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序数表

2025-08-15T06:52:34Z

Optimism：

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-Schütte 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)=\psi(\Omega^{\Omega^2})</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>
|-
| [[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
|<math>\psi(I(1,0,0))=\psi(M^M)</math>
|<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(2 \ \rm{aft} \ 4)</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>
|-
| [[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(\lambda\alpha.(\Omega_{\alpha+1})-\Pi_1)=\psi(a_2^{\Omega_{a+1}}+\psi_{a_2}(a_2^{\Omega_{a+1}})\times\omega)</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(\lambda\alpha.(\Omega_{\alpha+2})-\Pi_{1})=\psi(\Omega_{a_2+1}+\psi_{a_2}(\Omega_{a_2+1})\times\omega)</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}=\psi(\Omega_{a_2+1}\times\omega)</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.(\psi_{I_{\alpha+1}}(0))-\Pi_{0})=\psi(\Omega_{a_2+1}\times a_2)</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=\psi(a_3) </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=\psi(a_4) </math>
|<math>\mathrm{BMS}(0,0,0)(1,1,1)(2,2,1)(3,3,1)(4,4,0)</math>
|-
| [[LRO|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>
|-
|[[LRO|SBO]]
|Small Bashicu 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>
|-
|[[方括号稳定|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>
|-
|[[TSSO|SSPO]]
|Small Simple Projection Ordinal
|<math>\psi(\omega-\text{proj.})=\psi(\sigma S\times \omega)=\psi(H^\omega)</math>
|<math>\mathrm{BMS}(0)(1,1,1,1)</math>
|-
| [[TSSO]]|| Trio Sequence System Ordinal || <math>\psi(\omega-\text{proj.})=\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.})=\psi(\sigma S\times S) </math>
|<math>\mathrm{BMS}(0)(1,1,1,1)(2,1,1,1)(3,1)(2)</math>
|-
|[[Q0.5BGO]]
|QSS 0.5th Back Gear Ordinal
|<math>\psi(\psi_S(\sigma S\times S\times \omega+S_2)) </math>
|<math>\rm{BMS}(0)(1,1,1,1)(2,2)</math>
|-
|[[Q1BGO]]
|QSS 1st Back Gear Ordinal
| <math>\psi(\psi_S(\sigma S\times S\times \omega+S_\omega)) </math>
|<math>\mathrm{BMS}(0)(1,1,1,1)(2,2,2)</math>
|-
|[[ESPO]]
|Extend Simple Projection Ordinal
| <math>\psi(\psi_S(\sigma S\times S\times \omega^2)) </math>
|<math>\mathrm{BMS}(0)(1,1,1,1)(2,2,2,1)</math>
|-
|[[BOBO]]
|Big Omega Back Ordinal
| <math>\psi((\omega,0)-P)=\psi(\psi_{H}(H^{H\omega})) </math>
|<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>
|-
|[[QSSO|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>\psi(\psi_{H}(H^{H^{H^{\omega}}}))</math>
|<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)=\lim(\rm BMS)</math>
|-
|βO
|Beta Universe Ordinal
|<math>\rm PTO(Z_2)</math>
|<math>\geqslant Y(1,3)</math>
|-
| [[ΩSSO]]|| <math>\Omega</math> Sequence System Ordinal || <math>\psi(\psi_H(\varphi(\Omega,H+1)))?</math>|| <math>Y(1,3,4,2,5,8,10)</math>
|-
|[[LRPO]]
|Large Right Projection Ordinal
|<math>\psi(\psi_{H}(\varphi(H,1)))?</math>
|<math>Y(1,3,4,2,5,8,10,4,9,14,17,10)</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>\psi(\psi_H(\psi_T(T_2\times 2)))?</math>|| <math>Y(1,3,4,3)</math>
|-
|[[DCO]]
|Difference Catching Ordinal
|<math>\psi(\psi_H(\psi_T(T_2\times\psi_T(T_2^2))))?</math>
|<math>Y(1,3,5)</math>
|-
| [[SYO]]|| Small Yukito Ordinal || <math>\omega \rm{ MN}(0)(,,,1)</math>|| <math>\omega-Y(1,4)</math>
|-
| [[MHO]]/ωYO<ref name=":0" />|| Medium Hydra Ordinal || <math>\omega \rm{2MN}(0)(;1)</math>|| <math>\lim(\omega-Y)</math>
|-
| [[CKO]]|| Church-Kleene Ordinal || <math>\omega_{1}^{\rm CK}</math>
|-
| [[FUO]]|| First Uncountable Ordinal || <math>\omega_{1}</math>||
|}

=== 已弃用序数表 ===
{| class="wikitable"
!缩写
!英文全称
!定义
!大小
!命名者
|-
|SMDO
|Small Multidimensional Ordinal
|<math>\omega^\omega</math>
|<math>(0)(1)(2)</math>
|318`4
|-
|SHO
|Small Hydra Ordinal
|<math>\varepsilon_0=\psi(\Omega)</math>
|<math>(0)(1,1)</math>
|FataliS1024
|-
|ESVO
|Extended Small Veblen Ordinal
|<math>\psi\left(\Omega^{\Omega^{\Omega^{\omega}}}\right)</math>
|<math>(0)(1,1)(2,1)(3,1)(4,1)(5)</math>
|
|-
|ELVO
|Extended Large Veblen Ordinal
|<math>\psi\left(\Omega^{\Omega^{\Omega^{\Omega}}}\right)</math>
|<math>(0)(1,1)(2,1)(3,1)(4,1)(5,1)</math>
|
|-
|LDO(旧)
|Large Dropping Ordinal
|<math>\psi(\lambda\alpha.(\Omega_{\alpha\times 2})-\Pi_{0})</math>
|<math>(0)(1,1,1)(2,2,1)(3,1)(2)</math>
|
|-
|EDO
|Extended Dropping Ordinal
|<math>\psi(\lambda\alpha.\psi_{I_{\alpha+1}}(I_{\alpha+1})-\Pi_0)</math>
|<math>(0)(1,1,1)(2,2,1)(3,2)</math>
|
|-
|SEIO
|Small Eveog-Imagined Ordinal
|<math>\Sigma_1\text{ admissible}</math>
|
|
|-
|MEIO
|Medium Eveog-Imagined Ordinal
|<math>\Sigma_2\text{ admissible}</math>
|
|
|-
|LEIO
|Large Eveog-Imagined Ordinal
|<math>\Sigma_3\text{ admissible}</math>
|
|
|-
|SOSO
|Second Order Stable Ordinal
|<math>\psi(1-o-\Sigma_2-\text{stb.})</math>
|
|
|-
|EGO
|Eveog's Ordinal
|<math>\psi(\psi\sigma(\sigma_\omega))</math>
|<math>(0)(1,1,1,1)(2,2,2,1)(3,2,1)(4)</math>
|
|-
|MHO
|Medium Hydra Ordinal
|<math>\lim({\rm BMS})=\lim(0-Y)</math>
|<math>Y(1,3)</math>
|FataliS1024
|-
|LHO
|Large Hydra Ordinal
|<math>\lim(\omega-Y)</math>
|<math>\Omega-Y(1,3,12)</math>
|FataliS1024
|-
|ZDO
|Zeta Differenciating Ordinal
|<math>\text{FOS911 }\Theta(\zeta_0)</math>
|<math>\Omega-Y(1,3,12)</math>
|
|-
|WYO
|Omega Y Ordinal
|<math>\lim(\Omega-Y)</math>
|<math>\Omega-Y(1,\omega)</math>
|318`4
|-
|EYO
|Extended Y Ordinal
|<math>\text{bFOS }\Theta(\text{BHO})</math>
|<math>\text{bFOS }(0)(1)(\omega)(\varepsilon_0)(\text{BHO})</math>
|318`4
|-
|UCO
|Upgrade Catching Ordinal
|<math>\text{sFOS }\Theta(\text{BHO})</math>
|<math>\text{bFOS }\Theta(\text{BO})</math>
|318`4
|-
|XYO
|Extreme Y Ordinal
|<math>\text{bFOS }\Theta(\text{BO})</math>
|<math>\text{bFOS }(0)(1)(\omega)(\varepsilon_0)(\text{BO})</math>
|318`4
|-
|DMO
|Difference Matrix Ordinal
|<math>\text{sFOS }\Theta(\text{BO})</math>
|<math>\text{bFOS }\Theta(\text{SHO})</math>
|318`4
|-
|GYO / 😰O
|Grand Y-Sequence Ordinal / 😰 Ordinal
|<math>\lim(X-Y)</math>
|<math>\text{sFOS }(0)(1)(\omega)(\varepsilon_0)(\text{SHO})</math>
|318`4
|-
|LDCO
|Large Difference Catching Ordinal
|<math>\text{sFOS }\Theta(\text{SYO})</math>
|<math>\text{b2-FOS }\Theta(\zeta_1)</math>
|318`4
|-
|RHO
|Remaining Hydra Ordinal
|<math>\lim(\text{sFOS})</math>
|<math>\text{b2-FOS }\Theta(\varphi(\omega,0))</math>
|318`4
|-
|WFO
|Omega Fundamental Ordinal
|<math>\lim(\text{Weak 2-FOS})</math>
|<math>\text{b2-FOS }\Theta(\Gamma_0)</math>
|318`4
|-
|TMDO
|Tri-Multidimensional Ordinal
|<math>\text{s2-FOS }\Theta(\varphi(\omega,0))</math>
|<math>\omega2-\text{RD}</math>
|318`4
|-
|ERHO
|Extended Remaining hydra Ordinal
|<math>\lim(\text{b2-FOS})</math>
|<math>\omega2+1-\text{RD}</math>
|318`4
|-
|LMDO
|Large Multidimensional Ordinal
|<math>\lim(\omega-\text{FOS})</math>
|<math>\omega^2-\text{RD}</math>
|318`4
|-
|IFO
|Infintesimal Function Ordinal
|<math>\lim(\text{IFS})</math>
|<math>\varepsilon_0-\text{RD}</math>
|318`4
|-
|WRO
|Omega Remaining Ordinal
|<math>\lim(\text{ROS})</math>
|<math>R\ \Omega-Y</math>
|318`4
|-
|SCLO
|Small Code Lift Ordinal
|<math>\sup(n-\text{code})</math>
|
|
|-
|EHO
|Huge Hydra Ordinal
|<math>\lim(\text{pfffz})</math>
|
|夏夜星空
|-
|ROO
|Remaining Omega Ordinal
|<math>R_\omega\text{ remaining}</math>
|
|318`4
|-
|UHO
|Ultimate Hydra Ordinal
|<math>\lim(\text{RSAM})</math>
|
|夏夜星空
|-
|IHO
|Infinite Hydra Ordinal
|<math>\lim(\text{SAM})</math>
|
|夏夜星空
|}
本表取自 [https://docs.qq.com/sheet/DSnFWckliSU5TTE15 Worldly Sheet]：<blockquote>- （SCO/CO/LCO/HCO）谁起不重要，重要的是这是纪念康托尔的，如果没有他所有gggist今天(甚至永远)都走不到一起”

