序数表

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。

常用序数表

缩写	英文全称	BMS / Y	其他表示方法（OCF 等）
FTO	First Transfinite Ordinal	$(0)(1)$	$\omega$
LAO	Linear Array Ordinal	$(0)(1)(2)$	${\omega }^{\omega }$
SCO	Small Cantor Ordinal	$(0)(1,1)$	$\phi (1,0)={\epsilon }_0=\psi (\mathrm{\Omega })$
CO	Cantor Ordinal	$(0)(1,1)(2,1)$	$\phi (2,0)={\zeta }_0=\psi ({\mathrm{\Omega }}^2)$
LCO	Large Cantor Ordinal	$(0)(1,1)(2,1)(2,1)$	$\phi (3,0)={\eta }_0=\psi ({\mathrm{\Omega }}^3)$
HCO	Hyper Cantor Ordinal	$(0)(1,1)(2,1)(3)$	$\phi (\omega ,0)=\psi ({\mathrm{\Omega }}^{\omega })$
FSO	Feferman-Schütte Ordinal	$(0)(1,1)(2,1)(3,1)$	$\phi (1,0,0)={\mathrm{\Gamma }}_0=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})$
SVO	Small Veblen Ordinal	$(0)(1,1)(2,1)(3,1)(4)$	$\phi (1$@$\omega )=\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$
LVO	Large Veblen Ordinal	$(0)(1,1)(2,1)(3,1)(4,1)$	$\phi (1$@$(1,0))=\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
BHO	Bachmann-Howard Ordinal	$(0)(1,1)(2,2)$	$\psi ({\mathrm{\Omega }}_2)$
BO	Buchholz's Ordinal	$(0)(1,1,1)$	$\psi ({\mathrm{\Omega }}_{\omega })$
TFBO	Takeuti-Feferman-Buchholz Ordinal	$(0)(1,1,1)(2,1)(3,2)$	$\psi ({\mathrm{\Omega }}_{\omega +1})$
BIO	Bird's Ordinal	$(0)(1,1,1)(2,1,1)(3,1)$	$\psi ({\mathrm{\Omega }}_{\mathrm{\Omega }})$
EBO	Extended Buchholz Ordinal	$(0)(1,1,1)(2,1,1)(3,1)(2)$	$\psi (I)$
JO	Jager's Ordinal	$(0)(1,1,1)(2,1,1)(3,1)(4,2)$	$\psi ({\mathrm{\Omega }}_{I+1})$
SIO	Small Inaccessible Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)$	$\psi (I_{\omega })$
MBO	Mutiply Buchholz Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(3)$	$\psi (I(\omega ,0))=\psi (M^{\omega })$
TBO	Transfinitary Buchholz's Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2)$	$\psi (I(1,0,0))=\psi (M^M)$
SRO	Small Rathjen Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,2)$	$\psi ({\mathrm{\Omega }}_{M+1})$
SMO	Small Mahlo Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)$	$\psi (M_{\omega })$
SNO	Small Nonconvertible (N) Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1,1)$	$\psi (N_{\omega })$
RO	Rathjen's Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2)$	$\psi ({\mathrm{\Omega }}_{K+1})$
SKO	Small Weakly Compact (K) Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)$	$\psi (K_{\omega })$
