序数表：修订间差异

2025年7月4日 (五) 00:46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	$φ (2, 0) = ζ_{0} = ψ (Ω^{2})$	$B M S (0, 0) (1, 1) (2, 1)$
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	$φ (1, 0, 0) = Γ_{0} = ψ (Ω^{Ω})$	$B M S (0, 0) (1, 1) (2, 1) (3, 1)$
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	$ψ (I (ω, 0)) = ψ (M^{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 0, 0)$
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	$ψ (M_{ω})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (3, 1, 1)$
RO	Rathjen's Ordinal	$ψ (ε_{K + 1})$	$B M S (0, 0, 0) (1, 1, 1) (2, 1, 1) (3, 1, 1) (4, 1, 0) (5, 2, 0)$
SSO	Small Stegert Ordinal	$ψ (Π_{ω}) = ψ (a_{2})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0)$
LSO	Large Stegert Ordinal	$ψ (λ α . (α \times 2) - Π_{0}) = ψ (a_{2}^{a})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 0) (3, 2, 0) (4, 1, 0) (2, 0, 0)$
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	$ψ (ω - π - Π_{0})$	$B M S (0, 0, 0) (1, 1, 1) (2, 2, 2) = ψ (a_{ω})$
TSSO	Trio Sequence System Ordinal	$ψ (ω - p r o j e c t i o n) = ψ (σ S \times S) = ψ (H^{ω})$	$B M S (0, 0, 0, 0) (1, 1, 1, 1)$
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/BMO	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		$Y (1, 3, 4, 3)$
SYO	Small Yukito Ordinal		$ω - Y (1, 4)$
MHO/ωYO	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12行： / 第12行： @@
 | 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>|| <math>BMS(0,0)(1,1)</math>
+| [[SCO]]|| Small Cantor Ordinal || <math>\varphi(1,0)=\varepsilon_0=\psi(\Omega)</math>|| <math>BMS(0,0)(1,1)</math>
 |-
-| [[CO]]|| Cantor Ordinal || <math>\varphi(2,0)</math>|| <math>BMS(0,0)(1,1)(2,1)</math>
+| [[CO]]|| Cantor Ordinal || <math>\varphi(2,0)=\zeta_0=\psi(\Omega^2)</math>|| <math>BMS(0,0)(1,1)(2,1)</math>
 |-
-| [[HCO]]|| Hyper Cantor Ordinal || <math>\varphi(\omega,0)</math>|| <math>BMS(0,0)(1,1)(2,1)(3,0)</math>
+| [[HCO]]|| Hyper Cantor Ordinal || <math>\varphi(\omega,0)=\psi(\Omega^\omega)</math>|| <math>BMS(0,0)(1,1)(2,1)(3,0)</math>
 |-
-| [[FSO]]|| Feferman-Schutte Ordinal || <math>\varphi(1,0,0)</math>|| <math>BMS(0,0)(1,1)(2,1)(3,1)</math>
+| [[FSO]]|| Feferman-Schutte Ordinal || <math>\varphi(1,0,0)=\Gamma_0=\psi(\Omega^\Omega)</math>|| <math>BMS(0,0)(1,1)(2,1)(3,1)</math>
 |-
 | [[SVO]]|| Small Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\omega}})</math>|| <math>BMS(0,0)(1,1)(2,1)(3,1)(4,0)</math>
@@ 第38行： / 第38行： @@
 | [[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 || <math>\psi(I(\omega,0))</math>|| <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)</math>
+| [[MBO]]|| Mutiply Buchholz Ordinal || <math>\psi(I(\omega,0))=\psi(M^\omega)</math>|| <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)</math>
 |-
-| [[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>
+| [[SRO]]|| Small Rathjen Ordinal || <math>\psi(\varepsilon_{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 || <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>
+| [[SMO]]|| Small Mahlo Ordinal || <math>\psi(M_\omega)</math>|| <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)</math>
 |-
-| [[SSO]]|| Small Stegert Ordinal || <math>\psi(\Pi_{\omega})</math>|| <math>BMS(0,0,0)(1,1,1)(2,2,0)</math>
+| [[RO]]|| Rathjen's Ordinal || <math>\psi(\varepsilon_{K+1})</math>|| <math>BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,0)(5,2,0)</math>
 |-
-| [[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>
+| [[SSO]]|| Small Stegert Ordinal || <math>\psi(\Pi_{\omega})=\psi(a_2)</math>|| <math>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>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>BMS(0,0,0)(1,1,1)(2,2,1)</math>
@@ 第54行： / 第56行： @@
 | [[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 || <math>\psi(\omega-\pi-\Pi_{0})</math>|| <math>BMS(0,0,0)(1,1,1)(2,2,2)</math>
+| [[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)=\psi(a_\omega)</math>
 |-
-| [[TSSO]]|| Trio Sequence System Ordinal || <math>\psi(\omega-projection)</math>|| <math>BMS(0,0,0,0)(1,1,1,1)</math>
+| [[TSSO]]|| Trio Sequence System Ordinal || <math>\psi(\omega-projection)=\psi(\sigma S\times S)=\psi(H^\omega)</math>|| <math>BMS(0,0,0,0)(1,1,1,1)</math>
 |-
 | [[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/BMO|| Small Hydra Ordinal || <math>\psi(\psi_{H}(H\uparrow\uparrow\omega))?</math>|| <math>Y(1,3)</math>
+| SHO/BMO|| Small Hydra Ordinal || <math>\psi(\psi_{H}(\varepsilon_{H+1}))?</math>|| <math>Y(1,3)=BMS\text{极限}</math>
 |-
 | [[ΩSSO]]|| \Omega Sequence System Ordinal || || <math>Y(1,3,4,2,5,8,10)</math>
@@ 第72行： / 第74行： @@
 | [[CKO]]|| Church-Kleene Ordinal || <math>\omega_{1}^{CK}</math>
 |-
-| FUO || First Uncountable Ordinal || <math>\omega_{1}</math>||
+| [[FUO]] || First Uncountable Ordinal || <math>\omega_{1}</math>||
 |}