Omega^{\Omega+2})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,1)(3,0)</math>
|(0)(1,1)(2,1)(2,1)(2)(1,1)(2,1)(2,1)
|<math>\varphi(1,\omega,0)</math>
|ψ(Ω^3*(ω+1))
|<math>\psi(\Omega^{\Omega+\omega})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)(5,1)(6,0)</math>
|(0)(1,1)(2,1)(2,1)(2)(2)
|<math>\varphi(1,\varphi(1,\omega,0),0)</math>
|ψ(Ω^3*ω2)
|<math>\psi(\Omega^{\Omega+\psi(\Omega^{\Omega+\omega})})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,1)(3,1)</math>
|(0)(1,1)(2,1)(2,1)(2)(3)
|<math>\psi(2,0,0)</math>
|ψ(Ω^3*ω^2)
|<math>\psi(\Omega^{\Omega\times2})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)</math>
|(0)(1,1)(2,1)(2,1)(2)(3,1)
|<math>\psi(3,0,0)</math>
|ψ(Ω^3*ψ(Ω))
|<math>\psi(\Omega^{\Omega\times3})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(3,0)</math>
|(0)(1,1)(2,1)(2,1)(2)(3,1)(4,1)(4,1)
|<math>\varphi(\omega,0,0)</math>
|ψ(Ω^3+ψ(Ω^3))
|<math>\psi(\Omega^{\Omega\times\omega})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(3,1)</math>
|(0)(1,1)(2,1)(2,1)(2,1)
|<math>\varphi(1,0,0,0)</math>
|ψ(Ω^4)
|<math>\psi(\Omega^{\Omega^2})</math>
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(3,1)(3,1)</math>
|(0)(1,1)(2,1)(2,1)(2,1)(1,1)
|<math>\varphi(1,0,0,0,0)</math>
|ψ(Ω^4+Ω)
|<math>\psi(\Omega^{\Omega^3})</math>
|-
|}
|(0)(1,1)(2,1)(2,1)(2,1)(2)
{| class="wikitable"
|ψ(Ω^4*ω)
!BMS
|-
!MOCF
|(0)(1,1)(2,1)(2,1)(2,1)(2,1)
|ψ(Ω^5)
|-
|(0)(1,1)(2,1)(3)
|ψ(Ω^ω) = Hyper Cantor's Ordinal
|-
|(0)(1,1)(2,1)(3)(1)
|ψ(Ω^ω+1)
|-
|(0)(1,1)(2,1)(3)(1)(2)
|ψ(Ω^ω+ω)
|-
|(0)(1,1)(2,1)(3)(1)(2,1)
|ψ(Ω^ω+SCO)
|-
|(0)(1,1)(2,1)(3)(1)(2,1)(3,1)
|ψ(Ω^ω+CO)
|-
|(0)(1,1)(2,1)(3)(1)(2,1)(3,1)(4)
|ψ(Ω^ω+HCO)
|-
|(0)(1,1)(2,1)(3)(1,1)
|ψ(Ω^ω+Ω)
|-
|(0)(1,1)(2,1)(3)(1,1)(1,1)
|ψ(Ω^ω+Ω*2)
|-
|(0)(1,1)(2,1)(3)(1,1)(2)
|ψ(Ω^ω+Ω*ω)
|-
|(0)(1,1)(2,1)(3)(1,1)(2)(3,1)
|ψ(Ω^ω+Ω*SCO)
|-
|(0)(1,1)(2,1)(3)(1,1)(2,1)
|ψ(Ω^ω+Ω^2)
|-
|(0)(1,1)(2,1)(3)(1,1)(2,1)(3)
|ψ(Ω^ω*2)
|-
|(0)(1,1)(2,1)(3)(2)
|ψ(Ω^ω*ω)
|-
|(0)(1,1)(2,1)(3)(2)(3,1)
|ψ(Ω^ω*SCO)
|-
|(0)(1,1)(2,1)(3)(2,1)
|ψ(Ω^(ω+1))
|-
|(0)(1,1)(2,1)(3)(2,1)(2)
|ψ(Ω^(ω+1)*ω)
|-
|(0)(1,1)(2,1)(3)(2,1)(2,1)
|ψ(Ω^(ω+2))
|-
|(0)(1,1)(2,1)(3)(2,1)(3)
|ψ(Ω^(ω2))
|-
|(0)(1,1)(2,1)(3)(2,1)(3)(2,1)
|ψ(Ω^(ω2+1))
|-
|(0)(1,1)(2,1)(3)(3)
|ψ(Ω^(ω3))
|-
|(0)(1,1)(2,1)(3)(4)
|ψ(Ω^ω^2)
|-
|(0)(1,1)(2,1)(3)(4,1)
|ψ(Ω^SCO)
|-
|(0)(1,1)(2,1)(3)(4,1)(5,1)
|ψ(Ω^CO)
|-
|(0)(1,1)(2,1)(3)(4,1)(5,1)(6)
|ψ(Ω^HCO)
|-
|(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1)
|ψ(Ω^ψ(Ω^HCO))
|-
|(0)(1,1)(2,1)(3,1)
|ψ(Ω^Ω) = Feferman-Schütte Ordinal
|-
|(0)(1,1)(2,1)(3,1)(1)
|ψ(Ω^Ω+1)
|-
|(0)(1,1)(2,1)(3,1)(1)(2)
|ψ(Ω^Ω+ω)
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)
|ψ(Ω^Ω+SCO)
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)
|ψ(Ω^Ω+CO)
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4)
|ψ(Ω^Ω+HCO)
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)
|ψ(Ω^Ω+FSO)
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)
|ψ(Ω^Ω+ψ(Ω^Ω+1))
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)(3,1)
|ψ(Ω^Ω+ψ(Ω^Ω+SCO))
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)(3,1)(4,1)
|ψ(Ω^Ω+ψ(Ω^Ω+CO))
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)(3,1)(4,1)(5)
|ψ(Ω^Ω+ψ(Ω^Ω+HCO))
|-
|(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)(3,1)(4,1)(5,1)
|ψ(Ω^Ω+ψ(Ω^Ω+FSO))
|-
|(0)(1,1)(2,1)(3,1)(1,1)
|ψ(Ω^Ω+Ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(1)
|ψ(Ω^Ω+Ω+1)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(1)(2,1)
|ψ(Ω^Ω+Ω+SCO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(1)(2,1)(3,1)(4,1)
|ψ(Ω^Ω+Ω+FSO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(1,1)
|ψ(Ω^Ω+Ω*2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2)
|ψ(Ω^Ω+Ω*ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2)(3,1)
|ψ(Ω^Ω+Ω*SCO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2)(3,1)(4,1)(5,1)
|ψ(Ω^Ω+Ω*FSO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)
|ψ(Ω^Ω+Ω^2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1)(2,1)
|ψ(Ω^Ω+Ω^2+SCO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1)(2,1)(3,1)(4,1)
|ψ(Ω^Ω+Ω^2+FSO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1,1)
|ψ(Ω^Ω+Ω^2+Ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1,1)(1,1)
|ψ(Ω^Ω+Ω^2+Ω*2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1,1)(2,1)
|ψ(Ω^Ω+Ω^2*2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)
|ψ(Ω^Ω+Ω^2*3)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2)
|ψ(Ω^Ω+Ω^2*ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2)(3,1)
|ψ(Ω^Ω+Ω^2*SCO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2)(3,1)(4,1)(5,1)
|ψ(Ω^Ω+Ω^2*FSO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)
|ψ(Ω^Ω+Ω^3)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)(1,1)
|ψ(Ω^Ω+Ω^3+Ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)
|ψ(Ω^Ω+Ω^3*2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)(2)
|ψ(Ω^Ω+Ω^3*ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)(2,1)
|ψ(Ω^Ω+Ω^4)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)
|ψ(Ω^Ω+Ω^ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(1,1)(2,1)(3)
|ψ(Ω^Ω+Ω^ω*2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(2)
|ψ(Ω^Ω+Ω^ω*ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(2,1)
|ψ(Ω^Ω+Ω^(ω+1))
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(2,1)(3)
|ψ(Ω^Ω+Ω^(ω2))
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(3)
|ψ(Ω^Ω+Ω^ω^2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)
|ψ(Ω^Ω+Ω^SCO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)
|ψ(Ω^Ω+Ω^FSO)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2)
|ψ(Ω^Ω+Ω^FSO*ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)
|ψ(Ω^Ω+Ω^(FSO+1))
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(3)
|ψ(Ω^Ω+Ω^(FSO+ω))?
