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BMS分析Part3：EBO~SSO
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本词条展示[[BMS]]分析的第三部分 {| class="wikitable" |BMS |Standard([[序数坍缩函数#BOCF|BOCF]]/[[反射序数]]) |- |(0)(1,1,1)(2,1,1)(3,1)(2) |ψ(I) = Extented Buchholz's Ordinal |- |(0)(1,1,1)(2,1,1)(3,1)(2)(1,1) |ψ(I+Ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2)(1,1,1) |ψ(I+Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2)(1,1,1)(2,1,1) |ψ(I+Ω_(ω^2)) |- |(0)(1,1,1)(2,1,1)(3,1)(2)(1,1,1)(2,1,1)(3,1) |ψ(I+Ω_Ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+Λ) |- |(0)(1,1,1)(2,1,1)(3,1)(2)(2) |ψ(I+Λ*ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2)(3,1,1) |ψ(I+Λ*BO) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1) |ψ(I+Λ*Ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(1,1,1) |ψ(I+Λ*Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+Λ^2) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(2) |ψ(I+Λ^2*ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(2,1)(2) |ψ(I+Λ^3) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3) |ψ(I+Λ^ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,1) |ψ(I+Λ^Ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,1)(1,1,1) |ψ(I+Λ^Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+Λ^Λ) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2) |ψ(I+ψI(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1) |ψ(I+ψI(I+ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1) |ψ(I+ψI(I+Ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+ψI(I+ψI(I))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(2) |ψ(I+ψI(I+ψI(I))+ψ_ψI(I+1)(I+ψI(I+ψI(I))+1)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(2,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+ψI(I+ψI(I))+ψ_ψI(I+1)(I+ψI(I+ψI(I))+ψI(I))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(2,1)(2) |ψ(I+ψI(I+ψI(I))+ψ_ψI(I+1)(I+ψI(I+ψI(I))+ψI(I)*ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(2,1)(3,2) |ψ(I+ψI(I+ψI(I))+ψ_ψI(I+1)(I+ψI(I+ψI(I))+ψ_ψI(I+1)(I+ψI(I+ψI(I))))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+ψI(I+ψI(I))+ψ_ψI(I+1)(I+ψI(I+ψI(I))+ψ_ψI(I+1)(I+ψI(I+ψI(I))))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(3,2) |ψ(I+ψI(I+ψI(I))+ψI(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(3,2,1) |ψ(I+ψI(I+ψI(I))+ψI(I+ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+ψI(I+ψI(I))*2) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4) |ψ(I+ψI(I+ψI(I))*ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+ψI(I+ψI(I))*ψI(I)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,1)(5,2) |ψ(I+ψI(I+ψI(I))*ψ_ψI(I+1)(ψI(I+1))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2) |ψ(I+ψI(I+ψI(I))*ψI(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2)(3,2,1) |ψ(I+ψI(I+ψI(I))*ψI(I+ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2)(3,2,1)(4,2,1)(5,1)(2) |ψ(I+ψI(I+ψI(I))^2) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2)(4) |ψ(I+ψI(I+ψI(I))^ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2)(5,3) |ψ(I+ψI(I+ψI(I)+1)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2,1) |ψ(I+ψI(I+ψI(I)*ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2,1)(4,2,1) |ψ(I+ψI(I+ψI(I)*ω^2)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+ψI(I+ψI(I)^2)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(5) |ψ(I+ψI(I+ψI(I)^ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(5,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+ψI(I+ψI(I)^ψI(I))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,1)(6,2) |ψ(I+ψI(I+ψ_ψI(I+1)(ψI(I+1)))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2) |ψ(I+ψI(I+ψI(I+1))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(3,2,1) |ψ(I+ψI(I+ψI(I+ω))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I+ψI(I+ψI(I+ψI(I)))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(3,2,1)(4,2,1)(5,2) |ψ(I+ψI(I+ψI(I+ψI(I+1)))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(4) |ψ(I*2) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(4,2)(5,3,1)(6,3,1)(7,3)(6) |ψ(I*3) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1) |ψ(I*ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(2,1,1) |ψ(I*ω^2) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3) |ψ(I*ω^ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1) |ψ(I*Ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(1,1,1) |ψ(I*Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I*Λ) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(2,1,1) |ψ(I*ψI(I*ω))) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2) |ψ(I^2) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1,1) |ψ(I^2*ω) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1,1)(3,1)(2) |ψ(I^3) |- |(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1,1)(3,1)(2) |ψ(I^4) |- |(0)(1,1,1)(2,1,1)(3,1)(3) |ψ(I^ω) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1) |ψ(I^Ω) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