- 你们怎么把它弄成这样了，至少必要的（比如lim fffz/lim X-Y还是要的吧）

- fffz和X-Y公认理想之前搞这么多名字有什么用

- 不然MHO以上全都写成n-RD？- 不对 - 3184为什么要保留他造了那么多没用的序数缩写的黑历史？(bushi) - 不如还是加上毕竟fatalis的SHO/MHO/LHO都有了</blockquote>

==== DNAO ====
DNAO（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)
=== 脚注 ===
<references />

非递归BMS分析

2025-07-20T00:23:04Z

Optimism：Ω

本条目展示[[非递归BMS]]和其他非递归记号的列表分析

== 1 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>\varnothing</math>
|<math>0</math>
|-
|<math>(1)</math>
|<math>1</math>
|-
|<math>(1)(1)</math>
|<math>2</math>
|-
|<math>(1)(2)</math>
|<math>\omega</math>
|-
|<math>(1)(2)(1)</math>
|<math>\omega+1</math>
|-
|<math>(1)(2)(2)</math>
|<math>\omega^2</math>
|-
|<math>(1)(2)(3)</math>
|<math>\omega^\omega</math>
|-
|<math>(1)(2,1)</math>
|<math>\varepsilon_0</math>
|-
|<math>(1)(2,1,1)</math>
|<math>\psi(\Omega_\omega)</math>
|-
|<math>(1,1)</math>
|<math>\Omega</math>
|}

== 2 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1)</math>
|<math>\Omega</math>
|-
|<math>(1,1)(1)</math>
|<math>\Omega+1</math>
|-
|<math>(1,1)(1)(2)</math>
|<math>\Omega+\omega</math>
|-
|<math>(1,1)(1)(2,1)</math>
|<math>\Omega+\varepsilon_0</math>
|-
|<math>(1,1)(1)(2,1,1)</math>
|<math>\Omega+\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(1,1)</math>
|<math>\Omega\times2</math>
|-
|<math>(1,1)(1,1)(1,1)</math>
|<math>\Omega\times3</math>
|-
|<math>(1,1)(2)</math>
|<math>\Omega\times\omega</math>
|-
|<math>(1,1)(2)(1,1)</math>
|<math>\Omega\times\omega+\Omega</math>
|-
|<math>(1,1)(2)(2)</math>
|<math>\Omega\times\omega^2</math>
|-
|<math>(1,1)(2)(3,1,1)</math>
|<math>\Omega\times\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(2,1)</math>
|<math>\Omega^2</math>
|-
|<math>(1,1)(2,1)(1,1)(2,1)</math>
|<math>\Omega^2\times2</math>
|-
|<math>(1,1)(2,1)(2)</math>
|<math>\Omega^2\times\omega</math>
|-
|<math>(1,1)(2,1)(2)(3,1,1)</math>
|<math>\Omega^2\times\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(2,1)(2,1)</math>
|<math>\Omega^3</math>
|-
|<math>(1,1)(2,1)(2,1)(2,1)</math>
|<math>\Omega^4</math>
|-
|<math>(1,1)(2,1)(3)</math>
|<math>\Omega^\omega</math>
|-
|<math>(1,1)(2,1)(3)(4,1,1)</math>
|<math>\Omega^{\psi(\Omega_\omega)}</math>
|-
|<math>(1,1)(2,1)(3,1)</math>
|<math>\Omega^\Omega</math>
|-
|<math>(1,1)(2,1)(3,1)(4,1)</math>
|<math>\Omega^{\Omega^\Omega}</math>
|-
|<math>(1,1)(2,2)</math>
|<math>\psi_1(\Omega_2)</math>
|-
|<math>(1,1)(2,2)(2,1)</math>
|<math>\psi_1(\Omega_2+\Omega)</math>
|-
|<math>(1,1)(2,2)(2,1)(2,1)</math>
|<math>\psi_1(\Omega_2+\Omega\times2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3)</math>
|<math>\psi_1(\Omega_2+\Omega\times\omega)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,1)</math>
|<math>\psi_1(\Omega_2+\Omega^2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2))</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(2,1)(3,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2)\times2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(3)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2+1))</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))</math>
|-
|<math>(1,1)(2,2)(2,2)</math>
|<math>\psi_1(\Omega_2\times2)</math>
|-
|<math>(1,1)(2,2)(3)</math>
|<math>\psi_1(\Omega_2\times\omega)</math>
|-
|<math>(1,1)(2,2)(3,1)(4,2)</math>
|<math>\psi_1(\Omega_2\times\psi_1(\Omega_2))</math>
|-
|<math>(1,1)(2,2)(3,2)</math>
|<math>\psi_1(\Omega_2^2)</math>
|-
|<math>(1,1)(2,2)(3,2)(4,2)</math>
|<math>\psi_1(\Omega_2^{\Omega_2})</math>
|-
|<math>(1,1)(2,2)(3,3)</math>
|<math>\psi_1(\Omega_3)</math>
|-
|<math>(1,1)(2,2,1)</math>
|<math>\psi_1(\Omega_\omega)</math>
|-
|<math>(1,1)(2,2,1)(3,2,1)</math>
|<math>\psi_1(\Omega_{\omega^2})</math>
|-
|<math>(1,1)(2,2,1)(3,2,1)(4,1)</math>
|<math>\psi_1(\Omega_{\Omega})</math>
|-
|<math>(1,1)(2,2,1,1)</math>
|<math>\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)</math>
|<math>\Omega_2</math>
|}

== 3 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1,1)</math>
|<math>\Omega_2</math>
|-
|<math>(1,1,1)(1,1)</math>
|<math>\Omega_2+\Omega</math>
|-
|<math>(1,1,1)(1,1)(2,2,1,1)</math>
|<math>\Omega_2+\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)(1,1,1)</math>
|<math>\Omega_2\times2</math>
|-
|<math>(1,1,1)(2)</math>
|<math>\Omega_2\times\omega</math>
|-
|<math>(1,1,1)(2,1)</math>
|<math>\Omega_2\times\Omega</math>
|-
|<math>(1,1,1)(2,1)(2,1)</math>
|<math>\Omega_2\times\Omega^2</math>
|-
|<math>(1,1,1)(2,1)(3,2)</math>
|<math>\Omega_2\times\psi_1(\Omega_2)</math>
|-
|<math>(1,1,1)(2,1)(3,2,1,1)</math>
|<math>\Omega_2\times\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,1,1)</math>
|<math>\Omega_2^2</math>
|-
|<math>(1,1,1)(2,1,1)(2,1,1)</math>
|<math>\Omega_2^3</math>
|-
|<math>(1,1,1)(2,1,1)(3,1,1)</math>
|<math>\Omega_2^{\Omega_2}</math>
|-
|<math>(1,1,1)(2,2)</math>
|<math>\psi_2(\Omega_3)</math>
|-
|<math>(1,1,1)(2,2)(2,1,1)(3,2)</math>
|<math>\psi_2(\Omega_3+\psi_2(\Omega_3))</math>
|-
|<math>(1,1,1)(2,2)(2,2)</math>
|<math>\psi_2(\Omega_3\times2)</math>
|-
|<math>(1,1,1)(2,2)(3,1,1)(4,2)</math>
|<math>\psi_2(\Omega_3\times\psi_2(\Omega_3))</math>
|-
|<math>(1,1,1)(2,2)(3,2)</math>
|<math>\psi_2(\Omega_3^2)</math>
|-
|<math>(1,1,1)(2,2)(3,3)</math>
|<math>\psi_2(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2)(3,3,1)</math>
|<math>\psi_2(\Omega_\omega)</math>
|-
|<math>(1,1,1)(2,2)(3,3,1)(4,3,1)(5,1)</math>
|<math>\psi_2(\Omega_\Omega)</math>
|-
|<math>(1,1,1)(2,2)(3,3,1)(4,3,1)(5,1,1)</math>
|<math>\psi_2(\Omega_{\Omega_2})</math>
|-
|<math>(1,1,1)(2,2)(3,3,1,1)</math>
|<math>\psi_2(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)</math>
|<math>\Omega_3</math>
|-
|<math>(1,1,1)(2,2,1)(2,1)</math>
|<math>\Omega_3\times\Omega</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)</math>
|<math>\Omega_3\times\Omega_2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(2,1,1)</math>
|<math>\Omega_3\times\Omega_2^2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)</math>
|<math>\Omega_3\times\psi_2(\Omega_3)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(3,2)</math>
|<math>\Omega_3\times\psi_2(\Omega_3\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(4,3)</math>
|<math>\Omega_3\times\psi_2(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(4,3,1,1)</math>
|<math>\Omega_3\times\psi_2(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)</math>
|<math>\Omega_3^2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(2,1,1)(3,2,1)</math>
|<math>\Omega_3^3</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(3)</math>
|<math>\Omega_3^\omega</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(3,1,1)(4,2,1)</math>
|<math>\Omega_3^{\Omega_3}</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)</math>
|<math>\psi_3(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(2,2)</math>
|<math>\psi_3(\Omega_4\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(3,3,1)</math>
|<math>\psi_3(\Omega_\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(3,3,1,1)</math>
|<math>\psi_3(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2,1)</math>
|<math>\Omega_4</math>
|-
|<math>(1,1,1)(2,2,1)(2,2,1)(2,2,1)</math>
|<math>\Omega_5</math>
|-
|<math>(1,1,1)(2,2,1)(3)</math>
|<math>\Omega_\omega</math>
|}