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4)
|ψ(Ω^Ω+Ω^(FSO*ω))
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)
|ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω))
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)(5,1)(6)(7,1)(8,1)(9,1)
|ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω^ψ(Ω^Ω)))
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)(5,1)(6)(7,1)(8,1)(9,1)(5,1)
|ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω^(ψ(Ω^Ω)+1)))?
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
|ψ(Ω^Ω*2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)
|ψ(Ω^Ω*2+Ω)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)
|ψ(Ω^Ω*2+Ω^2)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)
|ψ(Ω^Ω*2+Ω^3)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
|ψ(Ω^Ω*3)
|-
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
|ψ(Ω^Ω*4)
|-
|(0)(1,1)(2,1)(3,1)(2)
|ψ(Ω^Ω*ω)
|-
|(0)(1,1)(2,1)(3,1)(2)(3,1)
|ψ(Ω^Ω*SCO)
|-
|(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)
|ψ(Ω^Ω*CO)
|-
|(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)
|ψ(Ω^Ω*FSO)
|-
|(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)(4)(5,1)(6,1)(7,1)
|ψ(Ω^Ω*ψ(Ω^Ω*FSO))
|-
|(0)(1,1)(2,1)(3,1)(2,1)
|ψ(Ω^(Ω+1))
|-
|(0)(1,1)(2,1)(3,1)(2,1)(1,1)
|ψ(Ω^(Ω+1)+Ω)
|-
|(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)
|ψ(Ω^(Ω+1)+Ω^Ω)
|-
|(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1)
|ψ(Ω^(Ω+1)*2)
|-
|(0)(1,1)(2,1)(3,1)(2,1)(2)
|ψ(Ω^(Ω+1)*ω)
|-
|(0)(1,1)(2,1)(3,1)(2,1)(2)(3,1)
|ψ(Ω^(Ω+1)*SCO)
|-
|(0)(1,1)(2,1)(3,1)(2,1)(2)(3,1)(4,1)(5,1)
|ψ(Ω^(Ω+1)*FSO)
|-
|(0)(1,1)(2,1)(3,1)(2,1)(2,1)
|ψ(Ω^(Ω+2))
|-
|(0)(1,1)(2,1)(3,1)(2,1)(3)
|ψ(Ω^(Ω+ω))
|-
|(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1)
|ψ(Ω^(Ω+SCO))
|-
|(0)(1,1)(2,1)(3,1)(2,1)(3,1)
|ψ(Ω^(Ω*2))
|-
|(0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)
|ψ(Ω^(Ω*3))
|-
|(0)(1,1)(2,1)(3,1)(3)
|ψ(Ω^(Ω*ω))
|-
|(0)(1,1)(2,1)(3,1)(3)(4,1)
|ψ(Ω^(Ω*SCO))
|-
|(0)(1,1)(2,1)(3,1)(3,1)
|ψ(Ω^Ω^2) = Ackerman's Ordinal
|-
|(0)(1,1)(2,1)(3,1)(3,1)(1,1)
|ψ(Ω^Ω^2+Ω)
|-
|(0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(6,1)
|ψ(Ω^Ω^2+Ω^ACO)
|-
|(0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3,1)
|ψ(Ω^Ω^2+Ω^Ω)
|-
|(0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3,1)(3)(4,1)(5,1)(6,1)(6,1)
|ψ(Ω^Ω^2+Ω^(Ω*ACO))
|-
|(0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3,1)(3,1)
|ψ(Ω^Ω^2*2)
|-
|(0)(1,1)(2,1)(3,1)(3,1)(2)(3,1)(4,1)(5,1)(5,1)
|ψ(Ω^Ω^2*ACO)
|-
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)
|ψ(Ω^(Ω^2+1))
|-
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)
|ψ(Ω^(Ω^2+Ω))
|-
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(7,1)
|ψ(Ω^(Ω^2+Ω*ACO))
|-
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1)
|ψ(Ω^(Ω^2*2))
|-
|(0)(1,1)(2,1)(3,1)(3,1)(3)
|ψ(Ω^(Ω^2*ω))
|-
|(0)(1,1)(2,1)(3,1)(3,1)(3)(4,1)(5,1)(6,1)(6,1)
|ψ(Ω^(Ω^2*ACO))
|-
|(0)(1,1)(2,1)(3,1)(3,1)(3,1)
|ψ(Ω^Ω^3)
|-
|(0)(1,1)(2,1)(3,1)(3,1)(3,1)(3,1)
|ψ(Ω^Ω^4)
|-
|(0)(1,1)(2,1)(3,1)(4)
|ψ(Ω^Ω^ω) = Small Veblen's Ordinal
|-
|(0)(1,1)(2,1)(3,1)(4)(1,1)
|ψ(Ω^Ω^ω+Ω)
|-
|(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)
|ψ(Ω^Ω^ω+Ω^2)
|-
|(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(2,1)
|ψ(Ω^Ω^ω+Ω^3)
|-
|(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(3)
|ψ(Ω^Ω^ω+Ω^ω)
|-
|(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(7)
|ψ(Ω^Ω^ω+Ω^SVO)
|-
|(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(3,1)
|ψ(Ω^Ω^ω+Ω^Ω)
|-
|(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(3,1)(4)
|ψ(Ω^Ω^ω*2)
|-
|(0)(1,1)(2,1)(3,1)(4)(2)
|ψ(Ω^Ω^ω*ω)
|-
|(0)(1,1)(2,1)(3,1)(4)(2,1)
|ψ(Ω^(Ω^ω+1))
|-
|(0)(1,1)(2,1)(3,1)(4)(2,1)(3)
|ψ(Ω^(Ω^ω+ω))
|-
|(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1)
|ψ(Ω^(Ω^ω+Ω))
|-
|(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1)(3)(4,1)(5,1)(6,1)(7)
|ψ(Ω^(Ω^ω+Ω*SVO))
|-
|(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1)(3,1)
|ψ(Ω^(Ω^ω+Ω^2))
|-
|(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1)(4)
|ψ(Ω^(Ω^ω*2))
|-
|(0)(1,1)(2,1)(3,1)(4)(3)
|ψ(Ω^(Ω^ω*ω))
|-
|(0)(1,1)(2,1)(3,1)(4)(3)(4,1)(5,1)(6,1)(7)
|ψ(Ω^(Ω^ω*SVO))
|-
|(0)(1,1)(2,1)(3,1)(4)(3,1)
|ψ(Ω^Ω^(ω+1))
|-
|(0)(1,1)(2,1)(3,1)(4)(3,1)(3,1)
|ψ(Ω^Ω^(ω+2))
|-
|(0)(1,1)(2,1)(3,1)(4)(3,1)(4)
|ψ(Ω^Ω^(ω2))
|-
|(0)(1,1)(2,1)(3,1)(4)(4)
|ψ(Ω^Ω^ω^2)
|-
|(0)(1,1)(2,1)(3,1)(4)(5)
|ψ(Ω^Ω^ω^ω)
|-
|(0)(1,1)(2,1)(3,1)(4)(5,1)
|ψ(Ω^Ω^SCO)
|-
|(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)
|ψ(Ω^Ω^FSO)
|-
|(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8)
|ψ(Ω^Ω^SVO)
|-
|(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8)(9,1)(A,1)(B,1)(C)
|ψ(Ω^Ω^ψ(Ω^Ω^SVO))
|-
|(0)(1,1)(2,1)(3,1)(4,1)
|ψ(Ω^Ω^Ω) = Large Veblen's Ordinal = the Tank
|-
|(0)(1,1)(2,1)(3,1)(4,1)(1,1)
|ψ(Ω^Ω^Ω+Ω)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(7,1)
|ψ(Ω^Ω^Ω+Ω^LVO)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1)
