序数表：修订间差异

2025年7月3日 (四) 14:12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	$φ (1, 0)$	$B M S (0, 0) (1, 1)$
CO	Cantor Ordinal	$φ (2, 0)$	$B M S (0, 0) (1, 1) (2, 1)$
HCO	Hyper Cantor Ordinal	$φ (ω, 0)$	$B M S (0, 0) (1, 1) (2, 1) (3, 0)$
FSO	Feferman-Schutte Ordinal	$φ (1, 0, 0)$	$B M S (0, 0) (1, 1) (2, 1) (3, 1)$
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	$ψ (Ω^{Ω^{Ω}})$	$B M S (0, 0) (1, 1) (2, 1) (3, 1) (4, 1)$
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	$ψ (I (ω, 0))$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 0, 0)$
SRO	Small Rathjen Ordinal	$ψ (Ω_{M + 1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 1, 0) (4, 2, 0)$
RO	Rathjen's Ordinal	$ψ (Ω_{K + 1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (4, 1, 0) (5, 2, 0)$
SSO	Small Stegert Ordinal	$ψ (Π_{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0)$
LSO	Large Stegert Ordinal	$ψ (λ α . (α \times 2) - Π_{0})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0) (3, 2, 0) (4, 1, 0) (2, 0, 0)$
BGO	TSS 1st Back Gear Ordinal^[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	$ψ (λ α . (O F P a f t α) - Π_{0})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 1) (3, 2, 0)$
pLRO	p.f.e.c. Large Rathjen Ordinal	$ψ (ω - π - Π_{0})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 2)$
TSSO	Trio Sequence System Ordinal	$ψ (ω - p r o j e c t i o n)$	$B M S (0, 0, 0, 0) (1, 1, 1, 1)$
QSSO	Quardo Sequence System Ordinal	$ψ (ψ_{H} (H^{H^{ω}})) ?$	$B M S (0, 0, 0, 0, 0) (1, 1, 1, 1, 1)$
SHO	Small Hydra Ordinal	$ψ (ψ_{H} (H ↑ ↑ ω)) ?$	$Y (1, 3)$
\OmegaSSO	\Omega Sequence System Ordinal		$Y (1, 3, 4, 2, 5, 8, 10)$
GHO	No-Go Hydra Ordinal		$Y (1, 3, 4, 3)$
SYO	Small Yukito Ordinal		$ω - Y (1, 4)$
MHO	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

[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

[1]

[2]

[3]