(1,1,1) |ψ(I^Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(2) |ψ(I^I) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(2,1,1) |ψ(I^I*ω) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(2,1,1)(3,1)(2) |ψ(I^(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(2,1,1)(3,1)(3,1)(2) |ψ(I^(I*2)) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(3) |ψ(I^(I*ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(3,1) |ψ(I^(I*Ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(3,1)(2) |ψ(I^(I^2)) |- |(0)(1,1,1)(2,1,1)(3,1)(3,1)(3,1)(3,1)(2) |ψ(I^(I^3)) |- |(0)(1,1,1)(2,1,1)(3,1)(4) |ψ(I^(I^ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,1) |ψ(I^(I^Ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,1)(2) |ψ(I^(I^I)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,1)(4) |ψ(I^(I^(I*ω))) |- |(0)(1,1,1)(2,1,1)(3,1)(4,1)(5) |ψ(I^(I^(I^ω))) |- |(0)(1,1,1)(2,1,1)(3,1)(4,1)(5,1)(2) |ψ(I^(I^(I^I))) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2) |ψ(Ω_(I+1)) = Jäger's Ordinal |- |(0)(1,1,1)(2,1,1)(3,1)(4,2)(5,3) |ψ(Ω_(I+2)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2,1) |ψ(Ω_(I+ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1) |ψ(Ω_(I+ω^2)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,1) |ψ(Ω_(I+Ω)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,1)(2) |ψ(Ω_(I2)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,1)(7,2) |ψ(Ω_ψ_Ω_(I+1)(Ω_(I+1))) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2) |ψ(Ω_Ω_(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(5) |ψ(I_2) |- |(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3)(8) |ψ(I_3) |- |(0)(1,1,1)(2,1,1)(3,1,1) |ψ(I_ω) = Small Inaccessible Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1) |ψ(I_ω+Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I_ω+ψI(I)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(2,1,1) |ψ(I_ω+ψI(I*ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2) |ψ(I_ω+ψI(Ω_(I+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1) |ψ(I_ω+ψI(Ω_(I+ω))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I_ω+ψI(Ω_(I+ψ_I(I)))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,1)(2) |ψ(I_ω+ψI(Ω_(I2))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2) |ψ(I_ω+ψI(Ω_Ω_(I+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(5) |ψ(I_ω+ψI(I_2)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ω+ψI(I_ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2) |ψ(I_ω+ψI(I_ω)*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2) |ψ(I_ω+ψI(I_ω+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1) |ψ(I_ω+ψI(I_ω+Ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I_ω+ψI(I_ω+ψI(I))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2) |ψ(I_ω+ψI(I_ω+ψI(I_ω))+ψI(I_ω)*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2) |ψ(I_ω+ψI(I_ω+ψI(I_ω))+ψ_ψI(I_ω+1)(ψI(I_ω+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I_ω+ψI(I_ω+ψI(I_ω))+ψ_ψI(I_ω+1)(ψI(I_ω+ψI(I)))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω))+ψ_ψI(I_ω+1)(ψI(I_ω+ψI(I_ω)))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(2) |ψ(I_ω+ψI(I_ω+ψI(I_ω))+ψ_ψI(I_ω+1)(ψI(I_ω+ψI(I_ω))+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(2,1)(3,2) |ψ(I_ω+ψI(I_ω+ψI(I_ω))+ψ_ψI(I_ω+1)(ψI(I_ω+ψI(I_ω)+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(3,2) |ψ(I_ω+ψI(I_ω+ψI(I_ω))+ψI(I_ω+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(3,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω))+ψI(I_ω+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω))*2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(4) |ψ(I_ω+ψI(I_ω+ψI(I_ω))*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω))*ψI(I_ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2) |ψ(I_ω+ψI(I_ω+ψI(I_ω))*ψI(I_ω+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2)(3,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω))*ψI(I_ω+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω))^2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2)(4) |ψ(I_ω+ψI(I_ω+ψI(I_ω))^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2)(5,3) |ψ(I_ω+ψI(I_ω+ψI(I_ω)+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,1)(4,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω)*ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2) |ψ(I_ω+ψI(I_ω+ψI(I_ω+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2)(3,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω+ω))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ω+ψI(I_ω+ψI(I_ω+ψI(I_ω)))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2)(4) |ψ(I_ω+I) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2)(4,2,1) |ψ(I_ω+I*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2)(6,3) |ψ(I_ω+ψ_Ω_(I+1)(Ω_(I+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1) |ψ(I_ω+ψ_Ω_(I+1)(I_ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1,1) |ψ(I_ω+ψ_Ω_(I+1)(I_ω+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