|ψ(Ω^Ω^Ω+Ω^Ω)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1)(4)
|ψ(Ω^Ω^Ω+Ω^Ω^ω)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8,1)
|ψ(Ω^Ω^Ω+Ω^Ω^LVO)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1)(4,1)
|ψ(Ω^Ω^Ω*2)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(2)
|ψ(Ω^Ω^Ω*ω)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(2,1)
|ψ(Ω^(Ω^Ω+1))
|-
|(0)(1,1)(2,1)(3,1)(4,1)(2,1)(3,1)(4,1)
|ψ(Ω^(Ω^Ω*2))
|-
|(0)(1,1)(2,1)(3,1)(4,1)(3)
|ψ(Ω^(Ω^Ω*ω))
|-
|(0)(1,1)(2,1)(3,1)(4,1)(3)(4,1)(5,1)(6,1)(7,1)
|ψ(Ω^(Ω^Ω*LVO))
|-
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)
|ψ(Ω^Ω^(Ω+1))
|-
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4)
|ψ(Ω^Ω^(Ω+ω))
|-
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1)
|ψ(Ω^Ω^(Ω*2))
|-
|(0)(1,1)(2,1)(3,1)(4,1)(4)
|ψ(Ω^Ω^(Ω*ω))
|-
|(0)(1,1)(2,1)(3,1)(4,1)(4,1)
|ψ(Ω^Ω^Ω^2)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(4,1)(4,1)
|ψ(Ω^Ω^Ω^3)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(5)
|ψ(Ω^Ω^Ω^ω)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(5)(6,1)
|ψ(Ω^Ω^Ω^SCO)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(5)(6,1)(7,1)
|ψ(Ω^Ω^Ω^FSO)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(5)(6,1)(7,1)(8,1)
|ψ(Ω^Ω^Ω^LVO)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)
|ψ(Ω^Ω^Ω^Ω)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6)
|ψ(Ω^Ω^Ω^Ω^ω)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)
|ψ(Ω^Ω^Ω^Ω^Ω)
|-
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1)
|ψ(Ω^^6)
|-
|(0)(1,1)(2,2)
|ψ(Ω_2) = Bachmann-Howard Ordinal
|-
|(0)(1,1)(2,2)(1)
|ψ(Ω_2+1)
|-
|(0)(1,1)(2,2)(1)(2,1)
|ψ(Ω_2+SCO)
|-
|(0)(1,1)(2,2)(1)(2,1)(3,1)
|ψ(Ω_2+CO)
|-
|(0)(1,1)(2,2)(1)(2,1)(3,2)
|ψ(Ω_2+BHO)
|-
|(0)(1,1)(2,2)(1)(2,1)(3,2)(2)(3,1)(4,2)
|ψ(Ω_2+ψ(Ω_2+BHO))
|-
|(0)(1,1)(2,2)(1,1)
|ψ(Ω_2+Ω)
|-
|(0)(1,1)(2,2)(1,1)(2,1)
|ψ(Ω_2+Ω^2)
|-
|(0)(1,1)(2,2)(1,1)(2,1)(3)(4,1)(5,2)
|ψ(Ω_2+Ω^BHO)
|-
|(0)(1,1)(2,2)(1,1)(2,1)(3,1)
|ψ(Ω_2+Ω^Ω)
|-
|(0)(1,1)(2,2)(1,1)(2,2)
|ψ(Ω_2+ψ_1(Ω_2))
|-
|(0)(1,1)(2,2)(1,1)(2,2)(1,1)
|ψ(Ω_2+ψ_1(Ω_2)+Ω)
|-
|(0)(1,1)(2,2)(1,1)(2,2)(1,1)(2,2)
|ψ(Ω_2+ψ_1(Ω_2)*2)
|-
|(0)(1,1)(2,2)(2)
|ψ(Ω_2+ψ_1(Ω_2+1))
|-
|(0)(1,1)(2,2)(2)(3,1)
|ψ(Ω_2+ψ_1(Ω_2+SCO))
|-
|(0)(1,1)(2,2)(2)(3,1)(4,2)
|ψ(Ω_2+ψ_1(Ω_2+BHO))
|-
|(0)(1,1)(2,2)(2,1)
|ψ(Ω_2+ψ_1(Ω_2+Ω))
|-
|(0)(1,1)(2,2)(2,1)(3)(4,1)(5,2)(4,1)(5,2)
|ψ(Ω_2+ψ_1(Ω_2+Ω^ψ(Ω_2+ψ_1(Ω_2)))
|-
|(0)(1,1)(2,2)(2,1)(3,1)
|ψ(Ω_2+ψ_1(Ω_2+Ω^Ω))
|-
|(0)(1,1)(2,2)(2,1)(3,2)
|ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2)))
|-
|(0)(1,1)(2,2)(2,1)(3,2)(1)
|ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2))+1)
|-
|(0)(1,1)(2,2)(2,1)(3,2)(2)
|ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2)+1))
|-
|(0)(1,1)(2,2)(2,1)(3,2)(3)
|ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+1)))
|-
|(0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)
|ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2))))
|-
|(0)(1,1)(2,2)(2,2)
|ψ(Ω_2*2)
|-
|(0)(1,1)(2,2)(2,2)(1,1)(2,2)
|ψ(Ω_2*2+ψ_1(Ω_2))
|-
|(0)(1,1)(2,2)(2,2)(1,1)(2,2)(2,2)
|ψ(Ω_2*2+ψ_1(Ω_2*2))
|-
|(0)(1,1)(2,2)(2,2)(2)
|ψ(Ω_2*2+ψ_1(Ω_2*2+1))
|-
|(0)(1,1)(2,2)(2,2)(2,1)
|ψ(Ω_2*2+ψ_1(Ω_2*2+Ω))
|-
|(0)(1,1)(2,2)(2,2)(2,2)
|ψ(Ω_2*3)
|-
|(0)(1,1)(2,2)(3)
|ψ(Ω_2*ω)
|-
|(0)(1,1)(2,2)(3,1)
|ψ(Ω_2*Ω)
|-
|(0)(1,1)(2,2)(3,1)(4,2)
|ψ(Ω_2*ψ_1(Ω_2))
|-
|(0)(1,1)(2,2)(3,2)
|ψ(Ω_2^2)
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(4,0)</math>
|(0)(1,1)(2,2)(3,2)(2,2)
|<math>\psi(\Omega^{\Omega^\omega})</math>
|ψ(Ω_2^2+Ω_2)
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(4,0)(5,1)</math>
|(0)(1,1)(2,2)(3,2)(2,2)(3,1)
|<math>\psi(\Omega^{\Omega^{\psi(0)}})</math>
|ψ(Ω_2^2+Ω_2*Ω)
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)</math>
|(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)
|<math>\psi(\Omega^{\Omega^{\psi(\Omega)}})</math>
|ψ(Ω_2^2+Ω_2*ψ_1(Ω_2^2))
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)(7,1)</math>
|(0)(1,1)(2,2)(3,2)(2,2)(3,2)
|<math>\psi(\Omega^{\Omega^{\psi(\Omega^\Omega)}})</math>
|ψ(Ω_2^2*2)
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(4,1)</math>
|(0)(1,1)(2,2)(3,2)(3)
|<math>\psi(\Omega^{\Omega^\Omega})</math>
|ψ(Ω_2^2*ω)
|-
|-
|<math>(0)(1,1)(2,1)(3,1)(4,1)(5,1)</math>
|(0)(1,1)(2,2)(3,2)(3,1)
|<math>\psi(\Omega^{\Omega^{\Omega^\Omega}})</math>
|ψ(Ω_2^2*Ω)
|-
|-
|<math>(0)(1,1)(2,2)</math>
|(0)(1,1)(2,2)(3,2)(3,2)
|<math>\psi(\psi_1(0))</math>
|ψ(Ω_2^3)
|-
|-
|<math>(0)(1,1)(2,2)(1,1)</math>
|(0)(1,1)(2,2)(3,2)(4)
|<math>\psi(\psi_1(0)+1)</math>
|ψ(Ω_2^ω)
|-
|-
|<math>(0)(1,1)(2,2)(1,1)(2,1)</math>
|(0)(1,1)(2,2)(3,2)(4,1)
|<math>\psi(\psi_1(0)+\Omega)</math>
|ψ(Ω_2^Ω)
|-
|-
|<math>(0)(1,1)(2,2)(1,1)(2,1)(3,1)</math>
|(0)(1,1)(2,2)(3,2)(4,2)
|<math>\psi(\psi_1(0)+\Omega^\Omega)</math>
|ψ(Ω_2^Ω_2)
|-
|-
|<math>(0)(1,1)(2,2)(1,1)(2,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(2,2)
|<math>\psi(\psi_1(0)\times2)</math>
