序数表：修订间差异

2025年7月5日 (六) 16:37的版本

本条目列举出一些有名字的序数，它们大多在 googology 中具有重大意义

需要注意的是，它们的命名很多来自 googology 爱好者而非专业数学研究者。

序数表


缩写	英文全称	常规表示方法(BOCF等)	BMS/Y
FTO	First Transfinite Ordinal	$ω$	$B M S (0) (1)$
LAO	Linar Array Ordinal^[1]	$ω^{ω}$	$B M S (0) (1) (2)$
SCO	Small Cantor Ordinal	$φ (1, 0) = ε_{0} = ψ (Ω)$	$B M S (0, 0) (1, 1)$
CO	Cantor Ordinal	$φ (2, 0) = ζ_{0} = ψ (Ω^{2})$	$B M S (0, 0) (1, 1) (2, 1)$
LCO	Large Cantor Ordinal	$φ (3, 0) = η_{0} = ψ (Ω^{3})$	$B M S (0, 0) (1, 1) (2, 1) (2, 1)$
HCO	Hyper Cantor Ordinal	$φ (ω, 0) = ψ (Ω^{ω})$	$B M S (0, 0) (1, 1) (2, 1) (3, 0)$
FSO	Feferman-Schutte Ordinal	$φ (1, 0, 0) = Γ_{0} = ψ (Ω^{Ω})$	$B M S (0, 0) (1, 1) (2, 1) (3, 1)$
ACO	Ackermann Ordinal	$φ (1, 0, 0, 0) = ψ (Ω^{Ω^{2}})$	$B M S (0, 0) (1, 1) (2, 1) (3, 1) (3, 1)$
SVO	Small Veblen Ordinal	$ψ (Ω^{Ω^{ω}})$	$B M S (0, 0) (1, 1) (2, 1) (3, 1) (4, 0)$
LVO	Large Veblen Ordinal	$ψ (Ω^{Ω^{Ω}})$	$B M S (0, 0) (1, 1) (2, 1) (3, 1) (4, 1)$
ESVO	Extended Small Veblen Ordinal	$ψ (Ω^{Ω^{Ω^{ω}}})$	$B M S (0, 0) (1, 1) (2, 1) (3, 1) (4, 1) (5, 0)$
ELVO	Extended Large Veblen Ordinal	$ψ (Ω^{Ω^{Ω^{Ω}}})$	$B M S (0, 0) (1, 1) (2, 1) (3, 1) (4, 1) (5, 1)$
BHO	Bachmann-Howard Ordinal	$ψ (Ω_{2})$	$B M S (0, 0) (1, 1) (2, 2)$
BO	Buchholz's Ordinal	$ψ (Ω_{ω})$	$B M S (0, 0, 0) (1, 1, 1)$
TFBO	Takeuti-Feferman-Buchholz Ordinal	$ψ (Ω_{ω + 1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 0) (3, 2, 0)$
BIO	Bird's Ordinal^[2]	$ψ (Ω_{Ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 0)$
EBO	Extended Buchholz Ordinal	$ψ (I)$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 0) (2, 0, 0)$
JO	Jager's Ordinal	$ψ (Ω_{I + 1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 0) (4, 2, 0)$
SIO	Small Inaccessible Ordinal	$ψ (I_{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1)$
MBO	Mutiply Buchholz Ordinal	$ψ (I (ω, 0)) = ψ (M^{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 0, 0)$
TBO	Transfinitary Buchholz's Ordinal	$ψ (I (1, 0, 0)) = ψ (M^{M})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 1, 0) (2, 0, 0)$
SRO	Small Rathjen Ordinal	$ψ (ε_{M + 1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 1, 0) (4, 2, 0)$
SMO	Small Mahlo Ordinal	$ψ (M_{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 1, 1)$
SNO	Small 1-Mahlo (N) Ordinal	$ψ (N_{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 1, 1) (3, 1, 1)$
RO	Rathjen's Ordinal	$ψ (ε_{K + 1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (4, 1, 0) (5, 2, 0)$
SKO	Small Weakly Compact (K) Ordinal	$ψ (K_{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (4, 1, 1)$
DO	Duchhart's Ordinal	$ψ (Ω (Π_{4} + 1))$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (4, 1, 1) (5, 1, 0) (6, 2, 0)$
SSO	Small Stegert Ordinal	$ψ (p s d . Π_{ω}) = ψ (a_{2})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0)$
BGO	TSS 1st Back Gear Ordinal	$ψ (p s d . Π_{ω}) = ψ (a_{2})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0)$
