查看“︁序数表”︁的源代码

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 ==
{| class="wikitable"
|+ 
|-
! 缩写 !! 英文全称 !! 常规表示方法(BOCF等) !! BMS/Y
|-
| FTO || First Transfinite Ordinal || <math>\omega</math>|| <math>\mathrm{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>|| <math>\mathrm{BMS}(0)(1)(2)</math>
|-
| [[SCO]]|| Small Cantor Ordinal || <math>\varphi(1,0)=\varepsilon_0=\psi(\Omega)</math>|| <math>\mathrm{BMS}(0,0)(1,1)</math>
|-
| [[CO]]|| Cantor Ordinal || <math>\varphi(2,0)=\zeta_0=\psi(\Omega^2)</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)</math>
|-
|LCO
|Large Cantor Ordinal
|<math>\varphi(3,0)=\eta_0=\psi(\Omega^3)</math>
|<math>\mathrm{BMS}(0,0)(1,1)(2,1)(2,1)</math>
|-
| [[HCO]]|| Hyper Cantor Ordinal || <math>\varphi(\omega,0)=\psi(\Omega^\omega)</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,0)</math>
|-
| [[FSO]]|| Feferman-Schutte Ordinal || <math>\varphi(1,0,0)=\Gamma_0=\psi(\Omega^\Omega)</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)</math>
|-
|ACO
|Ackermann Ordinal
|<math>\varphi(1,0,0,0)</math>
|<math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(3,1)</math>
|-
| [[SVO]]|| Small Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\omega}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,0)</math>
|-
| [[LVO]]|| Large Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\Omega}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,1)</math>
|-
| [[ESVO]]|| Extended Small Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\Omega^\omega}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,1)(5,0)</math>
|-
| [[ELVO]]|| Large Veblen Ordinal || <math>\psi(\Omega^{\Omega^{\Omega^{\Omega}}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,1)(5,1)</math>
|-
| [[BHO]]|| Bachmann-Howard Ordinal || <math>\psi(\Omega_{2})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,2)</math>
|-
| [[BO]]|| Buchholz's Ordinal || <math>\psi(\Omega_{\omega})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)</math>
|-
| [[TFBO]]|| Takeuti-Feferman-Buchholz Ordinal || <math>\psi(\Omega_{\omega+1})</math>|| <math>\mathrm{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>|| <math>\psi(\Omega_{\Omega})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,0)</math>
|-
| [[EBO]]|| Extended Buchholz Ordinal || <math>\psi(I)</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)</math>
|-
| [[JO]]|| Jager's Ordinal || <math>\psi(\Omega_{I+1})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)</math>
|-
| [[SIO]]|| Small Inaccessible Ordinal || <math>\psi(I_{\omega})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)</math>
|-
| [[MBO]]|| Mutiply Buchholz Ordinal || <math>\psi(I(\omega,0))=\psi(M^\omega)</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)</math>
|-
|TBO
|Transfinitary Buchholz's Ordinal
|\( \psi(I(1@(1,0))) \) 
|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)</math>
|-
| [[SRO]]|| Small Rathjen Ordinal || <math>\psi(\varepsilon_{M+1})</math>|| <math>\mathrm{BMS}(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)</math>
|-
| [[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>
|-
|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>
|-
|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>
|-
| [[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>
|-
|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>
|-
|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>
|-
| [[ΩSSO]]|| \Omega Sequence System Ordinal || || <math>Y(1,3,4,2,5,8,10)</math>
|-
| [[GHO]]|| No-Go Hydra Ordinal<ref>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</ref>|| || <math>Y(1,3,4,3)</math>
|-
| [[SYO]]|| Small Yukito Ordinal || || <math>\omega-Y(1,4)</math>
|-
| MHO/ωYO<ref name=":0" />|| Medium Hydra Ordinal || || <math>\omega-Y</math> 极限
|-
| [[CKO]]|| Church-Kleene Ordinal || <math>\omega_{1}^{\rm CK}</math>
|-
| [[FUO]] || First Uncountable Ordinal || <math>\omega_{1}</math>||
|}