|ψ(Ω_2^Ω_2+Ω_2)
|-
|-
|<math>(0)(1,1)(2,2)(2,0)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(2,2)(3,2)
|<math>\psi(\psi_1(0)\times\omega)</math>
|ψ(Ω_2^Ω_2+Ω_2^2)
|-
|-
|<math>(0)(1,1)(2,2)(2,1)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(2,2)(3,2)(4,1)(5,2)(6,2)(7,2)
|<math>\psi(\psi_1(0)\times\Omega)</math>
|ψ(Ω_2^Ω_2+Ω_2^ψ_1(Ω_2^Ω_2))
|-
|-
|<math>(0)(1,1)(2,2)(2,1)(3,0)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(2,2)(3,2)(4,2)
|<math>\psi(\psi_1(0)\times\Omega^\omega)</math>
|ψ(Ω_2^Ω_2*2)
|-
|-
|<math>(0)(1,1)(2,2)(2,1)(3,1)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(3)
|<math>\psi(\psi_1(0)\times\Omega^\Omega)</math>
|ψ(Ω_2^Ω_2*ω)
|-
|-
|<math>(0)(1,1)(2,2)(2,1)(3,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(3,1)
|<math>\psi(\psi_1(0)^2)</math>
|ψ(Ω_2^Ω_2*Ω)
|-
|-
|<math>(0)(1,1)(2,2)(2,1)(3,2)(3,0)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(3,2)
|<math>\psi(\psi_1(0)^\omega)</math>
|ψ(Ω_2^(Ω_2+1))
|-
|-
|<math>(0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(3,2)(4,2)
|<math>\psi(\psi_1(0)^{\psi_1(0)})</math>
|ψ(Ω_2^(Ω_2*2))
|-
|-
|<math>(0)(1,1)(2,2)(2,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(4)
|<math>\psi(\psi_1(1))</math>
|ψ(Ω_2^(Ω_2*ω))
|-
|-
|<math>(0)(1,1)(2,2)(3,0)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(4,1)
|<math>\psi(\psi_1(\omega))</math>
|ψ(Ω_2^(Ω_2*Ω))
|-
|-
|<math>(0)(1,1)(2,2)(3,1)(4,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(4,2)
|<math>\psi(\psi_1(\psi_1(0)))</math>
|ψ(Ω_2^Ω_2^2)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(4,2)(4,2)
|<math>\psi(\Omega_2)</math>
|ψ(Ω_2^Ω_2^3)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(1,1)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(5)
|<math>\psi(\Omega_2+\Omega)</math>
|ψ(Ω_2^Ω_2^ω)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(1,1)(2,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(5,1)
|<math>\psi(\Omega_2+\psi_1(0))</math>
|ψ(Ω_2^Ω_2^Ω)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(1,1)(2,2)(3,1)(4,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(5,2)
|<math>\psi(\Omega_2+\psi_1(\psi_1(0)))</math>
|ψ(Ω_2^^3)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(1,1)(2,2)(3,2)</math>
|(0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,2)
|<math>\psi(\Omega_2+\psi_1(\Omega_2))</math>
|ψ(Ω_2^^4)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,0)</math>
|(0)(1,1)(2,2)(3,3)
|<math>\psi(\Omega_2+\psi_1(\Omega_2)\times\omega)</math>
|ψ(Ω_3)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,1)(3,2)(4,2)</math>
|(0)(1,1)(2,2)(3,3)(1,1)
|<math>\psi(\Omega_2+\psi_1(\Omega_2)^2)</math>
|ψ(Ω_3+Ω)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,1)(3,2)(4,2)(3,0)</math>
|(0)(1,1)(2,2)(3,3)(1,1)(2,1)(3,1)
|<math>\psi(\Omega_2+\psi_1(\Omega_2)^\omega)</math>
|ψ(Ω_3+Ω^Ω)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,1)(3,2)(4,2)(3,1)(4,2)(5,2)</math>
|(0)(1,1)(2,2)(3,3)(1,1)(2,2)
|<math>\psi(\Omega_2+\psi_1(\Omega_2)^{\psi_1(\Omega_2)})</math>
|ψ(Ω_3+ψ_1(Ω_2))
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)</math>
|(0)(1,1)(2,2)(3,3)(2,1)(3,2)(4,3)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+1))</math>
|ψ(Ω_3+ψ_1(Ω_3))
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,0)</math>
|(0)(1,1)(2,2)(3,3)(2,2)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\omega))</math>
|ψ(Ω_3+Ω_2)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)</math>
|(0)(1,1)(2,2)(3,3)(2,2)(3,2)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))</math>
|ψ(Ω_3+Ω_2^2)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(3,0)</math>
|(0)(1,1)(2,2)(3,3)(2,2)(3,3)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)\times\omega))</math>
|ψ(Ω_3+ψ_2(Ω_3))
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,0)</math>
|(0)(1,1)(2,2)(3,3)(3)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)^\omega))</math>
|ψ(Ω_3+ψ_2(Ω_3+1))
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,1)(5,2)(6,2)</math>
|(0)(1,1)(2,2)(3,3)(3,1)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)^{\psi_1(\Omega_2)}))</math>
|ψ(Ω_3+ψ_2(Ω_3+Ω))
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2)</math>
|(0)(1,1)(2,2)(3,3)(3,1)(4,2)(5,3)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+1)))</math>
|ψ(Ω_3+ψ_2(Ω_3+ψ_1(Ω_3)))
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2)(5,0)</math>
|(0)(1,1)(2,2)(3,3)(3,2)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\omega)))</math>
|ψ(Ω_3+ψ_2(Ω_3+Ω_2))
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2)(5,0)(5,1)(6,2)(7,2)</math>
|(0)(1,1)(2,2)(3,3)(3,2)(4,3)
|<math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2))))</math>
|ψ(Ω_3+ψ_2(Ω_3+ψ_2(Ω_3)))
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(2,2)(3,2)</math>
|(0)(1,1)(2,2)(3,3)(3,3)
|<math>\psi(\Omega_2\times2)</math>