LSO	Large Stegert Ordinal	$ψ (λ α . (α \times 2) - Π_{0}) = ψ (a_{2}^{a})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0) (3, 2, 0) (4, 1, 0) (2, 0, 0)$
APO	Admissible-parameter free effective cardinal Ordinal	$ψ (1 - ((+) - Π_{1})) = ψ (a_{2}^{Ω_{a + 1}})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0) (3, 2, 0) (4, 1, 1)$
BGO	TSS 1st Back Gear Ordinal (CN ggg)^[3]	$ψ (Π_{1} - λ α . (Ω_{α + 2}) - Π_{1}) = ψ (Ω_{a_{2} + 1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1)$
SDO	Small Dropping Ordinal	$ψ (λ α . (Ω_{α + ω}) - Π_{0} = ψ (Ω_{a_{2} + ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 0, 0)$
LDO	Large Dropping Ordinal	$ψ (λ α . (O F P a f t α) - Π_{0}) = ψ (Ω_{a_{2} + 1} \times a_{2})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 2, 0)$
DSO	Doubly +1 Stable Ordinal	$ψ (λ α . (λ β . β + 1 - Π_{0}) - Π_{0} = ψ (a_{3})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 3, 0)$
TSO	Triply +1 Stable Ordinal	$ψ (λ α . (λ β . (λ γ . γ + 1 - Π_{0}) - Π_{0}) - Π_{0} = ψ (a_{4})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 3, 1) (4, 4, 0)$
pfec LRO	p.f.e.c. Large Rathjen Ordinal	$ψ (p f e c . ω - π - Π_{0}) = ψ (a_{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 2)$
SBO	Small Bashicu Ordinal	$ψ (p f e c . ω - π - Π_{0}) = ψ (a_{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 2)$
pfec M2O	pfec min Σ₂ Ordinal	$ψ (p f e c . \min (a ≺_{Σ_{1}} b ≺_{Σ_{2}} c))$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 2) (3, 2, 2) (4, 2, 2) (4, 2, 1)$ ？
LRO	Large Rathjen Ordinal	$F \cap ω_{1}^{CK}, θ = ω$	$⩽ B M S (0) (1, 1, 1, 1) ?$
SSPO	Small Simple Projection Ordinal	$ψ (ω - proj.) = ψ (σ S \times ω) = ψ (H^{ω})$	$B M S (0) (1, 1, 1, 1)$
TSSO	Trio Sequence System Ordinal	$ψ (ω - proj.) = ψ (σ S \times ω) = ψ (H^{ω})$	$B M S (0) (1, 1, 1, 1)$
LSPO	Large Simple Projection Ordinal	$ψ (\min α is α - proj.) = ψ (σ S \times S)$	$B M S (0) (1, 1, 1, 1) (2, 1, 1, 1) (3, 1) (2)$
EO	Eveog's Ordinal	$ψ (ψ_{σ} (σ_{n}))$	$B M S (0) (1, 1, 1, 1) (2, 2, 1, 1) (3)$
Q1BGO	Quadro Sequence System 1st Back Gear Ordinal	-	$B M S (0) (1, 1, 1, 1) (2, 2, 2)$
ESPO	Extend Simple Projection Ordinal	-	$B M S (0) (1, 1, 1, 1) (2, 2, 2, 1)$
BOBO	Big Omega Back Ordinal	-	$B M S (0) (1, 1, 1, 1) (2, 2, 2, 2)$
QSSO	Quardo Sequence System Ordinal	$ψ (ψ_{H} (H^{H^{ω}})) ?$	$B M S (0, 0, 0, 0, 0) (1, 1, 1, 1, 1)$
TCAO	Trio Comprehension Axiom Ordinal	$PTO ((Π_{3}^{1} - C A)_{0})$	$⩾ B M S (0) (1, 1, 1, 1, 1)$
QiSSO	Quinto Sequence System Ordinal	-	$B M S (0) (1, 1, 1, 1, 1, 1)$
SHO/BMO^[4]	Small Hydra Ordinal	$ψ (ψ_{H} (ε_{H + 1})) ?$	$Y (1, 3) = B M S 极限$
ΩSSO	\Omega Sequence System Ordinal		$Y (1, 3, 4, 2, 5, 8, 10)$
GHO	No-Go Hydra Ordinal^[5]		$Y (1, 3, 4, 3)$
SYO	Small Yukito Ordinal		$ω - Y (1, 4)$
MHO/ωYO^[4]	Medium Hydra Ordinal		$ω - Y$ 极限
CKO	Church-Kleene Ordinal	$ω_{1}^{C K}$
FUO	First Uncountable Ordinal	$ω_{1}$