SSO	Small Stegert Ordinal	$(0)(1,1,1)(2,2)$	$\psi (psd.{\mathrm{\Pi }}_{\omega })=\psi (a_2)$
LSO	Large Stegert Ordinal	$(0)(1,1,1)(2,2)(3,2)(4,1)(2)$	$\psi (\lambda \alpha .(\alpha \times 2)-{\mathrm{\Pi }}_0)=\psi (a_2^a)$
BGO	TSS 1st Back Gear Ordinal	$(0)(1,1,1)(2,2,1)$	$\psi (\lambda \alpha .({\mathrm{\Omega }}_{\alpha +2})-{\mathrm{\Pi }}_1)=\psi ({\mathrm{\Omega }}_{a_2+1}+{\psi }_{a_2}({\mathrm{\Omega }}_{a_2+1})\times \omega )$
SDO	Small Dropping Ordinal	$(0)(1,1,1)(2,2,1)(3)$	$\psi (\lambda \alpha .({\mathrm{\Omega }}_{\alpha +\omega })-{\mathrm{\Pi }}_0)=\psi ({\mathrm{\Omega }}_{a_2+1}\times \omega )$
LDO	Large Dropping Ordinal	$(0)(1,1,1)(2,2,1)(3,2)$	$\psi (\lambda \alpha .{\psi }_{I_{\alpha +1}}(I_{\alpha +1})-{\mathrm{\Pi }}_0)=\psi ({\mathrm{\Omega }}_{a_2+1}\times a_2)$
DSO	Doubly +1 Stable Ordinal	$(0)(1,1,1)(2,2,1)(3,3)$	$\psi (\lambda \alpha .(\lambda \beta .\beta +1-{\mathrm{\Pi }}_0)-{\mathrm{\Pi }}_0)=\psi (a_3)$
pfec LRO	pfec Large Rathjen Ordinal	$(0)(1,1,1)(2,2,2)$	$\psi (pfec.\omega -\pi -{\mathrm{\Pi }}_0)=\psi (a_{\omega })$
LRO	Large Rathjen Ordinal	$⩽(0)(1,1,1,1)?$	$F\cap {\omega }_1^{\text{CK}},\theta =\omega$
TSSO	Trio Sequence System Ordinal	$(0)(1,1,1,1)$	$\psi (\omega -\text{proj.})=\psi (\sigma S\times \omega )=\psi (H^{\omega })$
QSSO	Quardo Sequence System Ordinal	$(0)(1,1,1,1,1)$	$\psi ({\psi }_H(H^{H^{\omega }}))$
TCAO	Trio Comprehension Axiom Ordinal	$⩾(0)(1,1,1,1,1)$	$\text{PTO}({\mathrm{\Pi }}_3^1-C{A}_0)$
QiSSO	Quinto Sequence System Ordinal	$(0)(1,1,1,1,1,1)$	$\psi ({\psi }_H(H^{H^{H^{\omega }}}))$
SHO / BMO	Small Hydra Ordinal / Bashicu Matrix Ordinal	$Y(1,3)=\lim (\mathrm{B}\mathrm{M}\mathrm{S}\mathrm{)}$	$\psi ({\psi }_H({\epsilon }_{H+1}))$
βO	Beta Universe Ordinal	$⩾Y(1,3)$	$\mathrm{P}\mathrm{T}\mathrm{O}\mathrm{(}{\mathrm{Z}}_{\mathrm{2}}\mathrm{)}$
ΩSSO	Ω Sequence System Ordinal	$Y(1,3,4,2,5,8,10)$	$\psi ({\psi }_H(\phi (\mathrm{\Omega },H+1)))?$
GHO	No-Go Hydra Ordinal	$Y(1,3,4,3)$
SYO	Small Yukito Ordinal	$\lim (1-Y)=\omega -Y(1,4)$
MHO / ωYO	Medium Hydra Ordinal / ω-Y sequence Ordinal	$\lim (\omega -Y)$
CKO	Church-Kleene Ordinal		${\omega }_1^{\mathrm{C}\mathrm{K}}$
FUO	First Uncountable Ordinal		${\omega }_1$