|ψ(Ω_3*2)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(3,0)</math>
|(0)(1,1)(2,2)(3,3)(4)
|<math>\psi(\Omega_2\times\omega)</math>
|ψ(Ω_3*ω)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(3,1)(4,2)(5,2)</math>
|(0)(1,1)(2,2)(3,3)(4,3)
|<math>\psi(\Omega_2\times\psi_1(\Omega_2))</math>
|ψ(Ω_3^2)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(3,2)</math>
|(0)(1,1)(2,2)(3,3)(4,3)(4,3)
|<math>\psi(\Omega_2^2)</math>
|ψ(Ω_3^3)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(4,0)</math>
|(0)(1,1)(2,2)(3,3)(4,3)(5)
|<math>\psi(\Omega_2^\omega)</math>
|ψ(Ω_3^ω)
|-
|-
|<math>(0)(1,1)(2,2)(3,2)(4,2)</math>
|(0)(1,1)(2,2)(3,3)(4,3)(5,3)
|<math>\psi(\Omega_2^{\Omega_2})</math>
|ψ(Ω_3^Ω_3)
|-
|-
|<math>(0)(1,1)(2,2)(3,3)</math>
|(0)(1,1)(2,2)(3,3)(4,4)
|<math>\psi(\psi_2(0))</math>
|ψ(Ω_4)
|-
|-
|<math>(0)(1,1)(2,2)(3,3)(4,3)</math>
|(0)(1,1)(2,2)(3,3)(4,4)(5,5)
|<math>\psi(\Omega_3)</math>
|ψ(Ω_5)
|-
|-
|<math>(0)(1,1)(2,2)(3,3)(4,4)</math>
|(0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)
|<math>\psi(\psi_3(0))</math>
|ψ(Ω_6)
|-
|-
|<math>(0)(1,1,1)=(0)(1,1)(2,2)(3,3)\cdots</math>
|(0)(1,1,1)
|<math>\psi(\Omega_\omega)</math>
|ψ(Ω_ω) = Buchholz's Ordinal
|}
|}
[[分类:分析]]
[[分类:分析]]

2025年8月20日 (三) 20:24的版本

本条目展示BMS强度分析的第一部分

BMS Standard(BOCF)
(0) 1
(0)(0) 2
(0)(0)(0) 3
(0)(1) ω = First Transfinte Ordinal
(0)(1)(0) ω+1
(0)(1)(0)(0) ω+2
(0)(1)(0)(1) ω2
(0)(1)(0)(1)(0) ω2+1
(0)(1)(0)(1)(0)(1) ω3
(0)(1)(1) ω^2
(0)(1)(1)(0) ω^2+1
(0)(1)(1)(0)(1) ω^2+ω
(0)(1)(1)(0)(1)(0) ω^2+ω+1
(0)(1)(1)(0)(1)(0)(1) ω^2+ω2
(0)(1)(1)(0)(1)(1) ω^2*2
(0)(1)(1)(0)(1)(1)(0)(1)(1) ω^2*3
(0)(1)(1)(1) ω^3
(0)(1)(1)(1)(1) ω^4
(0)(1)(2) ω^ω = Linear Array Ordinal
(0)(1)(2)(0) ω^ω+1
(0)(1)(2)(0)(1)(2) ω^ω*2
(0)(1)(2)(1) ω^(ω+1)
(0)(1)(2)(1)(1) ω^(ω+2)
(0)(1)(2)(1)(2) ω^(ω2)
(0)(1)(2)(1)(2)(1)(2) ω^(ω3)
(0)(1)(2)(2) ω^ω^2
(0)(1)(2)(2)(2) ω^ω^3
(0)(1)(2)(3) ω^ω^ω
(0)(1)(2)(3)(1) ω^(ω^ω+1)
(0)(1)(2)(3)(2) ω^ω^(ω+1)
(0)(1)(2)(3)(3) ω^ω^(ω^2)
(0)(1)(2)(3)(4) ω^ω^ω^ω
(0)(1)(2)(3)(4)(5) ω^^5
(0)(1,1) ψ(Ω) = Small Cantor's Ordinal
(0)(1,1)(0) ψ(Ω)+1
(0)(1,1)(0)(1,1) ψ(Ω)*2
(0)(1,1)(1) ψ(Ω+1)
(0)(1,1)(1)(2) ψ(Ω+ω)
(0)(1,1)(1)(2,1) ψ(Ω+ψ(Ω))
(0)(1,1)(1)(2,1)(1) ψ(Ω+ψ(Ω)+1)
(0)(1,1)(1)(2,1)(2) ψ(Ω+ψ(Ω+1))
(0)(1,1)(1)(2,1)(2)(3,1) ψ(Ω+ψ(Ω+ψ(Ω)))
(0)(1,1)(1,1) ψ(Ω*2)
(0)(1,1)(1,1)(1) ψ(Ω*2+1)
(0)(1,1)(1,1)(1)(2) ψ(Ω*2+ω)
(0)(1,1)(1,1)(1)(2,1) ψ(Ω*2+ψ(Ω))
(0)(1,1)(1,1)(1)(2,1)(2,1) ψ(Ω*2+ψ(Ω*2))
(0)(1,1)(1,1)(1)(2,1)(2,1)(2) ψ(Ω*2+ψ(Ω*2+1))
(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)(3,1) ψ(Ω*2+ψ(Ω*2+ψ(Ω*2)))
(0)(1,1)(1,1)(1,1) ψ(Ω*3)
(0)(1,1)(1,1)(1,1)(1)(2,1)(2,1)(2,1) ψ(Ω*3+ψ(Ω*3))
(0)(1,1)(1,1)(1,1)(1,1) ψ(Ω*4)
(0)(1,1)(1,1)(1,1)(1,1)(1,1) ψ(Ω*5)
(0)(1,1)(2) ψ(Ω*ω)
(0)(1,1)(2)(1) ψ(Ω*ω+1)
(0)(1,1)(2)(1)(2) ψ(Ω*ω+ω)
(0)(1,1)(2)(1)(2,1) ψ(Ω*ω+ψ(Ω))
(0)(1,1)(2)(1)(2,1)(2,1) ψ(Ω*ω+ψ(Ω*2))
(0)(1,1)(2)(1)(2,1)(3) ψ(Ω*ω+ψ(Ω*ω))
(0)(1,1)(2)(1)(2,1)(3)(2) ψ(Ω*ω+ψ(Ω*ω+1))
(0)(1,1)(2)(1)(2,1)(3)(2)(3,1) ψ(Ω*ω+ψ(Ω*ω+ψ(Ω)))
(0)(1,1)(2)(1,1) ψ(Ω*(ω+1))
(0)(1,1)(2)(1,1)(1) ψ(Ω*(ω+1)+1)
(0)(1,1)(2)(1,1)(1)(2,1) ψ(Ω*(ω+1)+ψ(Ω))
(0)(1,1)(2)(1,1)(1)(2,1)(3)(2,1) ψ(Ω*(ω+1)+ψ(Ω*(ω+1)))
(0)(1,1)(2)(1,1)(1,1) ψ(Ω*(ω+2))
(0)(1,1)(2)(1,1)(1,1)(1,1) ψ(Ω*(ω+3))
(0)(1,1)(2)(1,1)(2) ψ(Ω*(ω2))
(0)(1,1)(2)(1,1)(2)(1,1) ψ(Ω*(ω2+1))
(0)(1,1)(2)(1,1)(2)(1,1)(2) ψ(Ω*(ω3))
(0)(1,1)(2)(2) ψ(Ω*ω^2)
(0)(1,1)(2)(2)(1,1) ψ(Ω*(ω^2+1))
(0)(1,1)(2)(2)(1,1)(2)(2) ψ(Ω*ω^2*2)
(0)(1,1)(2)(2)(2) ψ(Ω*ω^3)
(0)(1,1)(2)(3) ψ(Ω*ω^ω)
(0)(1,1)(2)(3)(4) ψ(Ω*ω^ω^ω)
(0)(1,1)(2)(3,1) ψ(Ω*ψ(Ω))
(0)(1,1)(2)(3,1)(3,1) ψ(Ω*ψ(Ω*2))
(0)(1,1)(2)(3,1)(4) ψ(Ω*ψ(Ω*ω))
(0)(1,1)(2)(3,1)(4)(5,1) ψ(Ω*ψ(Ω*ψ(Ω)))
(0)(1,1)(2,1) ψ(Ω^2) = Cantor's Ordinal
(0)(1,1)(2,1)(1) ψ(Ω^2+1)
(0)(1,1)(2,1)(1)(2,1) ψ(Ω^2+ψ(Ω))
(0)(1,1)(2,1)(1)(2,1)(3,1) ψ(Ω^2+ψ(Ω^2))
(0)(1,1)(2,1)(1,1) ψ(Ω^2+Ω)
(0)(1,1)(2,1)(1,1)(1,1) ψ(Ω^2+Ω*2)
(0)(1,1)(2,1)(1,1)(2) ψ(Ω^2+Ω*ω)
(0)(1,1)(2,1)(1,1)(2)(3,1) ψ(Ω^2+Ω*ψ(Ω))
(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1) ψ(Ω^2+Ω*ψ(Ω^2))
(0)(1,1)(2,1)(1,1)(2,1) ψ(Ω^2*2)
(0)(1,1)(2,1)(1,1)(2,1)(1,1) ψ(Ω^2*2+Ω)
(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2) ψ(Ω^2*2+Ω*ω)
(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1) ψ(Ω^2*3)
(0)(1,1)(2,1)(2) ψ(Ω^2*ω)
(0)(1,1)(2,1)(2)(1,1) ψ(Ω^2*ω+Ω)
(0)(1,1)(2,1)(2)(1,1)(2,1) ψ(Ω^2*(ω+1))
(0)(1,1)(2,1)(2)(2) ψ(Ω^2*ω2)
(0)(1,1)(2,1)(2)(3) ψ(Ω^2*ω^2)
(0)(1,1)(2,1)(2)(3)(4) ψ(Ω^2*ω^ω)
(0)(1,1)(2,1)(2)(3,1) ψ(Ω^2*ψ(Ω))
(0)(1,1)(2,1)(2)(3,1)(3,1) ψ(Ω^2*ψ(Ω*2))
(0)(1,1)(2,1)(2)(3,1)(4,1) ψ(Ω^2*ψ(Ω^2))
(0)(1,1)(2,1)(2)(3,1)(4,1)(4) ψ(Ω^2*ψ(Ω^2*ω))
(0)(1,1)(2,1)(2,1) ψ(Ω^3) = Large Cantor's Ordinal
(0)(1,1)(2,1)(2,1)(1) ψ(Ω^3+1)
(0)(1,1)(2,1)(2,1)(1)(2,1) ψ(Ω^3+ψ(Ω))
(0)(1,1)(2,1)(2,1)(1)(2,1)(3,1)(3,1) ψ(Ω^3+ψ(Ω^3))
(0)(1,1)(2,1)(2,1)(1,1) ψ(Ω^3+Ω)
(0)(1,1)(2,1)(2,1)(1,1)(1,1) ψ(Ω^3+Ω*2)
(0)(1,1)(2,1)(2,1)(1,1)(2) ψ(Ω^3+Ω*ω)
(0)(1,1)(2,1)(2,1)(1,1)(2)(3,1) ψ(Ω^3+Ω*ψ(Ω))