1,1)(3,1)(2) |ψ(I_ω+ψ_Ω_(I+1)(I_ω+I)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(3) |ψ(I_ω+ψ_Ω_(I+1)(I_ω+ψ_Ω_(I+1)(I_ω+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(4,2) |ψ(I_ω+Ω_(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(4,2,1) |ψ(I_ω+Ω_(I+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(4,2,1)(5,2,1)(6,2)(5) |ψ(I_ω+ψI_2(I_2)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(4,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1) |ψ(I_ω+ψI_2(I_ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(4,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1)(5,2)(6,3,1)(7,3,1)(8,3)(7) |ψ(I_ω+I_2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1,1) |ψ(I_ω*2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2) |ψ(I_ω*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1) |ψ(I_ω*Ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1) |ψ(I_ω*Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I_ω*ψI(I)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ω*ψI(I_ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,1)(2) |ψ(I_ω*I) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,1)(6,2) |ψ(I_ω*ψ_Ω_(I+1)(Ω_(I+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2) |ψ(I_ω*Ω_(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2)(4,2,1) |ψ(I_ω*Ω_(I+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2)(4,2,1)(5,2,1)(6,1)(2) |ψ(I_ω*Ω_(I2)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2)(4,2,1)(5,2,1)(6,2) |ψ(I_ω*Ω_Ω_(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2)(4,2,1)(5,2,1)(6,2)(5) |ψ(I_ω*ψI_2(I_2)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2)(4,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1) |ψ(I_ω*ψI_2(I_ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2)(4,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1)(8,2)(5) |ψ(I_ω*I_2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(1,1,1)(2,1,1)(3,1,1) |ψ(I_ω^2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(2) |ψ(I_ω^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(3,2) |ψ(Ω_(I_ω+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(3,2,1) |ψ(Ω_(I_ω+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(3,2,1)(4,2,1)(5,2)(4) |ψ(I_(ω+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1) |ψ(I_(ω2)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(4,2)(5,3,1)(6,3,1)(7,3,1) |ψ(I_(ω3)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1) |ψ(I_(ω^2)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(2,1,1) |ψ(I_(ω^3)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3) |ψ(I_(ω^ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1) |ψ(I_Ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1) |ψ(I_Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(2) |ψ(I_Λ) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_ψI(I_ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2) |ψ(I_I) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(5,2,1) |ψ(I_(I*ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(5,2,1)(6,1)(2) |ψ(I_(I^2)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(6) |ψ(I_(I^ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(6,1)(2) |ψ(I_(I^I)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(7,2) |ψ(I_ψ_Ω_(I+1)(Ω_(I+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(7,2,1) |ψ(I_ψ_Ω_(I+1)(Ω_(I+ω))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2) |ψ(I_Ω_(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2)(4,2,1) |ψ(I_Ω_(I+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2)(5) |ψ(I_Λ_(I+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1) |ψ(I_ψ_I_2(I_ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1)(8,3,1)(9,1)(2) |ψ(I_ψ_I_2(I_I)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1)(8,3,1)(9,2)(5) |ψ(I_I_2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1,1) |ψ(I_I_ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2) |ψ(I(1,0)) = Battery Memgent System Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1,1) |ψ(I(1,0)*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1,1) |ψ(I(1,0)*I_ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2) |ψ(I(1,0)^2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(3) |ψ(I(1,0)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4) |ψ(I(1,0)^I(1,0)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2) |ψ(Ω_(I(1,0)+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1) |ψ(Ω_(I(1,0)+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(5) |ψ(I_(I(1,0)+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_(I(1,0)+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_I_(I(1,0)+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5) |ψ(I(1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1)(8) |ψ(I(1,2)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1) |ψ(I(1,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2) |ψ(I(2,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1) |ψ(I(2,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3) |ψ(I(ω,0)) = Mutiple