序数表：修订间差异

2025年7月4日 (五) 21:39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	$φ (1, 0, 0, 0)$	$B M S (0, 0) (1, 1) (2, 1) (3, 1) (3, 1)$
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	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	$ \psi(I(1@(1,0))) $	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 1, 0) (2, 0, 0)$
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	-	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0)$
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	$ψ (1 - ((+) - Π_{1}))$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0) (3, 2, 0) (4, 1, 1)$
BGO	TSS 1st Back Gear Ordinal (CN ggg)^[3]	$ψ (Π_{1} - λ α . (Ω_{α + 2}) - Π_{1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1)$
SDO	Small Dropping Ordinal	$ψ (λ α . (Ω_{α + ω}) - Π_{0})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 0, 0)$
LDO	Large Dropping Ordinal	$ψ (λ α . (O F P a f t α) - Π_{0})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 2, 0)$
DSO	Doubly +1 Stable Ordinal	$ψ (λ α . (λ β . β + 1 - Π_{0}) - Π_{0}$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 3, 0)$
TSO	Triply +1 Stable Ordinal	$ψ (λ α . (λ β . (λ γ . γ + 1 - Π_{0}) - Π_{0}) - Π_{0}$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 3, 1) (4, 4, 0)$
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	-	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 2)$
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	$ψ (p s d . ω - proj.)$	$B M S (0) (1, 1, 1, 1)$
TSSO	Trio Sequence System Ordinal	$ψ (ω - p r o j e c t i o n) = ψ (σ S \times ω) = ψ (H^{ω})$	$B M S (0) (1, 1, 1, 1)$
LSPO	Large Simple Projection Ordinal	$ψ (\min α is α - proj.)$	$B M S (0) (1, 1, 1, 1) (2, 1, 1, 1) (3, 1) (2)$
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62行： / 第62行： @@
 |-
 | [[SMO]]|| Small Mahlo Ordinal || <math>\psi(M_\omega)</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)</math>
+|-
+|SNO
+|Small 1-Mahlo (N) Ordinal
+|<math>\psi(N_\omega)</math>
+|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1,1)</math>
 |-
 | [[RO]]|| Rathjen's Ordinal || <math>\psi(\varepsilon_{K+1})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,0)(5,2,0)</math>
 |-
-| [[SSO]]|| Small Stegert Ordinal || <math>\psi(\Pi_{\omega})=\psi(a_2)</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,0)</math>
+|SKO
+|Small Weakly Compact (K) Ordinal
+|<math>\psi(K_\omega)</math>
+|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)</math>
+|-
+|DO
+|Duchhart's Ordinal
+|<math>\psi(\Omega(\Pi_4+1))</math>
+|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,0)(6,2,0)</math>
+|-
+| [[SSO]]|| Small Stegert Ordinal || <math>\psi(psd.\Pi_{\omega})=\psi(a_2)</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,0)</math>
+|-
+|BGO
+|TSS 1st Back Gear Ordinal
+| -
+|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,0)</math>
 |-
 | [[LSO]]|| Large Stegert Ordinal || <math>\psi(\lambda\alpha.(\alpha\times 2)-\Pi_{0})=\psi(a_2^a)</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,1,0)(2,0,0)</math>
 |-
-| [[BGO]]|| TSS 1st Back Gear Ordinal<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>
+|APO
+|Admissible-parameter free effective cardinal Ordinal
+|<math>\psi(1-((+)-\Pi_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>
 |-
 | [[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>
@@ 第75行： / 第100行： @@
 | [[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>
 |-
-| [[pLRO]]|| p.f.e.c. Large Rathjen Ordinal || <math>\psi(\omega-\pi-\Pi_{0})=\psi(a_\omega) </math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,2)</math>
+|DSO
+|Doubly +1 Stable Ordinal
+|<math>\psi(\lambda\alpha.(\lambda\beta.\beta+1-\Pi_0)-\Pi_0 </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>\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>
+|-
+|SBO
+|Small Bashicu Ordinal
+| -
+|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,2)</math>
+|-
+|pfec M2O
+|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>
+|-
+|LRO
+|Large Rathjen Ordinal
+|<math>F\cap\omega_1^\text{CK},\theta=\omega </math>
+|<math>\leqslant\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}(0,0,0,0)(1,1,1,1)</math>
+|SSPO
+|Small Simple Projection Ordinal
+|<math>\psi(psd.\omega-\text{proj.}) </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>
+|-
+|LSPO
+|Large Simple Projection Ordinal
+|<math>\psi(\min\ \alpha\text{ is }\alpha-\text{proj.}) </math>
+|<math>\mathrm{BMS}(0)(1,1,1,1)(2,1,1,1)(3,1)(2)</math>
+|-
+|EO
+|Eveog's Ordinal
+|<math>\psi(\psi_\sigma(\sigma_n)) </math>
+|<math>\mathrm{BMS}(0)(1,1,1,1)(2,2,1,1)(3)</math>
+|-
+|Q1BGO
+|Quadro Sequence System 1st Back Gear Ordinal
+| -
+|<math>\mathrm{BMS}(0)(1,1,1,1)(2,2,2)</math>
+|-
+|ESPO
+|Extend Simple Projection Ordinal
+| -
+|<math>\mathrm{BMS}(0)(1,1,1,1)(2,2,2,1)</math>
+|-
+|BOBO
+|Big Omega Back Ordinal
+| -
+|<math>\mathrm{BMS}(0)(1,1,1,1)(2,2,2,2)</math>
 |-
 | [[QSSO]]|| Quardo Sequence System Ordinal || <math>\psi(\psi_{H}(H^{H^{\omega}}))?</math>|| <math>\mathrm{BMS}(0,0,0,0,0)(1,1,1,1,1)</math>
+|-
+|TCAO
+|Trio Comprehension Axiom Ordinal
+|<math>\text{PTO}((\Pi_3^1-CA)_0) </math>
+|<math>\geqslant\mathrm{BMS}(0)(1,1,1,1,1)</math>
+|-
+|QiSSO
+|Quinto Sequence System Ordinal
+| -
+|<math>\mathrm{BMS}(0)(1,1,1,1,1,1)</math>
 |-
 | SHO/BMO<ref name=":0">SHO,MHO的名字均来自FataliS1024.但原定义的SHO指的是<math>\varepsilon_0</math>,MHO指的是BMS极限。还有一个LHO指<math>\omega -Y</math>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</ref>|| Small Hydra Ordinal || <math>\psi(\psi_{H}(\varepsilon_{H+1}))?</math>|| <math>Y(1,3)=BMS\text{极限}</math>