(0)(1,1)(2,1)(2,1)(1,1)(2)(3,1)(4,1)(4,1) ψ(Ω^3+Ω*ψ(Ω^3))
(0)(1,1)(2,1)(2,1)(1,1)(2,1) ψ(Ω^3+Ω^2)
(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1) ψ(Ω^3*2)
(0)(1,1)(2,1)(2,1)(2) ψ(Ω^3*ω)
(0)(1,1)(2,1)(2,1)(2)(1,1)(2,1)(2,1) ψ(Ω^3*(ω+1))
(0)(1,1)(2,1)(2,1)(2)(2) ψ(Ω^3*ω2)
(0)(1,1)(2,1)(2,1)(2)(3) ψ(Ω^3*ω^2)
(0)(1,1)(2,1)(2,1)(2)(3,1) ψ(Ω^3*ψ(Ω))
(0)(1,1)(2,1)(2,1)(2)(3,1)(4,1)(4,1) ψ(Ω^3+ψ(Ω^3))
(0)(1,1)(2,1)(2,1)(2,1) ψ(Ω^4)
(0)(1,1)(2,1)(2,1)(2,1)(1,1) ψ(Ω^4+Ω)
(0)(1,1)(2,1)(2,1)(2,1)(2) ψ(Ω^4*ω)
(0)(1,1)(2,1)(2,1)(2,1)(2,1) ψ(Ω^5)
(0)(1,1)(2,1)(3) ψ(Ω^ω) = Hyper Cantor's Ordinal
(0)(1,1)(2,1)(3)(1) ψ(Ω^ω+1)
(0)(1,1)(2,1)(3)(1)(2) ψ(Ω^ω+ω)
(0)(1,1)(2,1)(3)(1)(2,1) ψ(Ω^ω+SCO)
(0)(1,1)(2,1)(3)(1)(2,1)(3,1) ψ(Ω^ω+CO)
(0)(1,1)(2,1)(3)(1)(2,1)(3,1)(4) ψ(Ω^ω+HCO)
(0)(1,1)(2,1)(3)(1,1) ψ(Ω^ω+Ω)
(0)(1,1)(2,1)(3)(1,1)(1,1) ψ(Ω^ω+Ω*2)
(0)(1,1)(2,1)(3)(1,1)(2) ψ(Ω^ω+Ω*ω)
(0)(1,1)(2,1)(3)(1,1)(2)(3,1) ψ(Ω^ω+Ω*SCO)
(0)(1,1)(2,1)(3)(1,1)(2,1) ψ(Ω^ω+Ω^2)
(0)(1,1)(2,1)(3)(1,1)(2,1)(3) ψ(Ω^ω*2)
(0)(1,1)(2,1)(3)(2) ψ(Ω^ω*ω)
(0)(1,1)(2,1)(3)(2)(3,1) ψ(Ω^ω*SCO)
(0)(1,1)(2,1)(3)(2,1) ψ(Ω^(ω+1))
(0)(1,1)(2,1)(3)(2,1)(2) ψ(Ω^(ω+1)*ω)
(0)(1,1)(2,1)(3)(2,1)(2,1) ψ(Ω^(ω+2))
(0)(1,1)(2,1)(3)(2,1)(3) ψ(Ω^(ω2))
(0)(1,1)(2,1)(3)(2,1)(3)(2,1) ψ(Ω^(ω2+1))
(0)(1,1)(2,1)(3)(3) ψ(Ω^(ω3))
(0)(1,1)(2,1)(3)(4) ψ(Ω^ω^2)
(0)(1,1)(2,1)(3)(4,1) ψ(Ω^SCO)
(0)(1,1)(2,1)(3)(4,1)(5,1) ψ(Ω^CO)
(0)(1,1)(2,1)(3)(4,1)(5,1)(6) ψ(Ω^HCO)
(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1) ψ(Ω^ψ(Ω^HCO))
(0)(1,1)(2,1)(3,1) ψ(Ω^Ω) = Feferman-Schütte Ordinal
(0)(1,1)(2,1)(3,1)(1) ψ(Ω^Ω+1)
(0)(1,1)(2,1)(3,1)(1)(2) ψ(Ω^Ω+ω)
(0)(1,1)(2,1)(3,1)(1)(2,1) ψ(Ω^Ω+SCO)
(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1) ψ(Ω^Ω+CO)
(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4) ψ(Ω^Ω+HCO)
(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1) ψ(Ω^Ω+FSO)
(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2) ψ(Ω^Ω+ψ(Ω^Ω+1))
(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)(3,1) ψ(Ω^Ω+ψ(Ω^Ω+SCO))
(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)(3,1)(4,1) ψ(Ω^Ω+ψ(Ω^Ω+CO))
(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)(3,1)(4,1)(5) ψ(Ω^Ω+ψ(Ω^Ω+HCO))
(0)(1,1)(2,1)(3,1)(1)(2,1)(3,1)(4,1)(2)(3,1)(4,1)(5,1) ψ(Ω^Ω+ψ(Ω^Ω+FSO))
(0)(1,1)(2,1)(3,1)(1,1) ψ(Ω^Ω+Ω)
(0)(1,1)(2,1)(3,1)(1,1)(1) ψ(Ω^Ω+Ω+1)
(0)(1,1)(2,1)(3,1)(1,1)(1)(2,1) ψ(Ω^Ω+Ω+SCO)
(0)(1,1)(2,1)(3,1)(1,1)(1)(2,1)(3,1)(4,1) ψ(Ω^Ω+Ω+FSO)
(0)(1,1)(2,1)(3,1)(1,1)(1,1) ψ(Ω^Ω+Ω*2)
(0)(1,1)(2,1)(3,1)(1,1)(2) ψ(Ω^Ω+Ω*ω)
(0)(1,1)(2,1)(3,1)(1,1)(2)(3,1) ψ(Ω^Ω+Ω*SCO)
(0)(1,1)(2,1)(3,1)(1,1)(2)(3,1)(4,1)(5,1) ψ(Ω^Ω+Ω*FSO)
(0)(1,1)(2,1)(3,1)(1,1)(2,1) ψ(Ω^Ω+Ω^2)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1)(2,1) ψ(Ω^Ω+Ω^2+SCO)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1)(2,1)(3,1)(4,1) ψ(Ω^Ω+Ω^2+FSO)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1,1) ψ(Ω^Ω+Ω^2+Ω)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1,1)(1,1) ψ(Ω^Ω+Ω^2+Ω*2)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1,1)(2,1) ψ(Ω^Ω+Ω^2*2)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1) ψ(Ω^Ω+Ω^2*3)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2) ψ(Ω^Ω+Ω^2*ω)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2)(3,1) ψ(Ω^Ω+Ω^2*SCO)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2)(3,1)(4,1)(5,1) ψ(Ω^Ω+Ω^2*FSO)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1) ψ(Ω^Ω+Ω^3)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)(1,1) ψ(Ω^Ω+Ω^3+Ω)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1) ψ(Ω^Ω+Ω^3*2)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)(2) ψ(Ω^Ω+Ω^3*ω)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1)(2,1) ψ(Ω^Ω+Ω^4)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3) ψ(Ω^Ω+Ω^ω)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(1,1)(2,1)(3) ψ(Ω^Ω+Ω^ω*2)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(2) ψ(Ω^Ω+Ω^ω*ω)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(2,1) ψ(Ω^Ω+Ω^(ω+1))
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(2,1)(3) ψ(Ω^Ω+Ω^(ω2))
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(3) ψ(Ω^Ω+Ω^ω^2)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1) ψ(Ω^Ω+Ω^SCO)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1) ψ(Ω^Ω+Ω^FSO)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2) ψ(Ω^Ω+Ω^FSO*ω)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1) ψ(Ω^Ω+Ω^(FSO+1))
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(3) ψ(Ω^Ω+Ω^(FSO+ω))?
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4) ψ(Ω^Ω+Ω^(FSO*ω))
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1) ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω))
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)(5,1)(6)(7,1)(8,1)(9,1) ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω^ψ(Ω^Ω)))
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)(5,1)(6)(7,1)(8,1)(9,1)(5,1) ψ(Ω^Ω+Ω^ψ(Ω^Ω+Ω^(ψ(Ω^Ω)+1)))?