== 4 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1,1)(2,2,1)(3)</math>
|<math>\Omega_\omega</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)</math>
|<math>\Omega_\Omega</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(2,2,1)(3,1)</math>
|<math>\Omega_{\Omega\times2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(3,1)</math>
|<math>\Omega_{\Omega^2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(4,2)</math>
|<math>\Omega_{\psi_1(\Omega_2)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(4,2,1,1)</math>
|<math>\Omega_{\psi_1(\omega-proj.)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)</math>
|<math>\Omega_{\Omega_2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2)</math>
|<math>\Omega_{\psi_2(\Omega_3)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2)(5,3,1,1)</math>
|<math>\Omega_{\psi_2(\omega-proj.)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)</math>
|<math>\Omega_{\Omega_3}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(4,2,1)</math>
|<math>\Omega_{\Omega_4}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5)</math>
|<math>\Omega_{\Omega_\omega}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)</math>
|<math>\Omega_{\Omega_{\Omega_2}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)(6,2,1)</math>
|<math>\Omega_{\Omega_{\Omega_3}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)(6,2,1)(7)</math>
|<math>\Omega_{\Omega_{\Omega_\omega}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+1)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\Omega_2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(2,2,1)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})\times\Omega_2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})^2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,1,1)(5,2,1)(6,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})^{\psi_\mathrm{I}(\mathrm{I})})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_{\psi_\mathrm{I}(\mathrm{I}+1)}(\psi_\mathrm{I}(\mathrm{I}+1)))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2)(5,3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_{\psi_\mathrm{I}(\mathrm{I}+1)}(\omega-proj.))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+1))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(4,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+\omega))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5,1,1)(6,2,1)(7,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2+1)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2+\psi_\mathrm{I}(\mathrm{I}\times2))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times3)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times\psi_\mathrm{I}(\mathrm{I}))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,2)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^3)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^\mathrm{I})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,3)</math>
|<math>\psi_\mathrm{I}(\Omega_{\mathrm{I}+1})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,3,1,1)</math>
|<math>\psi_\mathrm{I}(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)</math>
|<math>\mathrm{I}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,1,1)(3,2,1)(4,2,1)</math>
|<math>\mathrm{I^2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+1})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+2})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3,1)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+\omega})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3,1,1)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2,1)</math>
|<math>\Omega_{\mathrm{I}+1}</math>
|}
[[分类:分析]]

非递归BMS分析

2025-07-20T00:19:58Z

Optimism：Ω

本条目展示[[非递归BMS]]和其他非递归记号的列表分析

== 1 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>\varnothing</math>
|<math>0</math>
|-
|<math>(1)</math>
|<math>1</math>
|-
|<math>(1)(1)</math>
|<math>2</math>
|-
|<math>(1)(2)</math>
|<math>\omega</math>
|-
|<math>(1)(2)(1)</math>
|<math>\omega+1</math>
|-
|<math>(1)(2)(2)</math>
|<math>\omega^2</math>
|-
|<math>(1)(2)(3)</math>
|<math>\omega^\omega</math>
|-
|<math>(1)(2,1)</math>
|<math>\varepsilon_0</math>
|-
|<math>(1)(2,1,1)</math>
|<math>\psi(\Omega_\omega)</math>
|-
|<math>(1,1)</math>
|<math>\Omega</math>
|}

== 2 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1)</math>
|<math>\Omega</math>
|-
|<math>(1,1)(1)</math>
|<math>\Omega+1</math>
|-
|<math>(1,1)(1)(2)</math>
|<math>\Omega+\omega</math>
|-
|<math>(1,1)(1)(2,1)</math>
|<math>\Omega+\varepsilon_0</math>
|-
|<math>(1,1)(1)(2,1,1)</math>
|<math>\Omega+\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(1,1)</math>
|<math>\Omega\times2</math>
|-
|<math>(1,1)(1,1)(1,1)</math>
|<math>\Omega\times3</math>
|-
|<math>(1,1)(2)</math>
|<math>\Omega\times\omega</math>
|-
|<math>(1,1)(2)(1,1)</math>
|<math>\Omega\times\omega+\Omega</math>
|-
|<math>(1,1)(2)(2)</math>
|<math>\Omega\times\omega^2</math>
|-
|<math>(1,1)(2)(3,1,1)</math>
|<math>\Omega\times\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(2,1)</math>
|<math>\Omega^2</math>
|-
|<math>(1,1)(2,1)(1,1)(2,1)</math>
|<math>\Omega^2\times2</math>
|-
|<math>(1,1)(2,1)(2)</math>
|<math>\Omega^2\times\omega</math>
|-
|<math>(1,1)(2,1)(2)(3,1,1)</math>
|<math>\Omega^2\times\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(2,1)(2,1)</math>
|<math>\Omega^3</math>
|-
|<math>(1,1)(2,1)(2,1)(2,1)</math>
|<math>\Omega^4</math>
|-
|<math>(1,1)(2,1)(3)</math>
|<math>\Omega^\omega</math>
|-
|<math>(1,1)(2,1)(3)(4,1,1)</math>
|<math>\Omega^{\psi(\Omega_\omega)}</math>
|-
|<math>(1,1)(2,1)(3,1)</math>
|<math>\Omega^\Omega</math>
|-
|<math>(1,1)(2,1)(3,1)(4,1)</math>
|<math>\Omega^{\Omega^\Omega}</math>
|-
|<math>(1,1)(2,2)</math>
|<math>\psi_1(\Omega_2)</math>
|-
|<math>(1,1)(2,2)(2,1)</math>
|<math>\psi_1(\Omega_2+\Omega)</math>
|-
|<math>(1,1)(2,2)(2,1)(2,1)</math>
|<math>\psi_1(\Omega_2+\Omega\times2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3)</math>
|<math>\psi_1(\Omega_2+\Omega\times\omega)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,1)</math>
|<math>\psi_1(\Omega_2+\Omega^2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2))</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(2,1)(3,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2)\times2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(3)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2+1))</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))</math>
|-
|<math>(1,1)(2,2)(2,2)</math>
|<math>\psi_1(\Omega_2\times2)</math>
|-
|<math>(1,1)(2,2)(3)</math>
|<math>\psi_1(\Omega_2\times\omega)</math>
|-
|<math>(1,1)(2,2)(3,1)(4,2)</math>
|<math>\psi_1(\Omega_2\times\psi_1(\Omega_2))</math>
|-
|<math>(1,1)(2,2)(3,2)</math>
|<math>\psi_1(\Omega_2^2)</math>
|-
|<math>(1,1)(2,2)(3,2)(4,2)</math>
|<math>\psi_1(\Omega_2^{\Omega_2})</math>
|-
|<math>(1,1)(2,2)(3,3)</math>
|<math>\psi_1(\Omega_3)</math>
|-
|<math>(1,1)(2,2,1)</math>
|<math>\psi_1(\Omega_\omega)</math>
|-
|<math>(1,1)(2,2,1)(3,2,1)</math>
|<math>\psi_1(\Omega_{\omega^2})</math>
|-
|<math>(1,1)(2,2,1)(3,2,1)(4,1)</math>
|<math>\psi_1(\Omega_{\Omega})</math>
|-
|<math>(1,1)(2,2,1,1)</math>
|<math>\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)</math>
|<math>\Omega_2</math>
|}

== 3 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1,1)</math>
|<math>\Omega_2</math>
|-
|<math>(1,1,1)(1,1)</math>
|<math>\Omega_2+\Omega</math>
|-
|<math>(1,1,1)(1,1)(2,2,1,1)</math>
|<math>\Omega_2+\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)(1,1,1)</math>
|<math>\Omega_2\times2</math>
|-
|<math>(1,1,1)(2)</math>
|<math>\Omega_2\times\omega</math>
|-
|<math>(1,1,1)(2,1)</math>
|<math>\Omega_2\times\Omega</math>
|-
|<math>(1,1,1)(2,1)(2,1)</math>
|<math>\Omega_2\times\Omega^2</math>
|-
|<math>(1,1,1)(2,1)(3,2)</math>
|<math>\Omega_2\times\psi_1(\Omega_2)</math>
|-
|<math>(1,1,1)(2,1)(3,2,1,1)</math>
|<math>\Omega_2\times\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,1,1)</math>
|<math>\Omega_2^2</math>
|-
|<math>(1,1,1)(2,1,1)(2,1,1)</math>
|<math>\Omega_2^3</math>
|-
|<math>(1,1,1)(2,1,1)(3,1,1)</math>
|<math>\Omega_2^{\Omega_2}</math>
|-
|<math>(1,1,1)(2,2)</math>
|<math>\psi_2(\Omega_3)</math>
|-
|<math>(1,1,1)(2,2)(2,1,1)(3,2)</math>
|<math>\psi_2(\Omega_3+\psi_2(\Omega_3))</math>
|-
|<math>(1,1,1)(2,2)(2,2)</math>
|<math>\psi_2(\Omega_3\times2)</math>
|-
|<math>(1,1,1)(2,2)(3,1,1)(4,2)</math>
|<math>\psi_2(\Omega_3\times\psi_2(\Omega_3))</math>
|-
|<math>(1,1,1)(2,2)(3,2)</math>
|<math>\psi_2(\Omega_3^2)</math>
|-
|<math>(1,1,1)(2,2)(3,3)</math>
|<math>\psi_2(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2)(3,3,1)</math>
|<math>\psi_2(\Omega_\omega)</math>
|-
|<math>(1,1,1)(2,2)(3,3,1,1)</math>
|<math>\psi_2(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)</math>
|<math>\Omega_3</math>
|-
|<math>(1,1,1)(2,2,1)(2,1)</math>
|<math>\Omega_3\times\Omega</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)</math>
|<math>\Omega_3\times\Omega_2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(2,1,1)</math>
|<math>\Omega_3\times\Omega_2^2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)</math>
|<math>\Omega_3\times\psi_2(\Omega_3)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(3,2)</math>
|<math>\Omega_3\times\psi_2(\Omega_3\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(4,3)</math>
|<math>\Omega_3\times\psi_2(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(4,3,1,1)</math>
|<math>\Omega_3\times\psi_2(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)</math>
|<math>\Omega_3^2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(2,1,1)(3,2,1)</math>
|<math>\Omega_3^3</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(3)</math>
|<math>\Omega_3^\omega</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(3,1,1)(4,2,1)</math>
|<math>\Omega_3^{\Omega_3}</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)</math>
|<math>\psi_3(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(2,2)</math>
|<math>\psi_3(\Omega_4\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(3,3,1)</math>
|<math>\psi_3(\Omega_\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(3,3,1,1)</math>
|<math>\psi_3(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2,1)</math>
|<math>\Omega_4</math>
|-
|<math>(1,1,1)(2,2,1)(2,2,1)(2,2,1)</math>
|<math>\Omega_5</math>
|-
|<math>(1,1,1)(2,2,1)(3)</math>
|<math>\Omega_\omega</math>
|}