Buchholz's Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1) |ψ(I(Ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1) |ψ(I(Ω_ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(1,0,0)) = Transfinite Buchholz's Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2)(1,1,1)(2,1,1)(3,1,1)(3) |ψ(I(1,0,0)+I(ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(1,0,0)+ψI(1,0,0)(I(1,0,0))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1) |ψ(I(1,0,0)+ψI(1,0,0)(I(1,0,0))*Ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2) |ψ(I(1,0,0)+Ω(ψI(1,0,0)(I(1,0,0))+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0))+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5) |ψ(I(1,0,0)+I(ω,ψI(1,0,0)(I(1,0,0))+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(1,1,1)(2,1,1)(3,1,1)(3) |ψ(I(1,0,0)+I(I(ω,0),ψI(1,0,0)(I(1,0,0))+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)+ψ_Ω(ψI(1,0,0)(I(1,0,0))+1)(Ω(ψI(1,0,0)(I(1,0,0))))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)+ψ_Ω(ψI(1,0,0)(I(1,0,0))+1)(I(ψI(1,0,0)(I(1,0,0)),1)+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(3,2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)+Ω(ψI(1,0,0)(I(1,0,0))+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(3,2,1) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)+Ω(ψI(1,0,0)(I(1,0,0))+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(3,2,1)(4,2,1)(5,2,1) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)+I(ψI(1,0,0)(I(1,0,0))+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)*2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)*Ω(ψI(1,0,0)(I(1,0,0))+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2)(3,2,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),1)^2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2)(5,3) |ψ(I(1,0,0)+Ω_(I(ψI(1,0,0)(I(1,0,0)),1)+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),ψI(1,0,0)(I(1,0,0)))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,1)(5) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),ψI(1,0,0)(I(1,0,0))*ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,1)(6,2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),ψ_Ω(ψI(1,0,0)(I(1,0,0))+1)(Ω(ψI(1,0,0)(I(1,0,0))+1)))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),Ω_(I(ψI(1,0,0)(I(1,0,0)),1)+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,2)(3,2,1)(4,2,1)(5,2,1)(5,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0)),I(ψI(1,0,0)(I(1,0,0)),1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,2)(4) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0))+1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,2,1) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0))+ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(5) |ψ(I(1,0,0)+I(ψI(1,0,0)(I(1,0,0))*ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(6,2) |ψ(I(1,0,0)+I(ψ_Ω(ψI(1,0,0)(I(1,0,0))+1)(Ω(ψI(1,0,0)(I(1,0,0))+1)),0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,2) |ψ(I(1,0,0)+I(Ω_(I(ψI(1,0,0)(I(1,0,0)),1)+1),0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,2)(3,2,1)(4,2,1)(5,2,1)(5,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(1,0,0)+I(I(ψI(1,0,0)(I(1,0,0)),1),0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1) |ψ(I(1,0,0)*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(2) |ψ(I(1,0,0)^2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(2,1,1) |ψ(I(1,0,0)^2*ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(3) |ψ(I(1,0,0)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2) |ψ(Ω_(I(1,0,0)+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1) |ψ(I_(I(1,0,0)+ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6) |ψ(I(ω,I(1,0,0)+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(I(1,0,0),1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(5,2,1) |ψ(I(I(1,0,0),ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(5,2,1)(6,1)(2) |ψ(I(I(1,0,0),I(1,0,0))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(5,2,1)(6,2) |ψ(I(I(1,0,0),Ω_(I(1,0,0)+1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(5,2,1)(6,2)(4,2,1)(5,2,1)(6,2,1)(6,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(I(1,0,0),I(I(1,0,0),1))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(5,2,1)(6,2)(5) |ψ(I(I(1,0,0)+1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(5,2,1)(6,2,1) |ψ(I(I(1,0,0)+1,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(5,2,1)(6,2,1)(6) |ψ(I(I(1,0,0)+ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(6) |ψ(I(I(1,0,0)*ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,2) |ψ(I(Ω_(I(1,0,0)+1),0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,2)(4,2,1)(5,2,1)(6,2,1)(6) |ψ(I(I(ω,I(1,0,0)+1),0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,2)(4,2,1)(5,2,1)(6,2,1)(6,1)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(I(I(1,0,0),1),0