@@ 第8行： / 第8行： @@
 ! 缩写 !! 英文全称 !! 常规表示方法(BOCF等) !! BMS/Y
 |-
-| FTO || First Transfinite Ordinal || <math>\omega</math>|| BMS(0)(1)
+| FTO || First Transfinite Ordinal || <math>\omega</math>|| <math>BMS(0)(1)</math>
 |-
-| LAO || Linar Array Ordinal<ref>因为在googology一度经典的线性数阵的极限是它，因此得名</ref>|| <math>\omega^\omega</math>|| BMS(0)(1)(2)
+| LAO || Linar Array Ordinal<ref>因为在googology一度经典的线性数阵的极限是它，因此得名</ref>|| <math>\omega^\omega</math>|| <math>BMS(0)(1)(2)</math>
 |-
-| [[SCO]]|| Small Cantor Ordinal || <math>\varphi(1,0)</math>|| BMS(0,0)(1,1)
+| [[SCO]]|| Small Cantor Ordinal || <math>\varphi(1,0)</math>|| <math>BMS(0,0)(1,1)</math>
 |-
-| [[CO]]|| Cantor Ordinal || <math>\varphi(2,0)</math>|| BMS(0,0)(1,1)(2,1)
+| [[CO]]|| Cantor Ordinal || <math>\varphi(2,0)</math>|| <math>BMS(0,0)(1,1)(2,1)</math>
 |-
-| [[HCO]]|| Hyper Cantor Ordinal || <math>\varphi(\omega,0)</math>|| BMS(0,0)(1,1)(2,1)(3,0)
+| [[HCO]]|| Hyper Cantor Ordinal || <math>\varphi(\omega,0)</math>|| <math>BMS(0,0)(1,1)(2,1)(3,0)</math>
 |-
-| [[FSO]]|| Feferman-Schutte Ordinal || φ(1,0,0) || BMS(0,0)(1,1)(2,1)(3,1)
+| [[FSO]]|| Feferman-Schutte Ordinal || <math>\varphi(1,0,0)</math> || <math>BMS(0,0)(1,1)(2,1)(3,1)</math>
 |-
-| [[SVO]]|| Small Veblen Ordinal || ψ(Ω^Ω^ω) || BMS(0,0)(1,1)(2,1)(3,1)(4,0)
+| [[SVO]]|| Small Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\omega}})</math> || <math>BMS(0,0)(1,1)(2,1)(3,1)(4,0)</math>
 |-
-| [[LVO]]|| Large Veblen Ordinal || ψ(Ω^Ω^Ω) || BMS(0,0)(1,1)(2,1)(3,1)(4,1)
+| [[LVO]]|| Large Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\Omega}})</math> || <math>BMS(0,0)(1,1)(2,1)(3,1)(4,1)</math>
 |-
-| [[BHO]]|| Bachmann-Howard Ordinal || ψ(Ω_2) || BMS(0,0)(1,1)(2,2)
+| [[BHO]]|| Bachmann-Howard Ordinal || <math>\psi(\Omega_{2})</math> || <math>BMS(0,0)(1,1)(2,2)</math>
 |-
-| [[BO]]|| Buchholz's Ordinal || ψ(Ω_ω) || BMS(0,0,0)(1,1,1)
+| [[BO]]|| Buchholz's Ordinal || <math>\psi(\Omega_{\omega})</math> || <math>BMS(0,0,0)(1,1,1)</math>
 |-
-| [[TFBO]]|| Takeuti-Feferman-Buchholz Ordinal || ψ(Ω_(ω+1)) || BMS(0,0,0)(1,1,1)(2,1,0)(3,2,0)
+| [[TFBO]]|| Takeuti-Feferman-Buchholz Ordinal || <math>\psi(\Omega_{\omega+1})</math> || <math>BMS(0,0,0)(1,1,1)(2,1,0)(3,2,0)</math>
 |-
-| [[BIO]]|| Bird's Ordinal<ref>鸟之数阵第四版的极限是它，因此得名</ref>|| ψ(Ω_Ω) || BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)
+| [[BIO]]|| Bird's Ordinal<ref>鸟之数阵第四版的极限是它，因此得名</ref>|| <math>\psi(\Omega_{\Omega})</math> || <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)</math>
 |-
-| [[EBO]]|| Extended Buchholz Ordinal || ψ(I) || BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)
+| [[EBO]]|| Extended Buchholz Ordinal || <math>\psi(I)</math> || <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)</math>
 |-
-| [[JO]]|| Jager's Ordinal || ψ(Ω_(I+1)) || BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)
+| [[JO]]|| Jager's Ordinal || <math>\psi(\Omega_{I+1})</math> || <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)</math>
 |-
-| [[SIO]]|| Small Inaccessible Ordinal || ψ(I_ω) || BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)
+| [[SIO]]|| Small Inaccessible Ordinal || <math>\psi(I_{\omega})</math> || <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)</math>
 |-
-| [[MBO]]|| Mutiply Buchholz Ordinal || ψ(I(ω,0)) || BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)
+| [[MBO]]|| Mutiply Buchholz Ordinal || <math>\psi(I(\omega,0))</math> || <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)</math>