== 4 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1,1)(2,2,1)(3)</math>
|<math>\Omega_\omega</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)</math>
|<math>\Omega_\Omega</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(2,2,1)(3,1)</math>
|<math>\Omega_{\Omega\times2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(3,1)</math>
|<math>\Omega_{\Omega^2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(4,2)</math>
|<math>\Omega_{\psi_1(\Omega_2)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(4,2,1,1)</math>
|<math>\Omega_{\psi_1(\omega-proj.)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)</math>
|<math>\Omega_{\Omega_2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2)</math>
|<math>\Omega_{\psi_2(\Omega_3)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2)(5,3,1,1)</math>
|<math>\Omega_{\psi_2(\omega-proj.)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)</math>
|<math>\Omega_{\Omega_3}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(4,2,1)</math>
|<math>\Omega_{\Omega_4}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5)</math>
|<math>\Omega_{\Omega_\omega}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)</math>
|<math>\Omega_{\Omega_{\Omega_2}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)(6,2,1)</math>
|<math>\Omega_{\Omega_{\Omega_3}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)(6,2,1)(7)</math>
|<math>\Omega_{\Omega_{\Omega_\omega}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+1)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\Omega_2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(2,2,1)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})\times\Omega_2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})^2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,1,1)(5,2,1)(6,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})^{\psi_\mathrm{I}(\mathrm{I})})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_{\psi_\mathrm{I}(\mathrm{I}+1)}(\psi_\mathrm{I}(\mathrm{I}+1)))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2)(5,3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_{\psi_\mathrm{I}(\mathrm{I}+1)}(\omega-proj.))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+1))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(4,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+\omega))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5,1,1)(6,2,1)(7,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2+1)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2+\psi_\mathrm{I}(\mathrm{I}\times2))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times3)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times\psi_\mathrm{I}(\mathrm{I}))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,2)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^3)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^\mathrm{I})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,3)</math>
|<math>\psi_\mathrm{I}(\Omega_{\mathrm{I}+1})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,3,1,1)</math>
|<math>\psi_\mathrm{I}(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)</math>
|<math>\mathrm{I}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,1,1)(3,2,1)(4,2,1)</math>
|<math>\mathrm{I^2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+1})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+2})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3,1)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+\omega})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3,1,1)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2,1)</math>
|<math>\Omega_{\mathrm{I}+1}</math>
|}
[[分类:分析]]

非递归BMS分析

2025-07-20T00:17:31Z

Optimism：I

本条目展示[[非递归BMS]]和其他非递归记号的列表分析

== 1 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>\varnothing</math>
|<math>0</math>
|-
|<math>(1)</math>
|<math>1</math>
|-
|<math>(1)(1)</math>
|<math>2</math>
|-
|<math>(1)(2)</math>
|<math>\omega</math>
|-
|<math>(1)(2)(1)</math>
|<math>\omega+1</math>
|-
|<math>(1)(2)(2)</math>
|<math>\omega^2</math>
|-
|<math>(1)(2)(3)</math>
|<math>\omega^\omega</math>
|-
|<math>(1)(2,1)</math>
|<math>\varepsilon_0</math>
|-
|<math>(1)(2,1,1)</math>
|<math>\psi(\Omega_\omega)</math>
|-
|<math>(1,1)</math>
|<math>\Omega</math>
|}

== 2 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1)</math>
|<math>\Omega</math>
|-
|<math>(1,1)(1)</math>
|<math>\Omega+1</math>
|-
|<math>(1,1)(1)(2)</math>
|<math>\Omega+\omega</math>
|-
|<math>(1,1)(1)(2,1)</math>
|<math>\Omega+\varepsilon_0</math>
|-
|<math>(1,1)(1)(2,1,1)</math>
|<math>\Omega+\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(1,1)</math>
|<math>\Omega\times2</math>
|-
|<math>(1,1)(1,1)(1,1)</math>
|<math>\Omega\times3</math>
|-
|<math>(1,1)(2)</math>
|<math>\Omega\times\omega</math>
|-
|<math>(1,1)(2)(1,1)</math>
|<math>\Omega\times\omega+\Omega</math>
|-
|<math>(1,1)(2)(2)</math>
|<math>\Omega\times\omega^2</math>
|-
|<math>(1,1)(2)(3,1,1)</math>
|<math>\Omega\times\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(2,1)</math>
|<math>\Omega^2</math>
|-
|<math>(1,1)(2,1)(1,1)(2,1)</math>
|<math>\Omega^2\times2</math>
|-
|<math>(1,1)(2,1)(2)</math>
|<math>\Omega^2\times\omega</math>
|-
|<math>(1,1)(2,1)(2)(3,1,1)</math>
|<math>\Omega^2\times\psi(\Omega_\omega)</math>
|-
|<math>(1,1)(2,1)(2,1)</math>
|<math>\Omega^3</math>
|-
|<math>(1,1)(2,1)(2,1)(2,1)</math>
|<math>\Omega^4</math>
|-
|<math>(1,1)(2,1)(3)</math>
|<math>\Omega^\omega</math>
|-
|<math>(1,1)(2,1)(3)(4,1,1)</math>
|<math>\Omega^{\psi(\Omega_\omega)}</math>
|-
|<math>(1,1)(2,1)(3,1)</math>
|<math>\Omega^\Omega</math>
|-
|<math>(1,1)(2,1)(3,1)(4,1)</math>
|<math>\Omega^{\Omega^\Omega}</math>
|-
|<math>(1,1)(2,2)</math>
|<math>\psi_1(\Omega_2)</math>
|-
|<math>(1,1)(2,2)(2,1)</math>
|<math>\psi_1(\Omega_2+\Omega)</math>
|-
|<math>(1,1)(2,2)(2,1)(2,1)</math>
|<math>\psi_1(\Omega_2+\Omega\times2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3)</math>
|<math>\psi_1(\Omega_2+\Omega\times\omega)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,1)</math>
|<math>\psi_1(\Omega_2+\Omega^2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2))</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(2,1)(3,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2)\times2)</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(3)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2+1))</math>
|-
|<math>(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)</math>
|<math>\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))</math>
|-
|<math>(1,1)(2,2)(2,2)</math>
|<math>\psi_1(\Omega_2\times2)</math>
|-
|<math>(1,1)(2,2)(3)</math>
|<math>\psi_1(\Omega_2\times\omega)</math>
|-
|<math>(1,1)(2,2)(3,1)(4,2)</math>
|<math>\psi_1(\Omega_2\times\psi_1(\Omega_2))</math>
|-
|<math>(1,1)(2,2)(3,2)</math>
|<math>\psi_1(\Omega_2^2)</math>
|-
|<math>(1,1)(2,2)(3,2)(4,2)</math>
|<math>\psi_1(\Omega_2^{\Omega_2})</math>
|-
|<math>(1,1)(2,2)(3,3)</math>
|<math>\psi_1(\Omega_3)</math>
|-
|<math>(1,1)(2,2,1)</math>
|<math>\psi_1(\Omega_\omega)</math>
|-
|<math>(1,1)(2,2,1,1)</math>
|<math>\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)</math>
|<math>\Omega_2</math>
|}

== 3 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1,1)</math>
|<math>\Omega_2</math>
|-
|<math>(1,1,1)(1,1)</math>
|<math>\Omega_2+\Omega</math>
|-
|<math>(1,1,1)(1,1)(2,2,1,1)</math>
|<math>\Omega_2+\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)(1,1,1)</math>
|<math>\Omega_2\times2</math>
|-
|<math>(1,1,1)(2)</math>
|<math>\Omega_2\times\omega</math>
|-
|<math>(1,1,1)(2,1)</math>
|<math>\Omega_2\times\Omega</math>
|-
|<math>(1,1,1)(2,1)(2,1)</math>
|<math>\Omega_2\times\Omega^2</math>
|-
|<math>(1,1,1)(2,1)(3,2)</math>
|<math>\Omega_2\times\psi_1(\Omega_2)</math>
|-
|<math>(1,1,1)(2,1)(3,2,1,1)</math>
|<math>\Omega_2\times\psi_1(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,1,1)</math>
|<math>\Omega_2^2</math>
|-
|<math>(1,1,1)(2,1,1)(2,1,1)</math>
|<math>\Omega_2^3</math>
|-
|<math>(1,1,1)(2,1,1)(3,1,1)</math>
|<math>\Omega_2^{\Omega_2}</math>
|-
|<math>(1,1,1)(2,2)</math>
|<math>\psi_2(\Omega_3)</math>
|-
|<math>(1,1,1)(2,2)(2,1,1)(3,2)</math>
|<math>\psi_2(\Omega_3+\psi_2(\Omega_3))</math>
|-
|<math>(1,1,1)(2,2)(2,2)</math>
|<math>\psi_2(\Omega_3\times2)</math>
|-
|<math>(1,1,1)(2,2)(3,1,1)(4,2)</math>
|<math>\psi_2(\Omega_3\times\psi_2(\Omega_3))</math>
|-
|<math>(1,1,1)(2,2)(3,2)</math>
|<math>\psi_2(\Omega_3^2)</math>
|-
|<math>(1,1,1)(2,2)(3,3)</math>
|<math>\psi_2(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2)(3,3,1)</math>
|<math>\psi_2(\Omega_\omega)</math>
|-
|<math>(1,1,1)(2,2)(3,3,1,1)</math>
|<math>\psi_2(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)</math>
|<math>\Omega_3</math>
|-
|<math>(1,1,1)(2,2,1)(2,1)</math>
|<math>\Omega_3\times\Omega</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)</math>
|<math>\Omega_3\times\Omega_2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(2,1,1)</math>
|<math>\Omega_3\times\Omega_2^2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)</math>
|<math>\Omega_3\times\psi_2(\Omega_3)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(3,2)</math>
|<math>\Omega_3\times\psi_2(\Omega_3\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(4,3)</math>
|<math>\Omega_3\times\psi_2(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2)(4,3,1,1)</math>
|<math>\Omega_3\times\psi_2(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)</math>
|<math>\Omega_3^2</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(2,1,1)(3,2,1)</math>
|<math>\Omega_3^3</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(3)</math>
|<math>\Omega_3^\omega</math>
|-
|<math>(1,1,1)(2,2,1)(2,1,1)(3,2,1)(3,1,1)(4,2,1)</math>
|<math>\Omega_3^{\Omega_3}</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)</math>
|<math>\psi_3(\Omega_4)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(2,2)</math>
|<math>\psi_3(\Omega_4\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(3,3,1)</math>
|<math>\psi_3(\Omega_\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2)(3,3,1,1)</math>
|<math>\psi_3(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(2,2,1)</math>
|<math>\Omega_4</math>
|-
|<math>(1,1,1)(2,2,1)(2,2,1)(2,2,1)</math>
|<math>\Omega_5</math>
|-
|<math>(1,1,1)(2,2,1)(3)</math>
|<math>\Omega_\omega</math>
|}

