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。

序数表

缩写	英文全称	常规表示方法(BOCF等)	BMS/Y
FTO	First Transfinite Ordinal	$\omega$	$BMS(0)(1)$
LAO	Linar Array Ordinal	${\omega }^{\omega }$	$BMS(0)(1)(2)$
SCO	Small Cantor Ordinal	$\phi (1,0)={\epsilon }_0=\psi (\mathrm{\Omega })$	$BMS(0,0)(1,1)$
CO	Cantor Ordinal	$\phi (2,0)={\zeta }_0=\psi ({\mathrm{\Omega }}^2)$	$BMS(0,0)(1,1)(2,1)$
HCO	Hyper Cantor Ordinal	$\phi (\omega ,0)=\psi ({\mathrm{\Omega }}^{\omega })$	$BMS(0,0)(1,1)(2,1)(3,0)$
FSO	Feferman-Schutte Ordinal	$\phi (1,0,0)={\mathrm{\Gamma }}_0=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})$	$BMS(0,0)(1,1)(2,1)(3,1)$
SVO	Small Veblen Ordinal	$\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$	$BMS(0,0)(1,1)(2,1)(3,1)(4,0)$
LVO	Large Veblen Ordinal	$\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$	$BMS(0,0)(1,1)(2,1)(3,1)(4,1)$
BHO	Bachmann-Howard Ordinal	$\psi ({\mathrm{\Omega }}_2)$	$BMS(0,0)(1,1)(2,2)$
BO	Buchholz's Ordinal	$\psi ({\mathrm{\Omega }}_{\omega })$	$BMS(0,0,0)(1,1,1)$
TFBO	Takeuti-Feferman-Buchholz Ordinal	$\psi ({\mathrm{\Omega }}_{\omega +1})$	$BMS(0,0,0)(1,1,1)(2,1,0)(3,2,0)$
BIO	Bird's Ordinal	$\psi ({\mathrm{\Omega }}_{\mathrm{\Omega }})$	$BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)$
EBO	Extended Buchholz Ordinal	$\psi (I)$	$BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)$
JO	Jager's Ordinal	$\psi ({\mathrm{\Omega }}_{I+1})$	$BMS(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)$
SIO	Small Inaccessible Ordinal	$\psi (I_{\omega })$	$BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)$
MBO	Mutiply Buchholz Ordinal	$\psi (I(\omega ,0))=\psi (M^{\omega })$	$BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)$
SRO	Small Rathjen Ordinal	$\psi ({\epsilon }_{M+1})$	$BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)$
SMO	Small Mahlo Ordinal	$\psi (M_{\omega })$	$BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)$
RO	Rathjen's Ordinal	$\psi ({\epsilon }_{K+1})$	$BMS(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,0)(5,2,0)$
SSO	Small Stegert Ordinal	$\psi ({\mathrm{\Pi }}_{\omega })=\psi (a_2)$	$BMS(0,0,0)(1,1,1)(2,2,0)$
LSO	Large Stegert Ordinal	$\psi (\lambda \alpha .(\alpha \times 2)-{\mathrm{\Pi }}_0)=\psi (a_2^a)$	$BMS(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,1,0)(2,0,0)$
BGO	TSS 1st Back Gear Ordinal	$\psi ({\mathrm{\Pi }}_1-\lambda \alpha .({\mathrm{\Omega }}_{\alpha +2})-{\mathrm{\Pi }}_1)$	$BMS(0,0,0)(1,1,1)(2,2,1)$
SDO	Small Dropping Ordinal	$\psi (\lambda \alpha .({\mathrm{\Omega }}_{\alpha +\omega })-{\mathrm{\Pi }}_0)$	$BMS(0,0,0)(1,1,1)(2,2,1)(3,0,0)$
LDO	Large Dropping Ordinal	$\psi (\lambda \alpha .(\mathrm{O}\mathrm{F}\mathrm{P}\mathrm{a}\mathrm{f}\mathrm{t}\alpha )-{\mathrm{\Pi }}_0)$	$BMS(0,0,0)(1,1,1)(2,2,1)(3,2,0)$
pLRO	p.f.e.c. Large Rathjen Ordinal	$\psi (\omega -\pi -{\mathrm{\Pi }}_0)$	$BMS(0,0,0)(1,1,1)(2,2,2)=\psi (a_{\omega })$
TSSO	Trio Sequence System Ordinal	$\psi (\omega -projection)=\psi (\sigma S\times S)=\psi (H^{\omega })$	$BMS(0,0,0,0)(1,1,1,1)$
QSSO	Quardo Sequence System Ordinal	$\psi ({\psi }_H(H^{H^{\omega }}))?$	$BMS(0,0,0,0,0)(1,1,1,1,1)$
SHO/BMO	Small Hydra Ordinal	$\psi ({\psi }_H({\epsilon }_{H+1}))?$	$Y(1,3)=BMS\text{极限}$
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		$\omega -Y(1,4)$
MHO/ωYO	Medium Hydra Ordinal		$\omega -Y$ 极限
CKO	Church-Kleene Ordinal	${\omega }_1^{CK}$
FUO	First Uncountable Ordinal	${\omega }_1$