不常用序数、旧称与别称表

缩写	英文全称	BMS / Y	其他表示方法（OCF 等）
SMDO	Small Multidimensional Ordinal	$(0)(1)(2)$	${\omega }^{\omega }$
SHO (旧)	Small Hydra Ordinal	$(0)(1,1)$	$\phi (1,0)={\epsilon }_0=\psi (\mathrm{\Omega })$
ACO	Ackermann Ordinal	$(0)(1,1)(2,1)(3,1)(3,1)$	$\phi (1,0,0,0)=\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})$
ESVO	Extended Small Veblen Ordinal	$(0)(1,1)(2,1)(3,1)(4,1)(5)$	$\psi \left({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}}\right)$
ELVO	Extended Large Veblen Ordinal	$(0)(1,1)(2,1)(3,1)(4,1)(5,1)$	$\psi \left({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}}}\right)$
FBLO	First BMS Lifting Ordinal	$(0)(1,1,1)(2,1)(1,1,1)$	$\psi ({\mathrm{\Omega }}_{\omega }^2)$
BTBO	Brouwer Tree Barrier Ordinal	$(0)(1,1,1)(2,1)(3,2)$	$\psi ({\mathrm{\Omega }}_{\omega +1})$
SRO (国际)	Small Rathjen Ordinal	$(0)(1,1,1)(2,1,1)(3,1)(4,2)$	$\psi ({\mathrm{\Omega }}_{I+1})$
RO (国际)	Rathjen's Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,2)$	$\psi ({\mathrm{\Omega }}_{M+1})$
LRO (国际)	Large Rathjen Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2)$	$\psi ({\mathrm{\Omega }}_{K+1})$
DO	Duchhart's Ordinal	$(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(6,2)$	$\psi ({\mathrm{\Pi }}_2\mathrm{a}\mathrm{f}\mathrm{t}{\mathrm{\Pi }}_{\mathrm{4}}\mathrm{)}$
BGO (国际)	TSS 1st Back Gear Ordinal	$(0)(1,1,1)(2,2)$	$\psi (psd.{\mathrm{\Pi }}_{\omega })=\psi (a_2)$
APO	Admissible-parameter free effective cardinal Ordinal	$(0)(1,1,1)(2,2)(3,2)(4,1,1)$	$\psi (\lambda \alpha .({\mathrm{\Omega }}_{\alpha +1})-{\mathrm{\Pi }}_1)=\psi (a_2^{{\mathrm{\Omega }}_{a+1}}+{\psi }_{a_2}(a_2^{{\mathrm{\Omega }}_{a+1}})\times \omega )$
LDO (旧)	Large Dropping Ordinal	$(0)(1,1,1)(2,2,1)(3,1)(2)$	$\psi (\lambda \alpha .({\mathrm{\Omega }}_{\alpha \times 2})-{\mathrm{\Pi }}_0)$
EDO	Extended Dropping Ordinal	$(0)(1,1,1)(2,2,1)(3,2)$	$\psi (\lambda \alpha .{\psi }_{I_{\alpha +1}}(I_{\alpha +1})-{\mathrm{\Pi }}_0)=\psi ({\mathrm{\Omega }}_{a_2+1}\times a_2)$
TSO	Triply +1 Stable Ordinal	$(0)(1,1,1)(2,2,1)(3,3,1)(4,4)$	$\psi (\lambda \alpha .(\lambda \beta .(\lambda \gamma .\gamma +1-{\mathrm{\Pi }}_0)-{\mathrm{\Pi }}_0)-{\mathrm{\Pi }}_0)=\psi (a_4)$
SBO	Small Bashicu Ordinal	$(0)(1,1,1)(2,2,2)$	$\psi (pfec.\omega -\pi -{\mathrm{\Pi }}_0)=\psi (a_{\omega })$
pfec M2O	pfec min Σ2 Ordinal	$(0)(1,1,1)(2,2,2)(3,2,2)(4,2,2)(4,2,1)?$	$\psi (pfec.\min (a{\prec }_{{\mathrm{\Sigma }}_1}b{\prec }_{{\mathrm{\Sigma }}_2}c))$
SOSO	Second Order Stable Ordinal		$\psi (1-o-{\mathrm{\Sigma }}_2-\text{stb.})$
SSPO	Small Simple Projection Ordinal	$(0)(1,1,1,1)$	$\psi (\omega -\text{proj.})=\psi (\sigma S\times \omega )=\psi (H^{\omega })$
LSPO	Large Simple Projection Ordinal	$(0)(1,1,1,1)(2,1,1,1)(3,1)(2)$	$\psi (\min \alpha \text{ is }\alpha -\text{proj.})=\psi (\sigma S\times S)$
Q0.5BGO	QSS 0.5th Back Gear Ordinal	$(0)(1,1,1,1)(2,2)$
Q1BGO	QSS 1st Back Gear Ordinal	$(0)(1,1,1,1)(2,2,2)$
ESPO	Extend Simple Projection Ordinal	$(0)(1,1,1,1)(2,2,2,1)$
EGO	Eveog's Ordinal	$(0)(1,1,1,1)(2,2,2,1)(3,2,1)(4)$	$\psi ({\psi }_{\sigma }({\sigma }_{\omega }))$
BOBO	Big Omega Back Ordinal	$(0)(1,1,1,1)(2,2,2,2)$	$\psi ((\omega ,0)-\text{proj.})=\psi ({\psi }_H(H^{H\omega }))$
MHO (旧)	Medium Hydra Ordinal	$Y(1,3)$	$\lim (\mathrm{B}\mathrm{M}\mathrm{S})=\lim (0-Y)$
LRPO	Large Right Projection Ordinal	$Y(1,3,4,2,5,8,10,4,9,14,17,10)$	$\psi ({\psi }_H(\phi (H,1)))?$
DCO	Difference Catching Ordinal	$Y(1,3,5)$
LHO	Large Hydra Ordinal	$\lim (\omega -Y)$
GYO	Grand Y-Sequence Ordinal	$\lim (X-Y)$