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,2)(5) |ψ(I(1,0,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1) |ψ(I(1,0,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2) |ψ(I(1,1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2) |ψ(Ω_(I(1,1,0)+1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,1)(2) |ψ(I(I(1,1,0),1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,2) |ψ(I(Ω_(I(1,1,0)+1),0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,2)(5) |ψ(I(1,1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1) |ψ(I(1,1,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2) |ψ(I(1,2,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(3) |ψ(I(1,ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(I(2,0,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(3) |ψ(I(ω,0,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(3,1)(2) |ψ(M^M^2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4) |ψ(M^M^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,1)(2) |ψ(M^M^M) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,1)(5,1)(2) |ψ(M^M^M^M) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,2) |ψ(2 aft 2-2) = Small Rathjen's Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,2,1) |ψ(1-2 aft 2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(7,3) |ψ(2 aft 2nd 2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1) |ψ(1-2-2) = Small Mahlo Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(2,1)(3,2) |ψ(2 aft 1-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(2,1,1) |ψ(1-1-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(2,1,1)(3,1)(2) |ψ((1-)^(1,0) 2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(2,1,1)(3,1)(4,2) |ψ(2 aft 2 1-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(2,1,1)(3,1,1) |ψ(1-2 1-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(2,1,1)(3,1,1)(3,1,1) |ψ(1-2-2 1-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3) |ψ((2-2 1-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1) |ψ((2-2 1-)^Ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1)(2) |ψ((2-2 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1)(4,2) |ψ(2 aft 2-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(6,2,1)(6,2)(5) |ψ(2nd 2-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1,1) |ψ(1-2-2-2) = Small Nonconvertible Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1,1)(3,1)(2) |ψ(2-2-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1,1)(3,1)(4,2) |ψ(2 aft 2-2-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1,1)(3,1,1) |ψ(1-2-2-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1,1)(3,1,1)(3,1,1) |ψ(1-2-2-2-2-2) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4) |ψ((2-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1) |ψ((2-)^Ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(1,1,1) |ψ((2-)^Ω_ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2) |ψ((2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2,1,1) |ψ(1-(2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2,1,1)(3,1)(2) |ψ(2 1-(2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2,1,1)(3,1,1)(3) |ψ((2 1-)^ω (2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2,1,1)(3,1,1)(3,1)(2) |ψ((2 1-)^(1,0) (2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2,1,1)(3,1,1)(3,1)(4,2) |ψ(2 aft 2-2 1-(2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2,1,1)(3,1,1)(3,1,1) |ψ(1-2-2 1-(2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2,1,1)(3,1,1)(4) |ψ((2-)^ω 1-(2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(2,1,1)(3,1,1)(4,1)(2) |ψ((2-)^(1,0) 1-(2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3) |ψ(((2-)^(1,0) 1-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(2) |ψ(((2-)^(1,0) 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(2,1,1) |ψ(1-((2-)^(1,0) 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(2,1,1)(3,1,1) |ψ(1-2 1-((2-)^(1,0) 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(2,1,1)(3,1,1)(4) |ψ((2-)^ω 1-((2-)^(1,0) 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(2,1,1)(3,1,1)(4,1)(2) |ψ(((2-)^(1,0) 1-)^(1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(2,1,1)(3,1,1)(4,1)(3) |ψ(((2-)^(1,0) 1-)^(1,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(2,1,1)(3,1,1)(4,1)(3,1)(2) |ψ(((2-)^(1,0) 1-)^(2,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(3) |ψ(((2-)^(1,0) 1-)^(ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(3,1)(2) |ψ(((2-)^(1,0) 1-)^(1,0,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(4) |ψ(((2-)^(1,0) 1-)^(1@ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1)(4,2) |ψ(2 aft (2-)^(1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1,1) |ψ(1-(2-)^(1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1,1)(4) |ψ((2-)^(1,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(3,1,1)(4,1)(2) |ψ((2-)^(2,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(4) |ψ((2-)^(ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(4,1)(2) |ψ((2-)^(1,0,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5) |ψ((2-)^(1@ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,1)(2) |ψ((2-)^(1@(1,0))) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2) |ψ(2 