== 4 ==
{| class="wikitable"
|+
!非递归BMS
!其他记号
|-
|<math>(1,1,1)(2,2,1)(3)</math>
|<math>\Omega_\omega</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)</math>
|<math>\Omega_\Omega</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(2,2,1)(3,1)</math>
|<math>\Omega_{\Omega\times2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(3,1)</math>
|<math>\Omega_{\Omega^2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(4,2)</math>
|<math>\Omega_{\psi_1(\Omega_2)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1)(4,2,1,1)</math>
|<math>\Omega_{\psi_1(\omega-proj.)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)</math>
|<math>\Omega_{\Omega_2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2)</math>
|<math>\Omega_{\psi_2(\Omega_3)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2)(5,3,1,1)</math>
|<math>\Omega_{\psi_2(\omega-proj.)}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)</math>
|<math>\Omega_{\Omega_3}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(4,2,1)</math>
|<math>\Omega_{\Omega_4}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5)</math>
|<math>\Omega_{\Omega_\omega}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)</math>
|<math>\Omega_{\Omega_{\Omega_2}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)(6,2,1)</math>
|<math>\Omega_{\Omega_{\Omega_3}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5,1,1)(6,2,1)(7)</math>
|<math>\Omega_{\Omega_{\Omega_\omega}}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+1)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\Omega_2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(2,2,1)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})\times\Omega_2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})^2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,1,1)(5,2,1)(6,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})^{\psi_\mathrm{I}(\mathrm{I})})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_{\psi_\mathrm{I}(\mathrm{I}+1)}(\psi_\mathrm{I}(\mathrm{I}+1)))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2)(5,3,1,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_{\psi_\mathrm{I}(\mathrm{I}+1)}(\omega-proj.))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+1))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(4,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+\omega))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5,1,1)(6,2,1)(7,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I}+\psi_\mathrm{I}(\mathrm{I})))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2+1)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)(3,1,1)(4,2,1)(5,2)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times2+\psi_\mathrm{I}(\mathrm{I}\times2))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)(2,2,1)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times3)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,1,1)(4,2,1)(5,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}\times\psi_\mathrm{I}(\mathrm{I}))</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^2)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(3,2)(3,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^3)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^\omega)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,2)</math>
|<math>\psi_\mathrm{I}(\mathrm{I}^\mathrm{I})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,3)</math>
|<math>\psi_\mathrm{I}(\Omega_{\mathrm{I}+1})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2)(4,3,1,1)</math>
|<math>\psi_\mathrm{I}(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)</math>
|<math>\mathrm{I}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,1,1)(3,2,1)(4,2,1)</math>
|<math>\mathrm{I^2}</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+1})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+2})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3,1)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\Omega_{\mathrm{I}+\omega})</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2)(3,3,1,1)</math>
|<math>\psi_{\Omega_{\mathrm{I}+1}}(\omega-proj.)</math>
|-
|<math>(1,1,1)(2,2,1)(3,2,1)(2,2,1)</math>
|<math>\Omega_{\mathrm{I}+1}</math>
|}
[[分类:分析]]

BMS分析

2025-07-08T15:33:01Z

Optimism：/* 1：单行BMS（PrSS） */

目前使用的OCF为M型，后续补充BOCF

==== 1：单行BMS（PrSS） ====
<math>\varnothing=0</math>

<math>(0)=1</math>

<math>(0)(0)=2</math>

<math>(0)(0)(0)=3</math>

<math>(0)(1)=(0)(0)(0)(0)(0)...=\omega</math>

<math>(0)(1)(0)=\omega+1</math>

<math>(0)(1)(0)(0)=\omega+2</math>

<math>(0)(1)(0)(1)=\omega\times2</math>

<math>(0)(1)(0)(1)(0)(1)=\omega\times3</math>

<math>(0)(1)(1)=(0)(1)(0)(1)(0)(1)...=\omega^2</math>

<math>(0)(1)(1)(0)=\omega^2+1</math>

<math>(0)(1)(1)(0)(1)=\omega^2+\omega</math>

<math>(0)(1)(1)(0)(1)(0)(1)=\omega^2+\omega\times2</math>

<math>(0)(1)(1)(0)(1)(1)=\omega^2\times2</math>

<math>(0)(1)(1)(0)(1)(1)(0)(1)=\omega^2\times2+\omega</math>

<math>(0)(1)(1)(0)(1)(1)(0)(1)(1)=\omega^2\times3</math>

<math>(0)(1)(1)(1)=\omega^3</math>

<math>(0)(1)(1)(1)(1)=\omega^4</math>

<math>(0)(1)(2)=(0)(1)(1)(1)(1)...=\omega^\omega</math>

<math>(0)(1)(2)(0)(1)(2)=\omega^\omega\times2</math>

<math>(0)(1)(2)(1)=\omega^{\omega+1}</math>

<math>(0)(1)(2)(1)(1)=\omega^{\omega+2}</math>

<math>(0)(1)(2)(1)(1)(1)=\omega^{\omega+3}</math>

<math>(0)(1)(2)(1)(2)=\omega^{\omega\times2}</math>

<math>(0)(1)(2)(1)(2)(1)(2)=\omega^{\omega\times3}</math>

<math>(0)(1)(2)(2)=\omega^{\omega^2}</math>

<math>(0)(1)(2)(2)(1)=\omega^{\omega^2+1}</math>

<math>(0)(1)(2)(2)(1)(2)=\omega^{\omega^2+\omega}</math>

<math>(0)(1)(2)(2)(1)(2)(2)=\omega^{\omega^2\times2}</math>

<math>(0)(1)(2)(2)(1)(2)(2)(1)(2)(2)=\omega^{\omega^2\times3}</math>

<math>(0)(1)(2)(2)(2)=\omega^{\omega^3}</math>

<math>(0)(1)(2)(2)(2)(2)=\omega^{\omega^4}</math>

<math>(0)(1)(2)(3)=(0)(1)(2)(2)(2)(2)...=\omega^{\omega^\omega}</math>

<math>(0)(1)(2)(3)(4)=\omega^{\omega^{\omega^\omega}}</math>

<math>(0)(1)(2)(3)(4)(5)=\omega^{\omega^{\omega^{\omega^\omega}}}</math>

<math>(0)(1,1)=(0)(1)(2)(3)(4)(5)(6)...=\varepsilon_0</math>

==== 2：双行BMS ====
<math>(0)(1,1)=\varepsilon_0</math>

<math>(0)(1,1)(0)=\varepsilon_0+1</math>

<math>(0)(1,1)(0)(1)=\varepsilon_0+\omega</math>

<math>(0)(1,1)(0)(1,1)=\varepsilon_0\times2</math>

<math>(0)(1,1)(0)(1,1)(0)(1,1)=\varepsilon_0\times3</math>

<math>(0)(1,1)(1)=\varepsilon_0\times\omega</math>

<math>(0)(1,1)(1)(1)=\varepsilon_0\times\omega^2</math>

<math>(0)(1,1)(1)(2)=\varepsilon_0\times\omega^\omega</math>

<math>(0)(1,1)(1)(2)(3)=\varepsilon_0\times\omega^{\omega^\omega}</math>

<math>(0)(1,1)(1)(2,1)=\varepsilon_0^2</math>

<math>(0)(1,1)(1)(2,1)(1)(2)=\varepsilon_0^2\times\omega</math>

<math>(0)(1,1)(1)(2,1)(1)(2)(3)=\varepsilon_0^2\times\omega^\omega</math>

<math>(0)(1,1)(1)(2,1)(1)(2,1)=\varepsilon_0^3</math>

<math>(0)(1,1)(1)(2,1)(1)(2,1)(1)(2,1)=\varepsilon_0^4</math>

<math>(0)(1,1)(1)(2,1)(2)=\varepsilon_0^\omega</math>

<math>(0)(1,1)(1)(2,1)(2)(1)(2,1)=\varepsilon_0^{\omega+1}</math>

<math>(0)(1,1)(1)(2,1)(2)(1)(2,1)(1)(2,1)=\varepsilon_0^{\omega+2}</math>

<math>(0)(1,1)(1)(2,1)(2)(1)(2,1)(2)=\varepsilon_0^{\omega\times2}</math>

<math>(0)(1,1)(1)(2,1)(2)(1)(2,1)(2)(1)(2,1)(2)=\varepsilon_0^{\omega\times3}</math>

<math>(0)(1,1)(1)(2,1)(2)(2)=\varepsilon_0^{\omega^2}</math>

<math>(0)(1,1)(1)(2,1)(2)(3)=\varepsilon_0^{\omega^\omega}</math>

<math>(0)(1,1)(1)(2,1)(2)(3)(4)=\varepsilon_0^{\omega^{\omega^\omega}}</math>

<math>(0)(1,1)(1)(2,1)(2)(3,1)=\varepsilon_0^{\varepsilon_0}</math>

<math>(0)(1,1)(1)(2,1)(2)(3,1)(3)(4,1)=\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}</math>

<math>(0)(1,1)(1)(2,1)(2)(3,1)(3)(4,1)(4)(5,1)=\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}</math>

<math>(0)(1,1)(1,1)=\varepsilon_1</math>

<math>(0)(1,1)(1,1)(0)(1,1)(1,1)=\varepsilon_1\times2</math>

<math>(0)(1,1)(1,1)(1)=\varepsilon_1\times\omega</math>

<math>(0)(1,1)(1,1)(1)(2,1)=\varepsilon_1\times\varepsilon_0</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2)(3,1)=\varepsilon_1\times\varepsilon_0^2</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2)(3,1)(3)(4,1)=\varepsilon_1\times\varepsilon_0^{\varepsilon_0}</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)=\varepsilon_1^2</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)(2)=\varepsilon_1^\omega</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)=\varepsilon_1^{\varepsilon_0}</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)(3,1)=\varepsilon_1^{\varepsilon_1}</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)(3,1)(3)(4,1)(4,1)=\varepsilon_1^{\varepsilon_1^{\varepsilon_1}}</math>

<math>(0)(1,1)(1,1)(1,1)=\varepsilon_2</math>

<math>(0)(1,1)(1,1)(1,1)(1,1)=\varepsilon_3</math>

<math>(0)(1,1)(2)=\varepsilon_\omega</math>

<math>(0)(1,1)(2)(1)=\varepsilon_\omega\times\omega</math>

<math>(0)(1,1)(2)(1)(2,1)=\varepsilon_\omega\times\varepsilon_0</math>

<math>(0)(1,1)(2)(1)(2,1)(2,1)=\varepsilon_\omega\times\varepsilon_1</math>

<math>(0)(1,1)(2)(1)(2,1)(2,1)(2,1)=\varepsilon_\omega\times\varepsilon_2</math>

<math>(0)(1,1)(2)(1)(2,1)(2,1)(2,1)(2,1)=\varepsilon_\omega\times\varepsilon_3</math>

<math>(0)(1,1)(2)(1)(2,1)(3)=\varepsilon_\omega^2</math>

<math>(0)(1,1)(2)(1)(2,1)(3)(2)(3,1)(4)=\varepsilon_\omega^{\varepsilon_\omega}</math>