本表部分取自 Worldly Sheet

DNAO

DNAO（Deoxyribo Nucleic Acid Ordinal / Disgusting Nonsense Annoyance Ordinal）

定义：

(0)(1,1,1)(2,2,2)(3,3,3)(3,3,0)(4,4,1)(5,5,2)(6,6,2)(7,7,0)(8,8,1)(9,9,2)(10,9,2)(11,9,0)(12,10,1)(13,11,2)(13,11,2)(13,11,1)(14,12,2)(14,11,1)(15,12,2)(15,11,1)(16,12,0)(17,13,1)(18,14,2)(18,14,2)(18,14,1)(19,15,2)(19,14,1)(20,15,2)(20,14,1)(21,15,0)(22,16,1)(23,17,2)(23,17,2)(23,17,1)(24,18,2)(24,17,1)(25,18,2)(25,17,0)(26,18,1)(27,19,2)(27,19,2)(27,19,1)(28,20,2)(28,19,1)(29,20,2)(29,19,0)(30,20,1)(31,21,2)(31,21,2)(31,21,1)(32,22,2)(32,21,1)(33,22,2)(33,21,0)(34,22,1)(35,23,2)(35,23,2)(35,23,1)(36,24,2)(36,23,1)(37,24,2)(37,23,0)(38,24,1)(39,25,2)(40,25,2)(40,25,1)(41,26,2)(41,22,1)(42,23,2)(42,23,2)(42,23,1)(43,24,2)(43,23,1)(44,24,2)(44,23,0)(45,24,1)(46,25,2)(47,25,2)(47,25,1)(48,26,1)(49,27,0)(50,28,1)(51,29,2)(52,29,2)(52,29,1)(53,30,0)(54,31,1)(55,32,2)(56,32,2)(56,32,0)(57,33,1)(58,34,2)(59,34,2)(59,34,0)(60,35,1)(61,36,2)(62,36,2)(62,36,0)(63,37,1)(64,38,2)(65,38,2)(65,38,0)(66,39,1)(67,40,2)(68,40,2)(68,40,0)(69,41,1)(70,42,2)(71,42,2)(71,42,0)(72,43,1)(73,44,0)(74,45,1)(75,44,0)(76,45,1)(77,46,0)(78,47,0)(79,44,0)(80,45,1)(81,46,0)(82,47,0)(83,44,0)(84,45,1)(85,46,0)(86,47,0)(87,44,0)(88,45,1)(89,46,0)(90,47,0)(91,44,0)(92,45,1)(93,46,0)(94,47,0)(95,44,0)(96,45,1)(96,45,1)(96,45,1)(96,45,0)(97,46,0)(98,47,0)(99,48,0)(100,47,0)(101,48,0)(102,47,0)(103,48,0)(104,47,0)(105,48,0)(106,47,0)(107,48,0)(108,45,0)(109,46,0)(110,47,0)(111,47,0)(111,47,0)(111,47,0)(111,47,0)(111,47,0)(111,47,0)(111,46,0)(112,47,0)(113,47,0)(113,47,0)(113,47,0)(113,47,0)(113,47,0)(113,47,0)(113,46,0)(114,47,0)(115,46,0)(116,47,0)(117,46,0)(118,47,0)(119,46,0)(120,45,0)(121,46,0)(122,47,0)(123,47,0)(123,47,0)(123,47,0)(123,47,0)(123,47,0)(123,47,0)(123,46,0)(124,47,0)(125,47,0)(125,47,0)(125,47,0)(125,47,0)(125,47,0)(125,47,0)(125,46,0)(126,47,0)(127,46,0)(128,47,0)(129,46,0)(130,47,0)(131,46,0)(132,45,0)(133,46,0)(134,47,0)(135,47,0)(135,47,0)(135,47,0)(135,47,0)(135,47,0)(135,47,0)(135,46,0)(136,47,0)(137,47,0)(137,47,0)(137,47,0)(137,47,0)(137,47,0)(137,47,0)(137,46,0)(138,47,0)(139,46,0)(140,47,0)(141,46,0)(142,