aft 3) = Rathjen's Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1) |ψ(1-2 aft 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(7,2)(6) |ψ(2 1-2 aft 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(7,2,1) |ψ(1-2 1-2 aft 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(7,2,1)(7,2)(6) |ψ(2-2 aft 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(7,2,1)(7,2,1) |ψ(1-2-2 aft 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(7,2,1)(8) |ψ((2-)^ω aft 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(7,2,1)(8,1)(6) |ψ(2nd 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)(6,2,1)(7,2,1)(8,1)(9,2,1)(A,2,1)(B,2,1)(C,2,1)(B,1)(A) |ψ(3rd 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-3) = Small Weakly Compect Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1) |ψ(1-1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1)(2) |ψ(2 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1) |ψ(1-2 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(3,1)(2) |ψ(2-2 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(3,1,1) |ψ(1-2-2 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(4) |ψ((2-)^ω 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(4,1)(2) |ψ((2-)^(1,0) 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(4,1)(5,2) |ψ(2 aft 3 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-3 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-3 1-3 1-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3) |ψ((3 1-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(2) |ψ((3 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(2,1,1) |ψ(1-(3 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(2,1,1)(3,1,1) |ψ(2 1-(3 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(2,1,1)(3,1,1)(4) |ψ((2-)^ω 1-(3 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-(3 1-)^(1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(2,1,1)(3,1,1)(4,1,1)(3) |ψ((3 1-)^(1,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(2) |ψ((3 1-)^(2,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1)(3) |ψ((3 1-)^(ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1) |ψ(1-2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(2,1,1)(3,1)(2) |ψ(2 1-2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(2,1,1)(3,1,1) |ψ(1-2 1-2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-3 1-2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1) |ψ(1-2-3 1-2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3) |ψ((3 1-2-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1)(2) |ψ((3 1-2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1)(2,1,1) |ψ(1-(3 1-2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-3 1-(3 1-2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1) |ψ(1-(3 1-2-)^(1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3) |ψ(1-(3 1-2-)^(1,ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1)(3) |ψ(1-(3 1-2-)^(ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1)(4) |ψ(1-(3 1-2-)^(1@ω)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1)(4,2) |ψ(2 aft 2-2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1,1) |ψ(1-2-2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1,1)(3,1,1) |ψ(1-2-2-2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4) |ψ((2-)^ω 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(2) |ψ((2-)^(1,0) 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(2,1,1) |ψ(1-(2-)^(1,0) 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-3 1-(2-)^(1,0) 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(2) |ψ((2-)^(1,0) 3 1-(2-)^(1,0) 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(3) |ψ(((2-)^(1,0) 3 1-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(3,1)(2) |ψ(((2-)^(1,0) 3 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(3,1,1) |ψ(1-(2-)^(1,1) 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(3,1,1)(4,1)(2) |ψ(1-(2-)^(2,0) 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(4,1)(2) |ψ(1-(2-)^(1,0,0) 3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(5,2) |ψ(2 aft 3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1) |ψ(1-3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1)(2) |ψ(2 1-3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-3 1-3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(3) |ψ((3 1-)^ω 3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(3,1)(2) |ψ((3 1-)^(1,0) 3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(3,1,1) |ψ(1-2-3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(3,1,1)(3,1,1) |ψ(1-2-2-3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(3,1,1)(4) |ψ((2-)^ω 3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1)(2) |ψ((2-)^(1,0) 