↑ 因为在googology一度经典的线性数阵的极限是它，因此得名
↑ 鸟之数阵第四版的极限是它，因此得名
↑ Bashicu对BGO的原定义是BMS(0,0,0)(1,1,1)(2,2,0)。BGO指(0,0,0)(1,1,1)(2,2,1)是中文googology社区的重命名
↑ ^4.0 ^4.1 SHO,MHO的名字均来自FataliS1024.但原定义的SHO指的是 $ε_{0}$ ,MHO指的是BMS极限。还有一个LHO指 $ω - Y$ 极限。但后来不知为何变成了现在的这个版本，而LHO成为了无定义的名字
↑ 原名Guo bu qu de Hydra Ordinal，但过于口语化和非正式。而这个序数本身确实是一个重要的序数。曹知秋将名字改成了现在的版本

[1] 因为在googology一度经典的线性数阵的极限是它，因此得名

[2] 鸟之数阵第四版的极限是它，因此得名

[3] Bashicu对BGO的原定义是BMS(0,0,0)(1,1,1)(2,2,0)。BGO指(0,0,0)(1,1,1)(2,2,1)是中文googology社区的重命名

[:0-4] 4.0 ^4.1 SHO,MHO的名字均来自FataliS1024.但原定义的SHO指的是 $ε_{0}$ ,MHO指的是BMS极限。还有一个LHO指 $ω - Y$ 极限。但后来不知为何变成了现在的这个版本，而LHO成为了无定义的名字

[5] 原名Guo bu qu de Hydra Ordinal，但过于口语化和非正式。而这个序数本身确实是一个重要的序数。曹知秋将名字改成了现在的版本

[1]

[2]

[3]

[4]

[5]