 |-
-| [[SRO]]|| Small Rathjen Ordinal || ψ(Ω_(M+1)) || BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)
+| [[SRO]]|| Small Rathjen Ordinal || <math>\psi(\Omega_{M+1})</math> || <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)</math>
 |-
-| [[RO]]|| Rathjen's Ordinal || ψ(Ω_(K+1)) || BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,0)(5,2,0)
+| [[RO]]|| Rathjen's Ordinal || <math>\psi(\Omega_{K+1})</math> || <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,0)(5,2,0)</math>
 |-
-| [[SSO]]|| Small Stegert Ordinal || ψ(Π_ω) || BMS(0,0,0)(1,1,1)(2,2,0)
+| [[SSO]]|| Small Stegert Ordinal || <math>\psi(\Pi_{\omega})</math> || <math>BMS(0,0,0)(1,1,1)(2,2,0)</math>
 |-
-| [[LSO]]|| Large Stegert Ordinal || ψ(λα.(α×2)-Π<sub>0</sub>) || BMS(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,1,0)(2,0,0)
+| [[LSO]]|| Large Stegert Ordinal || <math>\psi(\lambda\alpha.(\alpha\times 2)-\Pi_{0})</math> || <math>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>|| ψ(Π<sub>1</sub>-λα.(Ω<sub>α+2</sub>)-Π<sub>1</sub>) || BMS(0,0,0)(1,1,1)(2,2,1)
+| [[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>BMS(0,0,0)(1,1,1)(2,2,1)</math>
 |-
-| [[SDO]]|| Small Dropping Ordinal || ψ(λα.(Ω<sub>α+ω</sub>)-Π<sub>0</sub>) || BMS(0,0,0)(1,1,1)(2,2,1)(3,0,0)
+| [[SDO]]|| Small Dropping Ordinal || <math>\psi(\lambda\alpha.(\Omega_{\alpha+\omega})-\Pi_{0})</math> || <math>BMS(0,0,0)(1,1,1)(2,2,1)(3,0,0)</math>
 |-
-| [[LDO]]|| Large Dropping Ordinal || ψ(λα.(OFP aft α)-Π<sub>0</sub>) || BMS(0,0,0)(1,1,1)(2,2,1)(3,2,0)
+| [[LDO]]|| Large Dropping Ordinal || <math>\psi(\lambda\alpha.(\mathrm{OFP}\ \mathrm{aft}\ \alpha)-\Pi_{0})</math> || <math>BMS(0,0,0)(1,1,1)(2,2,1)(3,2,0)</math>
 |-
-| [[pLRO]]|| p.f.e.c. Large Rathjen Ordinal || ψ(ω-π-Π<sub>0</sub>) || BMS(0,0,0)(1,1,1)(2,2,2)
+| [[pLRO]]|| p.f.e.c. Large Rathjen Ordinal || <math>\psi(\omega-\pi-\Pi_{0})</math> || <math>BMS(0,0,0)(1,1,1)(2,2,2)</math>
 |-
-| [[TSSO]]|| Trio Sequence System Ordinal || ψ(ω-projection) || BMS(0,0,0,0)(1,1,1,1)
+| [[TSSO]]|| Trio Sequence System Ordinal || <math>\psi(\omega-projection)</math> || <math>BMS(0,0,0,0)(1,1,1,1)</math>
 |-
-| [[QSSO]]|| Quardo Sequence System Ordinal || ψ(ψ<sub>H</sub>(H^H^ω))？ || BMS(0,0,0,0,0)(1,1,1,1,1)
+| [[QSSO]]|| Quardo Sequence System Ordinal || <math>\psi(\psi_{H}(H^{H^{\omega}}))?</math> || <math>BMS(0,0,0,0,0)(1,1,1,1,1)</math>
 |-
-| [[SHO]]|| Small Hydra Ordinal || ψ(ψ<sub>H</sub>(H^^ω))？ || Y(1,3)
+| [[SHO]]|| Small Hydra Ordinal || <math>\psi(\psi_{H}(H\uparrow\uparrow\omega))?</math> || <math>Y(1,3)</math>
 |-
-| [[ΩSSO]]|| Ω Sequence System Ordinal || || Y(1,3,4,2,5,8,10)
+| [[\OmegaSSO]]|| \Omega Sequence System Ordinal || || <math>Y(1,3,4,2,5,8,10)</math>
 |-
-| [[GHO]]|| No-Go Hydra Ordinal || || Y(1,3,4,3)
+| [[GHO]]|| No-Go Hydra Ordinal || || <math>Y(1,3,4,3)</math>
 |-
-| [[SYO]]|| Small Yukito Ordinal || || ω-Y(1,4)
+| [[SYO]]|| Small Yukito Ordinal || || <math>\omega-Y(1,4)</math>
 |-
-| [[MHO]]|| Medium Hydra Ordinal || || ω-Y极限
+| [[MHO]]|| Medium Hydra Ordinal || || <math>\omega-Y</math> 极限
 |-
-| [[CKO]]|| Church-Kleene Ordinal || ω<sub>1</sub><sup>CK</sup>||
+| [[CKO]]|| Church-Kleene Ordinal || <math>\omega_{1}^{CK}</math>
 |-
-| FUO || First Uncountable Ordinal || ω<sub>1</sub> ||
+| FUO || First Uncountable Ordinal || <math>\omega_{1}</math> ||
 |}