3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1) |ψ(1-3 2-3 2-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4) |ψ((3 2-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(2) |ψ((3 2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(2,1,1) |ψ(1-(3 2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(2,1,1)(3,1,1)(4,1,1) |ψ(1-3 1-(3 2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(2) |ψ((3 2-)^(1,0) 1-(3 2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(3) |ψ(((3 2-)^(1,0) 1-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(3,1)(2) |ψ(((3 2-)^(1,0) 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(3,1,1) |ψ(2-(3 2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(3,1,1)(4) |ψ((2-)^ω (3 2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(3,1,1)(4,1)(2) |ψ((2-)^(1,0) (3 2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(3,1,1)(4,1,1) |ψ(1-(3 2-)^(1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(3,1,1)(4,1,1)(4,1)(2) |ψ(1-(3 2-)^(2,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(4,1)(2) |ψ(1-(3 2-)^(1,0,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1)(5,2) |ψ(2 aft 3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1) |ψ(1-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1) |ψ(1-3-3 1-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3) |ψ((3-3 1-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3,1)(2) |ψ((3-3 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3,1,1) |ψ(1-2-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3,1,1)(4) |ψ((2-)^ω 3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3,1,1)(4,1)(2) |ψ((2-)^(1,0) 3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3,1,1)(4,1,1) |ψ(1-3 2-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3,1,1)(4,1,1)(3,1,1)(4,1,1) |ψ(1-3 2-3 2-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3,1,1)(4,1,1)(4,1)(2) |ψ((3 2-)^(1,0) 2-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(3,1,1)(4,1,1)(4,1,1) |ψ(1-3-3 2-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(4) |ψ((3-3 2)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(4,1)(2) |ψ((3-3 2)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(4,1,1) |ψ(1-3-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(4,1,1)(4,1,1)(4,1,1) |ψ(1-3-3-3-3) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5) |ψ((3-)^ω) = Weak UNOCF Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(2) |ψ((3-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(2,1,1) |ψ(1-(3-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(2) |ψ((3-)^(1,0) 1-(3-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(3,1)(2) |ψ(((3-)^(1,0) 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(3,1,1) |ψ(1-2-(3-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(3,1,1)(4,1,1)(5,1)(2) |ψ((3-)^(1,0) 2-(3-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(4,1)(2) |ψ(((3-)^(1,0) 2-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(4,1,1) |ψ(1-(3-)^(1,1)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(4,1,1)(5,1)(2) |ψ(1-(3-)^(2,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(5) |ψ(1-(3-)^(ω,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(5,1)(2) |ψ(1-(3-)^(1,0,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(6,2) |ψ(2 aft 4) = Duchhart's Ordinal |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(6,2,1)(7,2,1)(8,2,1)(9,2,1)(A,2)(7) |ψ(2nd 4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1) |ψ(1-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(2) |ψ(4 1-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(3,1)(2) |ψ((4 1-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(3,1,1) |ψ(1-2-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(3,1,1)(3,1,1) |ψ(1-2-2-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(3,1,1)(4,1,1) |ψ(1-3 2-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(3,1,1)(4,1,1)(5,1)(2) |ψ((3 2-)^(1,0) 4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(3,1,1)(4,1,1)(5,1,1) |ψ(1-4 2-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(4) |ψ((4 2-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(4,1,1) |ψ(1-3-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(4,1,1)(5,1)(2) |ψ((4 3-)^(1,0)) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(4,1,1)(5,1,1) |ψ(1-4 3-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(5) |ψ((3 4-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(5,1,1) |ψ(1-4-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(5,1,1)(5,1,1) |ψ(1-4-4-4) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(6) |ψ((4-)^ω) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(6,1,1) |ψ(1-5) |- |(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)(6,1,1)(7,1,1) |ψ(1-6) |- |(0)(1,1,1)(2,2) |ψ(λa.a+1-Π0) = Small Sterget's Ordinal |} [[分类:分析]]
返回
BMS分析Part3：EBO~SSO
。
查看“︁BMS分析Part3：EBO~SSO”︁的源代码
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