<math>(0)(1,1)(2)(1,1)=\varepsilon_{\omega+1}</math>

<math>(0)(1,1)(2)(1,1)(1,1)=\varepsilon_{\omega+2}</math>

<math>(0)(1,1)(2)(1,1)(1,1)(1,1)=\varepsilon_{\omega+3}</math>

<math>(0)(1,1)(2)(1,1)(2)=\varepsilon_{\omega\times2}</math>

<math>(0)(1,1)(2)(1,1)(2)(1,1)(2)=\varepsilon_{\omega\times3}</math>

<math>(0)(1,1)(2)(2)=\varepsilon_{\omega^2}</math>

<math>(0)(1,1)(2)(2)(2)=\varepsilon_{\omega^3}</math>

<math>(0)(1,1)(2)(3)=\varepsilon_{\omega^\omega}</math>

<math>(0)(1,1)(2)(3)(4)=\varepsilon_{\omega^{\omega^\omega}}</math>

<math>(0)(1,1)(2)(3,1)=\varepsilon_{\varepsilon_0}</math>

<math>(0)(1,1)(2)(3,1)(1,1)=\varepsilon_{\varepsilon_0+1}</math>

<math>(0)(1,1)(2)(3,1)(3,1)=\varepsilon_{\varepsilon_1}</math>

<math>(0)(1,1)(2)(3,1)(4)=\varepsilon_{\varepsilon_\omega}</math>

<math>(0)(1,1)(2)(3,1)(4)(5,1)=\varepsilon_{\varepsilon_{\varepsilon_0}}</math>

<math>(0)(1,1)(2)(3,1)(4)(5,1)(6)(7,1)=\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}</math>

<math>(0)(1,1)(2,1)=\zeta_0</math>

<math>(0)(1,1)(2,1)(1)(2,1)(3,1)=\zeta_0^2</math>

<math>(0)(1,1)(2,1)(1)(2,1)(3,1)(2)(3,1)(4,1)=\zeta_0^{\zeta_0}</math>

<math>(0)(1,1)(2,1)(1,1)=\varepsilon_{\zeta_0+1}</math>

<math>(0)(1,1)(2,1)(1,1)(1)(2,1)(3,1)(2,1)=\varepsilon_{\zeta_0+1}^2</math>

<math>(0)(1,1)(2,1)(1,1)(1,1)=\varepsilon_{\zeta_0+2}</math>

<math>(0)(1,1)(2,1)(1,1)(2)=\varepsilon_{\zeta_0+\omega}</math>

<math>(0)(1,1)(2,1)(1,1)(2)(3,1)=\varepsilon_{\zeta_0+\varepsilon_0}</math>

<math>(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)=\varepsilon_{\zeta_0\times2}</math>

<math>(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)=\varepsilon_{\varepsilon_{\zeta_0+1}}</math>

<math>(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)(4)(5,1)(6,1)(5,1)=\varepsilon_{\varepsilon_{\varepsilon_{\zeta+1}}}</math>

<math>(0)(1,1)(2,1)(1,1)(2,1)=\zeta_1</math>

<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)=\varepsilon_{\zeta_1+1}</math>

<math>(0)(1,1)(2,1)(1,1)(2)=\varepsilon_{\zeta_1+\omega}</math>

<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)=\zeta_2</math>

<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)=\zeta_3</math>

<math>(0)(1,1)(2,1)(2)=\zeta_\omega</math>

<math>(0)(1,1)(2,1)(2)(2)=\zeta_{\omega^2}</math>

<math>(0)(1,1)(2,1)(2)(3,1)=\zeta_{\varepsilon_0}</math>

<math>(0)(1,1)(2,1)(2)(3,1)(4,1)=\zeta_{\zeta_0}</math>

<math>(0)(1,1)(2,1)(2)(3,1)(4,1)(4)(5,1)(6,1)=\zeta_{\zeta_{\zeta_0}}</math>

<math>(0)(1,1)(2,1)(2,1)=\eta_0</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)=\varepsilon_{\eta_0+1}</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)=\zeta_{\eta_0+1}</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2)(3,1)(4,1)(4,1)(3,1)(4,1)=\zeta_{\zeta_{\eta_0+1}}</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)=\eta_1</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)=\eta_2</math>

<math>(0)(1,1)(2,1)(2,1)(2)=\eta_\omega</math>

<math>(0)(1,1)(2,1)(2,1)(2,1)=\varphi(4,0)=\psi(\Omega^3)</math>

<math>(0)(1,1)(2,1)(3)=\varphi(\omega,0)=\psi(\Omega^\omega)</math>

<math>(0)(1,1)(2,1)(3)(1,1)=\varphi(1,\varphi(\omega,0)+1)=\psi(\Omega^\omega+1)</math>

<math>(0)(1,1)(2,1)(3)(1,1)(2,1)=\varphi(2,\varphi(\omega,0)+1)=\psi(\Omega^\omega+\Omega)</math>

<math>(0)(1,1)(2,1)(3)(1,1)(2,1)(2,1)=\varphi(3,\varphi(\omega,0)+1)=\psi(\Omega^\omega+\Omega^2)</math>

<math>(0)(1,1)(2,1)(3)(1,1)(2,1)(3)=\varphi(\omega,1)=\psi(\Omega^\omega\times2)</math>

<math>(0)(1,1)(2,1)(3)(1,1)(2,1)(3)(1,1)(2,1)(3)=\varphi(\omega,2)=\psi(\Omega^\omega\times3)</math>

<math>(0)(1,1)(2,1)(3)(2)=\varphi(\omega,\omega)=\psi(\Omega^\omega\times\omega)</math>

<math>(0)(1,1)(2,1)(3)(2,1)=\varphi(\omega+1,0)=\psi(\Omega^{\omega+1})</math>

<math>(0)(1,1)(2,1)(3)(2,1)(2,1)=\varphi(\omega+2,0)=\psi(\Omega^{\omega+2})</math>

<math>(0)(1,1)(2,1)(3)(2,1)(2,1)(2,1)=\varphi(\omega+3,0)=\psi(\Omega^{\omega+3})</math>

<math>(0)(1,1)(2,1)(3)(2,1)(3)=\varphi(\omega\times2,0)=\psi(\Omega^{\omega\times2})</math>

<math>(0)(1,1)(2,1)(3)(2,1)(3)(2,1)(3)=\varphi(\omega\times3,0)=\psi(\Omega^{\omega\times3})</math>

<math>(0)(1,1)(2,1)(3)(3)=\varphi(\omega^2,0)=\psi(\Omega^{\omega^2})</math>

<math>(0)(1,1)(2,1)(3)(4,1)=\varphi(\varphi(1,0),0)=\psi(\Omega^{\psi(0)})</math>

<math>(0)(1,1)(2,1)(3)(4,1)(5,1)=\varphi(\varphi(2,0),0)=\psi(\Omega^{\psi(\Omega)})</math>

<math>(0)(1,1)(2,1)(3)(4,1)(5,1)(6)=\varphi(\varphi(\omega,0),0)=\psi(\Omega^{\psi(\Omega^\omega)})</math>

<math>(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1)(9)=\varphi(\varphi(\varphi(\omega,0),0),0)=\psi(\Omega^{\psi(\Omega^{\psi(\Omega^\omega)})})</math>

<math>(0)(1,1)(2,1)(3,1)=\varphi(1,0,0)=\Gamma_0=\psi(\Omega^\Omega)</math>

<math>(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)=\varphi(1,0,1)=\Gamma_1=\psi(\Omega^\Omega\times2)</math>

<math>(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)=\varphi(1,0,2)=\Gamma_2=\psi(\Omega^\Omega\times3)</math>

<math>(0)(1,1)(2,1)(3,1)(2)=\varphi(1,0,\omega)=\Gamma_\omega=\psi(\Omega^\Omega\times\omega)</math>

<math>(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)=\varphi(1,0,\varphi(1,0,0))=\Gamma_{\Gamma_0}=\psi(\Omega^\Omega\times\psi(\Omega^\Omega))</math>

<math>(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)(4)(5,1)(6,1)(7,1)=\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))=\Gamma_{\Gamma_{\Gamma_0}}=\psi(\Omega^\Omega\times\psi(\Omega^\Omega\times\psi(\Omega^\Omega)))</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)=\varphi(1,1,0)=\psi(\Omega^{\Omega+1})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(2,1)=\varphi(1,2,0)=\psi(\Omega^{\Omega+2})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(3)=\varphi(1,\omega,0)=\psi(\Omega^{\Omega+\omega})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1)(5,1)(6,1)(5,1)(6)=\varphi(1,\varphi(1,\omega,0),0)=\psi(\Omega^{\Omega+\psi(\Omega^{\Omega+\omega})})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(3,1)=\psi(2,0,0)=\psi(\Omega^{\Omega\times2})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)=\psi(3,0,0)=\psi(\Omega^{\Omega\times3})</math>

<math>(0)(1,1)(2,1)(3,1)(3)=\varphi(\omega,0,0)=\psi(\Omega^{\Omega\times\omega})</math>

<math>(0)(1,1)(2,1)(3,1)(3,1)=\varphi(1,0,0,0)=\psi(\Omega^{\Omega^2})</math>

<math>(0)(1,1)(2,1)(3,1)(3,1)(3,1)=\varphi(1,0,0,0,0)=\psi(\Omega^{\Omega^3})</math>

<math>(0)(1,1)(2,1)(3,1)(4)=\psi(\Omega^{\Omega^\omega})</math>

<math>(0)(1,1)(2,1)(3,1)(4)(5,1)=\psi(\Omega^{\Omega^{\psi(0)}})</math>

<math>(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)=\psi(\Omega^{\Omega^{\psi(\Omega)}})</math>

<math>(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)=\psi(\Omega^{\Omega^{\psi(\Omega^\Omega)}})</math>

<math>(0)(1,1)(2,1)(3,1)(4,1)=\psi(\Omega^{\Omega^\Omega})</math>

<math>(0)(1,1)(2,1)(3,1)(4,1)(5,1)=\psi(\Omega^{\Omega^{\Omega^\Omega}})</math>

<math>(0)(1,1)(2,2)=\psi(\psi_1(0))</math>
[[分类:分析]]