@@ 第30行： / 第30行： @@
 |<math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(3,1)</math>
 |-
-| [[SVO]]|| Small Veblen Ordinal || <nowiki>\( \varphi(1@\omega) =\psi(\Omega^{\Omega^{\omega}})  \)</nowiki>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,0)</math>
+| [[SVO]]|| Small Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\omega}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,0)</math>
 |-
-| [[LVO]]|| Large Veblen Ordinal || <nowiki>\( \varphi(1@(1,0)) =\psi(\Omega^{\Omega^{\Omega}})  \)</nowiki>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,1)</math>
+| [[LVO]]|| Large Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\Omega}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,1)</math>
 |-
 | [[ESVO]]|| Extended Small Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\Omega^\omega}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,1)(5,0)</math>
@@ 第56行： / 第56行： @@
 |TBO
 |Transfinitary Buchholz's Ordinal
-|\( \psi(I(1@(1,0))) \)
+|<math>\psi(I(1,0,0))=\psi(M^M)</math>
 |<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)</math>
 |-
@@ 第84行： / 第84行： @@
 |BGO
 |TSS 1st Back Gear Ordinal
-| -
+| <math>\psi(psd.\Pi_{\omega})=\psi(a_2)</math>
 |<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,0)</math>
 |-
@@ 第91行： / 第91行： @@
 |APO
 |Admissible-parameter free effective cardinal Ordinal
-|<math>\psi(1-((+)-\Pi_1)) </math>
+|<math>\psi(1-((+)-\Pi_1))=\psi(a_2^{\Omega_{a+1}})</math>
 |<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,1,1)</math>
 |-
-| [[BGO]]|| TSS 1st Back Gear Ordinal (CN ggg)<ref>Bashicu对BGO的原定义是BMS(0,0,0)(1,1,1)(2,2,0)。BGO指(0,0,0)(1,1,1)(2,2,1)是中文googology社区的重命名</ref>|| <math>\psi(\Pi_{1}-\lambda\alpha.(\Omega_{\alpha+2})-\Pi_{1})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)</math>
+| [[BGO]]|| TSS 1st Back Gear Ordinal (CN ggg)<ref>Bashicu对BGO的原定义是BMS(0,0,0)(1,1,1)(2,2,0)。BGO指(0,0,0)(1,1,1)(2,2,1)是中文googology社区的重命名</ref>|| <math>\psi(\Pi_{1}-\lambda\alpha.(\Omega_{\alpha+2})-\Pi_{1})=\psi(\Omega_{a_2+1})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)</math>
 |-
-| [[SDO]]|| Small Dropping Ordinal || <math>\psi(\lambda\alpha.(\Omega_{\alpha+\omega})-\Pi_{0})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)(3,0,0)</math>
+| [[SDO]]|| Small Dropping Ordinal || <math>\psi(\lambda\alpha.(\Omega_{\alpha+\omega})-\Pi_{0}=\psi(\Omega_{a_2+\omega})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)(3,0,0)</math>
 |-
-| [[LDO]]|| Large Dropping Ordinal || <math>\psi(\lambda\alpha.(\mathrm{OFP}\ \mathrm{aft}\ \alpha)-\Pi_{0})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)(3,2,0)</math>
+| [[LDO]]|| Large Dropping Ordinal || <math>\psi(\lambda\alpha.(\mathrm{OFP}\ \mathrm{aft}\ \alpha)-\Pi_{0})=\psi(\Omega_{a_2+1}\times a_2)</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)(3,2,0)</math>
 |-
 |DSO
 |Doubly +1 Stable Ordinal
-|<math>\psi(\lambda\alpha.(\lambda\beta.\beta+1-\Pi_0)-\Pi_0 </math>
+|<math>\psi(\lambda\alpha.(\lambda\beta.\beta+1-\Pi_0)-\Pi_0=\psi(a_3) </math>
 |<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)(3,3,0)</math>
 |-
 |TSO
 |Triply +1 Stable Ordinal
-|<math>\psi(\lambda\alpha.(\lambda\beta.(\lambda\gamma.\gamma+1-\Pi_0)-\Pi_0)-\Pi_0 </math>
+|<math>\psi(\lambda\alpha.(\lambda\beta.(\lambda\gamma.\gamma+1-\Pi_0)-\Pi_0)-\Pi_0=\psi(a_4) </math>
 |<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,4,0)</math>
 |-
-| pfec LRO|| p.f.e.c. Large Rathjen Ordinal || <math>\psi(pfec.\omega-\pi-\Pi_{0})=\psi(a_\omega) </math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,2)</math>
+| pfec LRO|| p.f.e.c. Large Rathjen Ordinal || <math>\psi(pfec.\omega-\pi-\Pi_{0})=\psi(a_\omega)</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,2)</math>
 |-
 |SBO
 |Small Bashicu Ordinal
-| -
+| <math>\psi(pfec.\omega-\pi-\Pi_{0})=\psi(a_\omega) </math>
 |<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,2)</math>
 |-
@@ 第120行： / 第120行： @@
 |pfec min Σ<sub>2</sub> Ordinal
 |<math>\psi(pfec.\min(a\prec_{\Sigma_1}b\prec_{\Sigma_2}c)) </math>
-|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,2,1)</math>
+|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,2,1)</math>？
 |-
 |LRO
@@ 第129行： / 第129行： @@
 |SSPO
 |Small Simple Projection Ordinal
-|<math>\psi(psd.\omega-\text{proj.}) </math>
+|<math>\psi(\omega-\text{proj.})=\psi(\sigma S\times \omega)=\psi(H^\omega)</math>
 |<math>\mathrm{BMS}(0)(1,1,1,1)</math>
 |-
-| [[TSSO]]|| Trio Sequence System Ordinal || <math>\psi(\omega-projection)=\psi(\sigma S\times \omega)=\psi(H^\omega)</math>|| <math>\mathrm{BMS}(0)(1,1,1,1)</math>
+| [[TSSO]]|| Trio Sequence System Ordinal || <math>\psi(\omega-\text{proj.})=\psi(\sigma S\times \omega)=\psi(H^\omega)</math>|| <math>\mathrm{BMS}(0)(1,1,1,1)</math>
 |-
 |LSPO
 |Large Simple Projection Ordinal
-|<math>\psi(\min\ \alpha\text{ is }\alpha-\text{proj.}) </math>
+|<math>\psi(\min\ \alpha\text{ is }\alpha-\text{proj.})=\psi(\sigma S\times S) </math>
 |<math>\mathrm{BMS}(0)(1,1,1,1)(2,1,1,1)(3,1)(2)</math>
 |-