47,0)(143,46,0)(144,45,0)(145,46,0)(146,47,0)(147,47,0)(147,47,0)(147,47,0)(147,47,0)(147,47,0)(147,47,0)(147,46,0)(147,46,0)(147,46,0)(147,46,0)(147,46,0)(147,45,0)(148,46,0)(148,45,0)(149,46,0)(149,45,0)(150,46,0)(150,45,0)(151,45,0)(151,45,0)(150,45,0)(151,45,0)(150,45,0)(151,45,0)(148,45,0)(149,46,0)(149,45,0)(150,46,0)(150,45,0)(151,45,0)(151,45,0)(150,45,0)(151,45,0)(150,45,0)(151,45,0)(148,45,0)(149,46,0)(149,45,0)(150,46,0)(150,45,0)(151,45,0)(151,45,0)(150,45,0)(151,45,0)(150,45,0)(151,45,0)(148,45,0)(149,45,0)(149,45,0)(148,45,0)(149,45,0)(149,45,0)(148,45,0)(149,45,0)(147,45,0)(148,46,0)(148,45,0)(149,46,0)(149,45,0)(150,46,0)(150,45,0)(151,45,0)(151,45,0)(150,45,0)(151,45,0)(148,45,0)(149,46,0)(149,45,0)(150,45,0)(150,45,0)(149,45,0)(150,45,0)(149,45,0)(150,45,0)(149,45,0)(149,45,0)(149,45,0)(146,47,0)(147,47,0)(147,47,0)(147,47,0)(147,47,0)(147,47,0)(147,47,0)(147,46,0)(147,46,0)(147,46,0)(147,46,0)(147,45,0)(148,46,0)(149,47,0)(150,47,0)(150,47,0)(150,47,0)(150,47,0)(150,47,0)(150,47,0)(150,46,0)(150,46,0)(150,46,0)(150,46,0)(150,46,0)(150,45,0)(151,46,0)(151,45,0)(152,46,0)(152,45,0)(153,46,0)(153,45,0)(154,45,0)(154,45,0)(153,45,0)(154,45,0)(153,45,0)(154,45,0)(151,45,0)(152,46,0)(152,45,0)(153,46,0)(153,45,0)(154,45,0)(154,45,0)(153,45,0)(154,45,0)(153,45,0)(154,45,0)(151,45,0)(152,46,0)(152,45,0)(153,46,0)(153,45,0)(154,45,0)(154,45,0)(153,45,0)(154,45,0)(153,45,0)(154,45,0)(151,45,0)(152,45,0)(152,45,0)(151,45,0)(152,45,0)(152,45,0)(151,45,0)(152,45,0)(150,45,0)(151,46,0)(151,45,0)(152,46,0)(152,45,0)(153,46,0)(153,45,0)(154,45,0)(154,45,0)(153,45,0)(154,45,0)(151,45,0)(152,46,0)(152,45,0)(153,45,0)(153,45,0)(152,45,0)(153,45,0)(152,45,0)(153,45,0)(152,45,0)(152,45,0)(152,45,0)(149,47,0)(150,47,0)(150,47,0)(150,47,0)(150,47,0)(150,47,0)(150,47,0)(150,46,0)(150,46,0)(150,46,0)(150,46,0)(150,45,0)(151,46,0)(152,47,0)(152,47,0)(152,47,0)(151,45,0)(152,46,0)(153,47,0)(150,45,0)(151,46,0)(152,47,0)(152,47,0)(152,47,0)(151,45,0)(152,46,0)(153,47,0)(150,45,0)(151,46,0)(152,47,0)(150,45,0)(151,46,0)(152,47,0)(150,45,0)(151,45,0)(152,45,0)(152,45,0)(151,45,0)(152,45,0)(152,45,0)(150,45,0)(149,45,0)