序数表

2025-07-05T13:29:03Z

Optimism：

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)=\psi(\Omega^{\Omega^2})</math>
|<math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(3,1)</math>
|-
| [[SVO]]|| Small Veblen Ordinal || <math>\varphi(1</math>@<math>\omega)=\psi(\Omega^{\Omega^{\omega}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,0)</math>
|-
| [[LVO]]|| Large Veblen Ordinal || <math>\varphi(1</math>@<math>(1,0))=\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]]|| Extended 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
|<math>\psi(I(1,0,0))=\psi(M^M)</math>
|<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>
|-
| [[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(\lambda\alpha.(\Omega_{\alpha+1})-\Pi_1)=\psi(a_2^{\Omega_{a+1}}+\psi_a(a_2^{\Omega_{a+1}})\times\omega)</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(\lambda\alpha.(\Omega_{\alpha+2})-\Pi_{1})=\psi(\Omega_{a_2+1}+\psi_a(\Omega_{a_2+1})\times\omega)</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}=\psi(\Omega_{a_2+1}\times\omega)</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})=\psi(\Omega_{a_2+1}\times a_2)</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=\psi(a_3) </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=\psi(a_4) </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>\psi(pfec.\omega-\pi-\Pi_{0})=\psi(a_\omega) </math>
|<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(\omega-\text{proj.})=\psi(\sigma S\times \omega)=\psi(H^\omega)</math>
|<math>\mathrm{BMS}(0)(1,1,1,1)</math>
|-
| [[TSSO]]|| Trio Sequence System Ordinal || <math>\psi(\omega-\text{proj.})=\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.})=\psi(\sigma S\times S) </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>||
|}

序数表

2025-07-05T13:27:03Z

Optimism：/* 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 ==
{| 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)=\psi(\Omega^{\Omega^2})</math>
|<math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(3,1)</math>
|-
| [[SVO]]|| Small Veblen Ordinal || <math>\varphi(1</math>@<math>\omega)=\psi(\Omega^{\Omega^{\omega}})</math>|| <math>\mathrm{BMS}(0,0)(1,1)(2,1)(3,1)(4,0)</math>
|-
| [[LVO]]|| Large Veblen Ordinal || <math>\varphi(1</math>@<math>(1,0))=\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]]|| Extended 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
|<math>\psi(I(1,0,0))=\psi(M^M)</math>
|<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>
|-
| [[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(\lambda\alpha.(\Omega_{\alpha+1})-\Pi_1)=\psi(a_2^{\Omega_{a+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(\lambda\alpha.(\Omega_{\alpha+2})-\Pi_{1})=\psi(\Omega_{a_2+1}+\psi_a(\Omega_{a_2+1})\times\omega)</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}=\psi(\Omega_{a_2+1}\times\omega)</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})=\psi(\Omega_{a_2+1}\times a_2)</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=\psi(a_3) </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=\psi(a_4) </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>\psi(pfec.\omega-\pi-\Pi_{0})=\psi(a_\omega) </math>
|<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(\omega-\text{proj.})=\psi(\sigma S\times \omega)=\psi(H^\omega)</math>
|<math>\mathrm{BMS}(0)(1,1,1,1)</math>
|-
| [[TSSO]]|| Trio Sequence System Ordinal || <math>\psi(\omega-\text{proj.})=\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.})=\psi(\sigma S\times S) </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>||
|}

首页

2025-07-03T16:59:33Z

Optimism：撤销Optimism（讨论）的修订版本682

<div class="googology-home">
<h1 class="main-title"><span class="mw-headline" id="Googology_Wiki">Googology Wiki</span></h1>
<div class="tagline">Mapping the Universe of Monumental Numbers.</div>

<div class="fixed-grid">
<div class="fixed-btn">
[[:Category:记号|<span class="btn-text blue">记号</span>]]
</div>
<div class="fixed-btn">
[[:Category:分析|<span class="btn-text blue">分析</span>]]
</div>
<div class="fixed-btn">
[[:Category:入门|<span class="btn-text blue">入门</span>]]
</div>
<div class="fixed-btn">
[[:序数表|<span class="btn-text blue">序数表</span>]]
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[[:前沿研究|<span class="btn-text blue">前沿</span>]]
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[[:非递归|<span class="btn-text blue">非递归</span>]]
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[[:大数简史|<span class="btn-text blue">大数简史</span>]]
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[[:Googology_社区|<span class="btn-text blue">社区</span>]]
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[[:记号展开器|<span class="btn-text blue">展开器</span>]]
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[[:证明|<span class="btn-text blue">证明</span>]]
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<div class="home-card">
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<p>Googology Wiki 欢迎你的编辑！</p>

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首页

2025-07-03T16:58:24Z

Optimism：

<div class="googology-home">
<h1 class="main-title"><span class="mw-headline" id="Googology_Wiki">Googology Wiki</span></h1>
<div class="tagline">Mapping the Universe of Monumental Numbers.</div>

<div class="fixed-grid">
<div class="fixed-btn">
[[:Category:记号|<span class="btn-text blue">记号</span>]]
</div>
<div class="fixed-btn">
[[:Category:分析|<span class="btn-text blue">分析</span>]]
</div>
<div class="fixed-btn">
[[:Category:入门|<span class="btn-text blue">入门</span>]]
</div>
<div class="fixed-btn">
[[:序数表|<span class="btn-text blue">序数表</span>]]
</div>
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[[:前沿研究|<span class="btn-text blue">前沿</span>]]
</div>
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[[:非递归|<span class="btn-text blue">非递归</span>]]
</div>
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[[:大数简史|<span class="btn-text blue">大数简史</span>]]
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[[:Googology_社区|<span class="btn-text blue">社区</span>]]
</div>
<div class="fixed-btn">
[[:Category:记号展开器|<span class="btn-text blue">展开器</span>]]
</div>
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[[:证明|<span class="btn-text blue">证明</span>]]
</div>
</div>


<div class="wiki-stats-container">
<div class="home-card">
<h2 style="margin-top: 0">内容统计</h2>
<p>Googology Wiki 欢迎你的编辑！</p>

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记号展开器

2025-07-03T16:55:17Z

Optimism：创建页面，内容为“[https://hypcos.github.io/notation-explorer/ 此处]包含了许多常见记号的展开器”

[https://hypcos.github.io/notation-explorer/ 此处]包含了许多常见记号的展开器

首页

2025-07-03T16:50:51Z

Optimism：

序数表

2025-07-03T16:47:24Z

Optimism：

序数表

2025-07-03T16:46:12Z

Optimism：/* 序数表 */

序数表

2025-07-03T16:26:11Z

Optimism：/* 序数表 */

高德纳箭头

2025-07-03T16:22:22Z

Optimism：糕糕

'''高德纳箭头(Knuth's arrow notation, 亦称"上箭头记号")'''，一种满足'''右结合律'''的二元运算。其定义如下：

* <math>a \uparrow b = a^{b}</math>
* <math>a \uparrow^{c} 0 = 1</math>
* <math>a \uparrow^{c+1} (b+1) = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

其中，<math>a,c</math>均为'''正整数，'''<math>b</math>为'''自然数'''，<math>a \uparrow^{c} b = a\ \underbrace{ \uparrow\uparrow\cdots\uparrow }_{c}\ b</math>.

在计算高德纳箭头时，如无括号，按照从右往左的顺序计算，即：

<math>a \uparrow^{m} b\uparrow^{n} c=a \uparrow^{m} (b\uparrow^{n} c)</math>

若将高德纳箭头的右结合律更替为左结合律，其余定义不变，将得到[[下箭号表示法|下箭头记号]]。

==== 性质 ====

高德纳箭头有如下性质：

===== '''展开''' =====
<math>n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>

===== 恒等律 =====

<math>2 \uparrow^{c+1} 2 = 2\uparrow^c2=.. .=2\uparrow2=4</math>

<math>1 \uparrow^{c+1} b = 1\uparrow^c1\uparrow^{c+1}(b-1)=1\uparrow^cb_2=.. .=1\uparrow b_k=1</math>

<math>a \uparrow^{c+1} 1 = a \uparrow^{c}a\uparrow^{c+1}0 =a\uparrow^{c} 1=...=a\uparrow1=a</math>

===== 增长率 =====

高德纳箭头的[[FGH]]增长率为<math>\omega</math>，特别地，

<math>a \uparrow^{c} b \approx f_{c+1}(b) </math>，

该推论可通过审视以下三组式子得到：

<math>a\uparrow b\approx f_2(b)=2^b\times b</math>

<math>a \uparrow^{c+1} (b+1) = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

<math>f_{c+1}(b+1)=f_{c}^{b+1}(b+1)=f_c(f_c^b(b+1))\approx f_c(f_c^b(b))=f_c(f_{c+1}(b))</math>

===== 超运算 =====

高德纳箭头是目前已被广泛认可、基本采用<ref>曹知秋. 大数理论: Vol.1[EB/OL]. (2025-05-16) [2025-07-02]: 35-36.</ref>的超运算记号。

若定义后继运算的运算等级为<math>0</math>，那么 <math>n</math> 个高德纳箭头的运算等级为 <math>n+2</math>

==== 历史 ====

高德纳箭头是由 <math>\mathrm{Donald\ Ervin\ Knuth}</math> 在1976年发明的大数记号<ref>Donald E. Knuth, ''Mathematics and Computer Science: Coping with Finiteness'', Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations, Science 194, pp. 1235--1242, 1976.https://cse-robotics.engr.tamu.edu/dshell/cs625/finiteness.pdf</ref>，曾在1977年被 <math> \mathrm{Martin\ Gardner} </math> 用于递归地定义[[葛立恒数]]。<ref>GARDNER M. Mathematical games[J]. Scientific American, 1977, 237(3): 28-38.https://raw.githubusercontent.com/AllenDowney/ModSimPy/master/papers/scientific_american_nov_77.pdf</ref>

而<math>\mathrm{Ronald\ Graham}</math>本人并未在论文中使用高德纳箭头或超运算来估计 [[葛立恒数#葛立恒问题|Graham问题]] 的上界，而是使用了类似 [[ACKERMANN函数]] 的递归函数 <math>F(m,n)</math>，和分别近似为 <math>2 \uparrow\uparrow n,2 \uparrow\uparrow\uparrow n</math> 的函数<math>TOWER(n),WOW(n)</math>。<ref>GRAHAM R L, ROTHSCHILD B L. Ramsey’s theorem for ff-parameter sets[J]. Transactions of the American Mathematical Society, 1971, 159: 257-292. https://www.ams.org/journals/tran/1971-159-00/S0002-9947-1971-0284352-8/S0002-9947-1971-0284352-8.pdf
</ref><ref>GRAHAM R L, ROTHSCHILD B L, SPENCER J H. Ramsey theory: Vol. 20[M]. John
Wiley & Sons, 1991.
https://people.dm.unipi.it/dinasso/ULTRABIBLIO/Graham_Rothschild_Spencer%20-%20Ramsey%20Theory%20(2nd%20edition).pdf</ref>

==== '''直观理解''' ====
高德纳箭头本质上是一种高级运算“折叠”低级运算的记号。

后继是最基础的运算，表现为 <math>n+1</math>.

在<math>n+m</math>中，运算 <math>+</math> 折叠了对 <math>n</math> 的 <math>m</math> 次后继运算，即

<math>\ n+m=n+\underbrace{1+1+\cdots+1}_{\text{m个1}}</math>.