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1) ψ(Ω^Ω*2)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1) ψ(Ω^Ω*2+Ω)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1) ψ(Ω^Ω*2+Ω^2)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(2,1) ψ(Ω^Ω*2+Ω^3)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1) ψ(Ω^Ω*3)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1) ψ(Ω^Ω*4)
(0)(1,1)(2,1)(3,1)(2) ψ(Ω^Ω*ω)
(0)(1,1)(2,1)(3,1)(2)(3,1) ψ(Ω^Ω*SCO)
(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1) ψ(Ω^Ω*CO)
(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1) ψ(Ω^Ω*FSO)
(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)(4)(5,1)(6,1)(7,1) ψ(Ω^Ω*ψ(Ω^Ω*FSO))
(0)(1,1)(2,1)(3,1)(2,1) ψ(Ω^(Ω+1))
(0)(1,1)(2,1)(3,1)(2,1)(1,1) ψ(Ω^(Ω+1)+Ω)
(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1) ψ(Ω^(Ω+1)+Ω^Ω)
(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1) ψ(Ω^(Ω+1)*2)
(0)(1,1)(2,1)(3,1)(2,1)(2) ψ(Ω^(Ω+1)*ω)
(0)(1,1)(2,1)(3,1)(2,1)(2)(3,1) ψ(Ω^(Ω+1)*SCO)
(0)(1,1)(2,1)(3,1)(2,1)(2)(3,1)(4,1)(5,1) ψ(Ω^(Ω+1)*FSO)
(0)(1,1)(2,1)(3,1)(2,1)(2,1) ψ(Ω^(Ω+2))
(0)(1,1)(2,1)(3,1)(2,1)(3) ψ(Ω^(Ω+ω))
(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1) ψ(Ω^(Ω+SCO))
(0)(1,1)(2,1)(3,1)(2,1)(3,1) ψ(Ω^(Ω*2))
(0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1) ψ(Ω^(Ω*3))
(0)(1,1)(2,1)(3,1)(3) ψ(Ω^(Ω*ω))
(0)(1,1)(2,1)(3,1)(3)(4,1) ψ(Ω^(Ω*SCO))
(0)(1,1)(2,1)(3,1)(3,1) ψ(Ω^Ω^2) = Ackerman's Ordinal
(0)(1,1)(2,1)(3,1)(3,1)(1,1) ψ(Ω^Ω^2+Ω)
(0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(6,1) ψ(Ω^Ω^2+Ω^ACO)
(0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3,1) ψ(Ω^Ω^2+Ω^Ω)
(0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3,1)(3)(4,1)(5,1)(6,1)(6,1) ψ(Ω^Ω^2+Ω^(Ω*ACO))
(0)(1,1)(2,1)(3,1)(3,1)(1,1)(2,1)(3,1)(3,1) ψ(Ω^Ω^2*2)
(0)(1,1)(2,1)(3,1)(3,1)(2)(3,1)(4,1)(5,1)(5,1) ψ(Ω^Ω^2*ACO)
(0)(1,1)(2,1)(3,1)(3,1)(2,1) ψ(Ω^(Ω^2+1))
(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1) ψ(Ω^(Ω^2+Ω))
(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(7,1) ψ(Ω^(Ω^2+Ω*ACO))
(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1) ψ(Ω^(Ω^2*2))
(0)(1,1)(2,1)(3,1)(3,1)(3) ψ(Ω^(Ω^2*ω))
(0)(1,1)(2,1)(3,1)(3,1)(3)(4,1)(5,1)(6,1)(6,1) ψ(Ω^(Ω^2*ACO))
(0)(1,1)(2,1)(3,1)(3,1)(3,1) ψ(Ω^Ω^3)
(0)(1,1)(2,1)(3,1)(3,1)(3,1)(3,1) ψ(Ω^Ω^4)
(0)(1,1)(2,1)(3,1)(4) ψ(Ω^Ω^ω) = Small Veblen's Ordinal
(0)(1,1)(2,1)(3,1)(4)(1,1) ψ(Ω^Ω^ω+Ω)
(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1) ψ(Ω^Ω^ω+Ω^2)
(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(2,1) ψ(Ω^Ω^ω+Ω^3)
(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(3) ψ(Ω^Ω^ω+Ω^ω)
(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(7) ψ(Ω^Ω^ω+Ω^SVO)
(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(3,1) ψ(Ω^Ω^ω+Ω^Ω)
(0)(1,1)(2,1)(3,1)(4)(1,1)(2,1)(3,1)(4) ψ(Ω^Ω^ω*2)
(0)(1,1)(2,1)(3,1)(4)(2) ψ(Ω^Ω^ω*ω)
(0)(1,1)(2,1)(3,1)(4)(2,1) ψ(Ω^(Ω^ω+1))
(0)(1,1)(2,1)(3,1)(4)(2,1)(3) ψ(Ω^(Ω^ω+ω))
(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1) ψ(Ω^(Ω^ω+Ω))
(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1)(3)(4,1)(5,1)(6,1)(7) ψ(Ω^(Ω^ω+Ω*SVO))
(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1)(3,1) ψ(Ω^(Ω^ω+Ω^2))
(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1)(4) ψ(Ω^(Ω^ω*2))
(0)(1,1)(2,1)(3,1)(4)(3) ψ(Ω^(Ω^ω*ω))
(0)(1,1)(2,1)(3,1)(4)(3)(4,1)(5,1)(6,1)(7) ψ(Ω^(Ω^ω*SVO))
(0)(1,1)(2,1)(3,1)(4)(3,1) ψ(Ω^Ω^(ω+1))
(0)(1,1)(2,1)(3,1)(4)(3,1)(3,1) ψ(Ω^Ω^(ω+2))
(0)(1,1)(2,1)(3,1)(4)(3,1)(4) ψ(Ω^Ω^(ω2))
(0)(1,1)(2,1)(3,1)(4)(4) ψ(Ω^Ω^ω^2)
(0)(1,1)(2,1)(3,1)(4)(5) ψ(Ω^Ω^ω^ω)
(0)(1,1)(2,1)(3,1)(4)(5,1) ψ(Ω^Ω^SCO)
(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1) ψ(Ω^Ω^FSO)
(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8) ψ(Ω^Ω^SVO)
(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8)(9,1)(A,1)(B,1)(C) ψ(Ω^Ω^ψ(Ω^Ω^SVO))
(0)(1,1)(2,1)(3,1)(4,1) ψ(Ω^Ω^Ω) = Large Veblen's Ordinal = the Tank
(0)(1,1)(2,1)(3,1)(4,1)(1,1) ψ(Ω^Ω^Ω+Ω)
(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(7,1) ψ(Ω^Ω^Ω+Ω^LVO)
(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1) ψ(Ω^Ω^Ω+Ω^Ω)
(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1)(4) ψ(Ω^Ω^Ω+Ω^Ω^ω)
(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8,1) ψ(Ω^Ω^Ω+Ω^Ω^LVO)
(0)(1,1)(2,1)(3,1)(4,1)(1,1)(2,1)(3,1)(4,1) ψ(Ω^Ω^Ω*2)
(0)(1,1)(2,1)(3,1)(4,1)(2) ψ(Ω^Ω^Ω*ω)
(0)(1,1)(2,1)(3,1)(4,1)(2,1) ψ(Ω^(Ω^Ω+1))
(0)(1,1)(2,1)(3,1)(4,1)(2,1)(3,1)(4,1) ψ(Ω^(Ω^Ω*2))
(0)(1,1)(2,1)(3,1)(4,1)(3) ψ(Ω^(Ω^Ω*ω))
(0)(1,1)(2,1)(3,1)(4,1)(3)(4,1)(5,1)(6,1)(7,1) ψ(Ω^(Ω^Ω*LVO))
(0)(1,1)(2,1)(3,1)(4,1)(3,1) ψ(Ω^Ω^(Ω+1))
(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4) ψ(Ω^Ω^(Ω+ω))
(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1) ψ(Ω^Ω^(Ω*2))
(0)(1,1)(2,1)(3,1)(4,1)(4) ψ(Ω^Ω^(Ω*ω))
(0)(1,1)(2,1)(3,1)(4,1)(4,1) ψ(Ω^Ω^Ω^2)
(0)(1,1)(2,1)(3,1)(4,1)(4,1)(4,1) ψ(Ω^Ω^Ω^3)
(0)(1,1)(2,1)(3,1)(4,1)(5) ψ(Ω^Ω^Ω^ω)
(0)(1,1)(2,1)(3,1)(4,1)(5)(6,1) ψ(Ω^Ω^Ω^SCO)
(0)(1,1)(2,1)(3,1)(4,1)(5)(6,1)(7,1) ψ(Ω^Ω^Ω^FSO)
(0)(1,1)(2,1)(3,1)(4,1)(5)(6,1)(7,1)(8,1) ψ(Ω^Ω^Ω^LVO)
(0)(1,1)(2,1)(3,1)(4,1)(5,1) ψ(Ω^Ω^Ω^Ω)
(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6) ψ(Ω^Ω^Ω^Ω^ω)
(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1) ψ(Ω^Ω^Ω^Ω^Ω)
(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1) ψ(Ω^^6)
(0)(1,1)(2,2) ψ(Ω_2) = Bachmann-Howard Ordinal
(0)(1,1)(2,2)(1) ψ(Ω_2+1)
(0)(1,1)(2,2)(1)(2,1) ψ(Ω_2+SCO)
(0)(1,1)(2,2)(1)(2,1)(3,1) ψ(Ω_2+CO)
(0)(1,1)(2,2)(1)(2,1)(3,2) ψ(Ω_2+BHO)
(0)(1,1)(2,2)(1)(2,1)(3,2)(2)(3,1)(4,2) ψ(Ω_2+ψ(Ω_2+BHO))
(0)(1,1)(2,2)(1,1) ψ(Ω_2+Ω)
(0)(1,1)(2,2)(1,1)(2,1) ψ(Ω_2+Ω^2)
(0)(1,1)(2,2)(1,1)(2,1)(3)(4,1)(5,2) ψ(Ω_2+Ω^BHO)
(0)(1,1)(2,2)(1,1)(2,1)(3,1) ψ(Ω_2+Ω^Ω)
(0)(1,1)(2,2)(1,1)(2,2) ψ(Ω_2+ψ_1(Ω_2))
(0)(1,1)(2,2)(1,1)(2,2)(1,1) ψ(Ω_2+ψ_1(Ω_2)+Ω)
(0)(1,1)(2,2)(1,1)(2,2)(1,1)(2,2) ψ(Ω_2+ψ_1(Ω_2)*2)
(0)(1,1)(2,2)(2) ψ(Ω_2+ψ_1(Ω_2+1))
(0)(1,1)(2,2)(2)(3,1) ψ(Ω_2+ψ_1(Ω_2+SCO))
(0)(1,1)(2,2)(2)(3,1)(4,2) ψ(Ω_2+ψ_1(Ω_2+BHO))
(0)(1,1)(2,2)(2,1) ψ(Ω_2+ψ_1(Ω_2+Ω))
(0)(1,1)(2,2)(2,1)(3)(4,1)(5,2)(4,1)(5,2) ψ(Ω_2+ψ_1(Ω_2+Ω^ψ(Ω_2+ψ_1(Ω_2)))
(0)(1,1)(2,2)(2,1)(3,1) ψ(Ω_2+ψ_1(Ω_2+Ω^Ω))
(0)(1,1)(2,2)(2,1)(3,2) ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2)))
(0)(1,1)(2,2)(2,1)(3,2)(1) ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2))+1)
(0)(1,1)(2,2)(2,1)(3,2)(2) ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2)+1))
(0)(1,1)(2,2)(2,1)(3,2)(3) ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+1)))
(0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2) ψ(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2+ψ_1(Ω_2))))
(0)(1,1)(2,2)(2,2) ψ(Ω_2*2)
(0)(1,1)(2,2)(2,2)(1,1)(2,2) ψ(Ω_2*2+ψ_1(Ω_2))
(0)(1,1)(2,2)(2,2)(1,1)(2,2)(2,2) ψ(Ω_2*2+ψ_1(Ω_2*2))
(0)(1,1)(2,2)(2,2)(2) ψ(Ω_2*2+ψ_1(Ω_2*2+1))
(0)(1,1)(2,2)(2,2)(2,1) ψ(Ω_2*2+ψ_1(Ω_2*2+Ω))
(0)(1,1)(2,2)(2,2)(2,2) ψ(Ω_2*3)
(0)(1,1)(2,2)(3) ψ(Ω_2*ω)
(0)(1,1)(2,2)(3,1) ψ(Ω_2*Ω)
(0)(1,1)(2,2)(3,1)(4,2) ψ(Ω_2*ψ_1(Ω_2))
(0)(1,1)(2,2)(3,2) ψ(Ω_2^2)
(0)(1,1)(2,2)(3,2)(2,2) ψ(Ω_2^2+Ω_2)
(0)(1,1)(2,2)(3,2)(2,2)(3,1) ψ(Ω_2^2+Ω_2*Ω)
(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2) ψ(Ω_2^2+Ω_2*ψ_1(Ω_2^2))
(0)(1,1)(2,2)(3,2)(2,2)(3,2) ψ(Ω_2^2*2)
(0)(1,1)(2,2)(3,2)(3) ψ(Ω_2^2*ω)
(0)(1,1)(2,2)(3,2)(3,1) ψ(Ω_2^2*Ω)
(0)(1,1)(2,2)(3,2)(3,2) ψ(Ω_2^3)
(0)(1,1)(2,2)(3,2)(4) ψ(Ω_2^ω)
(0)(1,1)(2,2)(3,2)(4,1) ψ(Ω_2^Ω)
(0)(1,1)(2,2)(3,2)(4,2) ψ(Ω_2^Ω_2)
(0)(1,1)(2,2)(3,2)(4,2)(2,2) ψ(Ω_2^Ω_2+Ω_2)
(0)(1,1)(2,2)(3,2)(4,2)(2,2)(3,2) ψ(Ω_2^Ω_2+Ω_2^2)
(0)(1,1)(2,2)(3,2)(4,2)(2,2)(3,2)(4,1)(5,2)(6,2)(7,2) ψ(Ω_2^Ω_2+Ω_2^ψ_1(Ω_2^Ω_2))
(0)(1,1)(2,2)(3,2)(4,2)(2,2)(3,2)(4,2) ψ(Ω_2^Ω_2*2)
(0)(1,1)(2,2)(3,2)(4,2)(3) ψ(Ω_2^Ω_2*ω)
(0)(1,1)(2,2)(3,2)(4,2)(3,1) ψ(Ω_2^Ω_2*Ω)
(0)(1,1)(2,2)(3,2)(4,2)(3,2) ψ(Ω_2^(Ω_2+1))
(0)(1,1)(2,2)(3,2)(4,2)(3,2)(4,2) ψ(Ω_2^(Ω_2*2))
(0)(1,1)(2,2)(3,2)(4,2)(4) ψ(Ω_2^(Ω_2*ω))
(0)(1,1)(2,2)(3,2)(4,2)(4,1) ψ(Ω_2^(Ω_2*Ω))
(0)(1,1)(2,2)(3,2)(4,2)(4,2) ψ(Ω_2^Ω_2^2)
(0)(1,1)(2,2)(3,2)(4,2)(4,2)(4,2) ψ(Ω_2^Ω_2^3)
(0)(1,1)(2,2)(3,2)(4,2)(5) ψ(Ω_2^Ω_2^ω)
(0)(1,1)(2,2)(3,2)(4,2)(5,1) ψ(Ω_2^Ω_2^Ω)
(0)(1,1)(2,2)(3,2)(4,2)(5,2) ψ(Ω_2^^3)
(0)(1,1)(2,2)(3,2)(4,2)(5,2)(6,2) ψ(Ω_2^^4)
(0)(1,1)(2,2)(3,3) ψ(Ω_3)
(0)(1,1)(2,2)(3,3)(1,1) ψ(Ω_3+Ω)
(0)(1,1)(2,2)(3,3)(1,1)(2,1)(3,1) ψ(Ω_3+Ω^Ω)
(0)(1,1)(2,2)(3,3)(1,1)(2,2) ψ(Ω_3+ψ_1(Ω_2))
(0)(1,1)(2,2)(3,3)(2,1)(3,2)(4,3) ψ(Ω_3+ψ_1(Ω_3))
(0)(1,1)(2,2)(3,3)(2,2) ψ(Ω_3+Ω_2)
(0)(1,1)(2,2)(3,3)(2,2)(3,2) ψ(Ω_3+Ω_2^2)
(0)(1,1)(2,2)(3,3)(2,2)(3,3) ψ(Ω_3+ψ_2(Ω_3))
(0)(1,1)(2,2)(3,3)(3) ψ(Ω_3+ψ_2(Ω_3+1))
(0)(1,1)(2,2)(3,3)(3,1) ψ(Ω_3+ψ_2(Ω_3+Ω))
(0)(1,1)(2,2)(3,3)(3,1)(4,2)(5,3) ψ(Ω_3+ψ_2(Ω_3+ψ_1(Ω_3)))
(0)(1,1)(2,2)(3,3)(3,2) ψ(Ω_3+ψ_2(Ω_3+Ω_2))
(0)(1,1)(2,2)(3,3)(3,2)(4,3) ψ(Ω_3+ψ_2(Ω_3+ψ_2(Ω_3)))
(0)(1,1)(2,2)(3,3)(3,3) ψ(Ω_3*2)
(0)(1,1)(2,2)(3,3)(4) ψ(Ω_3*ω)
(0)(1,1)(2,2)(3,3)(4,3) ψ(Ω_3^2)
(0)(1,1)(2,2)(3,3)(4,3)(4,3) ψ(Ω_3^3)
(0)(1,1)(2,2)(3,3)(4,3)(5) ψ(Ω_3^ω)
(0)(1,1)(2,2)(3,3)(4,3)(5,3) ψ(Ω_3^Ω_3)
(0)(1,1)(2,2)(3,3)(4,4) ψ(Ω_4)
(0)(1,1)(2,2)(3,3)(4,4)(5,5) ψ(Ω_5)
(0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6) ψ(Ω_6)
(0)(1,1,1) ψ(Ω_ω) = Buchholz's Ordinal
检索自“http://wiki.googology.top/index.php?title=BMS分析Part1：0~BO&oldid=2177”
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