在<math>n\times m</math>中，运算 <math>\times</math> 折叠了对 <math>n</math> 的 <math>m</math> 次 <math>+</math> 运算，即

<math>\ n\times m=\underbrace{n+n+\cdots+n}_{\text{m个n}}</math>.

在 <math>n^{m}</math> ( <math>n\uparrow m</math> ) 中，运算<math>\uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\times</math> 运算，即

<math>\ n^{m}=\underbrace{n\times n\times \cdots \times n}_{\text{m个n}}</math>.

在 <math>{}^m\!n</math> ( <math>n\uparrow \uparrow m</math> ) 中，运算<math>\uparrow \uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow</math> 运算，即

<math>\ n\uparrow \uparrow m=\underbrace{n\uparrow n\uparrow \cdots \uparrow n}_{\text{m个n}}=\underbrace{n^{n^{n^{\cdots}}}}_{\text{m个n}}</math>. （注意是右结合）

以此类推。最终，我们得到了高德纳箭头的形式化定义：

在<math>n \uparrow^{k} m</math>中，运算<math>\uparrow^{k}</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow^{k-1}</math> 运算，即

<math>\ n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>
==== 计算示例 ====

<math>\begin{align} 3 \uparrow\uparrow\uparrow 3\ & = 3\uparrow\uparrow3\uparrow\uparrow 3 \\ & = 3\uparrow\uparrow (3\uparrow 3\uparrow 3) \\ & = 3\uparrow\uparrow(3\uparrow27) \\ & = 3\uparrow\uparrow 7625597484987 \\ & = \underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \end{align}</math>

<math>\begin{align} 3 \uparrow\uparrow\uparrow\uparrow 4\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3 \\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \\ & = 3\uparrow\uparrow\uparrow(\ \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}\ ) \\ & = \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}}\end{align}</math>
==== 参考资料 ====
<references />
[[分类:记号]]
[[分类:入门]]

高德纳箭头

2025-07-03T16:20:12Z

Optimism：

'''高德纳箭头(Knuth's arrow notation, 亦称"上箭头记号")'''，一种满足'''右结合律'''的二元运算。其定义如下：

* <math>a \uparrow b = a^{b}</math>
* <math>a \uparrow^{c} 0 = 1</math>
* <math>a \uparrow^{c+1} b+1 = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

其中，<math>a,c</math>均为'''正整数，'''<math>b</math>为'''自然数'''，<math>a \uparrow^{c} b = a\ \underbrace{ \uparrow\uparrow\cdots\uparrow }_{c}\ b</math>.

在计算高德纳箭头时，如无括号，按照从右往左的顺序计算，即：

<math>a \uparrow^{m} b\uparrow^{n} c=a \uparrow^{m} (b\uparrow^{n} c)</math>

若将高德纳箭头的右结合律更替为左结合律，其余定义不变，将得到[[下箭号表示法|下箭头记号]]。

==== 性质 ====

高德纳箭头有如下性质：

===== '''展开''' =====
<math>n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>

===== 恒等律 =====

<math>2 \uparrow^{c+1} 2 = 2\uparrow^c2=.. .=2\uparrow2=4</math>

<math>1 \uparrow^{c+1} b = 1\uparrow^c1\uparrow^{c+1}(b-1)=1\uparrow^cb_2=.. .=1\uparrow b_k=1</math>

<math>a \uparrow^{c+1} 1 = a \uparrow^{c}a\uparrow^{c+1}0 =a\uparrow^{c} 1=...=a\uparrow1=a</math>

===== 增长率 =====

高德纳箭头的[[FGH]]增长率为<math>\omega</math>，特别地，

<math>a \uparrow^{c} b \approx f_{c+1}(b) </math>，

该推论可通过审视以下三组式子得到：

<math>a\uparrow b\approx f_2(b)=2^b\times b</math>

<math>a \uparrow^{c+1} (b+1) = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

<math>f_{c+1}(b+1)=f_{c}^{b+1}(b+1)=f_c(f_c^b(b+1))\approx f_c(f_c^b(b))=f_c(f_{c+1}(b))</math>

===== 超运算 =====

高德纳箭头是目前已被广泛认可、基本采用<ref>曹知秋. 大数理论: Vol.1[EB/OL]. (2025-05-16) [2025-07-02]: 35-36.</ref>的超运算记号。

若定义后继运算的运算等级为<math>0</math>，那么 <math>n</math> 个高德纳箭头的运算等级为 <math>n+2</math>

==== 历史 ====

高德纳箭头是由 <math>\mathrm{Donald\ Ervin\ Knuth}</math> 在1976年发明的大数记号<ref>Donald E. Knuth, ''Mathematics and Computer Science: Coping with Finiteness'', Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations, Science 194, pp. 1235--1242, 1976.https://cse-robotics.engr.tamu.edu/dshell/cs625/finiteness.pdf</ref>，曾在1977年被 <math> \mathrm{Martin\ Gardner} </math> 用于递归地定义[[葛立恒数]]。<ref>GARDNER M. Mathematical games[J]. Scientific American, 1977, 237(3): 28-38.https://raw.githubusercontent.com/AllenDowney/ModSimPy/master/papers/scientific_american_nov_77.pdf</ref>

而<math>\mathrm{Ronald\ Graham}</math>本人并未在论文中使用高德纳箭头或超运算来估计 [[葛立恒数#葛立恒问题|Graham问题]] 的上界，而是使用了类似 [[ACKERMANN函数]] 的递归函数 <math>F(m,n)</math>，和分别近似为 <math>2 \uparrow\uparrow n,2 \uparrow\uparrow\uparrow n</math> 的函数<math>TOWER(n),WOW(n)</math>。<ref>GRAHAM R L, ROTHSCHILD B L. Ramsey’s theorem for ff-parameter sets[J]. Transactions of the American Mathematical Society, 1971, 159: 257-292. https://www.ams.org/journals/tran/1971-159-00/S0002-9947-1971-0284352-8/S0002-9947-1971-0284352-8.pdf
</ref><ref>GRAHAM R L, ROTHSCHILD B L, SPENCER J H. Ramsey theory: Vol. 20[M]. John
Wiley & Sons, 1991.
https://people.dm.unipi.it/dinasso/ULTRABIBLIO/Graham_Rothschild_Spencer%20-%20Ramsey%20Theory%20(2nd%20edition).pdf</ref>

==== '''直观理解''' ====
高德纳箭头本质上是一种高级运算“折叠”低级运算的记号。

后继是最基础的运算，表现为 <math>n+1</math>.

在<math>n+m</math>中，运算 <math>+</math> 折叠了对 <math>n</math> 的 <math>m</math> 次后继运算，即

<math>\ n+m=n+\underbrace{1+1+\cdots+1}_{\text{m个1}}</math>.

在<math>n\times m</math>中，运算 <math>\times</math> 折叠了对 <math>n</math> 的 <math>m</math> 次 <math>+</math> 运算，即

<math>\ n\times m=\underbrace{n+n+\cdots+n}_{\text{m个n}}</math>.

在 <math>n^{m}</math> ( <math>n\uparrow m</math> ) 中，运算<math>\uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\times</math> 运算，即

<math>\ n^{m}=\underbrace{n\times n\times \cdots \times n}_{\text{m个n}}</math>.

在 <math>{}^m\!n</math> ( <math>n\uparrow \uparrow m</math> ) 中，运算<math>\uparrow \uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow</math> 运算，即

<math>\ n\uparrow \uparrow m=\underbrace{n\uparrow n\uparrow \cdots \uparrow n}_{\text{m个n}}=\underbrace{n^{n^{n^{\cdots}}}}_{\text{m个n}}</math>. （注意是右结合）

以此类推。最终，我们得到了高德纳箭头的形式化定义：

在<math>n \uparrow^{k} m</math>中，运算<math>\uparrow^{k}</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow^{k-1}</math> 运算，即

<math>\ n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>
==== 计算示例 ====

<math>\begin{align} 3 \uparrow\uparrow\uparrow 3\ & = 3\uparrow\uparrow3\uparrow\uparrow 3 \\ & = 3\uparrow\uparrow (3\uparrow 3\uparrow 3) \\ & = 3\uparrow\uparrow(3\uparrow27) \\ & = 3\uparrow\uparrow 7625597484987 \\ & = \underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \end{align}</math>

<math>\begin{align} 3 \uparrow\uparrow\uparrow\uparrow 4\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3 \\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \\ & = 3\uparrow\uparrow\uparrow(\ \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}\ ) \\ & = \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}}\end{align}</math>
==== 参考资料 ====
<references />
[[分类:记号]]
[[分类:入门]]

高德纳箭头

2025-07-03T16:09:54Z

Optimism：

'''高德纳箭头(Knuth's arrow notation, 亦称"上箭头记号")'''，一种满足'''右结合律'''的二元运算。其定义如下：

* <math>a \uparrow b = a^{b}</math>
* <math>a \uparrow^{c} 0 = 1</math>
* <math>a \uparrow^{c+1} b+1 = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

其中，<math>a,c</math>均为'''正整数，'''<math>b</math>为'''自然数'''，<math>a \uparrow^{c} b = a\ \underbrace{ \uparrow\uparrow\cdots\uparrow }_{c}\ b</math>.

在计算高德纳箭头时，如无括号，按照从右往左的顺序计算，即：

<math>a \uparrow^{m} b\uparrow^{n} c=a \uparrow^{m} (b\uparrow^{n} c)</math>

若将高德纳箭头的右结合律更替为左结合律，其余定义不变，将得到[[下箭号表示法|下箭头记号]]。

==== 性质 ====

高德纳箭头有如下性质：

'''展开'''

<math>n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>

===== 恒等律 =====

<math>2 \uparrow^{c+1} 2 = 2\uparrow^c2=.. .=2\uparrow2=4</math>

<math>1 \uparrow^{c+1} b = 1\uparrow^c1\uparrow^{c+1}(b-1)=1\uparrow^cb_2=.. .=1\uparrow b_k=1</math>

<math>a \uparrow^{c+1} 1 = a \uparrow^{c}a\uparrow^{c+1}0 =a\uparrow^{c} 1=...=a\uparrow1=a</math>

===== 增长率 =====

高德纳箭头的[[FGH]]增长率为<math>\omega</math>，特别地，

<math>a \uparrow^{c} b \approx f_{c+1}(b) </math>，

该推论可通过审视以下三组式子得到：

<math>a\uparrow b\approx f_2(b)=2^b\times b</math>

<math>a \uparrow^{c+1} (b+1) = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

<math>f_{c+1}(b+1)=f_{c}^{b+1}(b+1)=f_c(f_c^b(b+1))\approx f_c(f_c^b(b))=f_c(f_{c+1}(b))</math>

===== 超运算 =====

高德纳箭头是目前已被广泛认可、基本采用<ref>曹知秋. 大数理论: Vol.1[EB/OL]. (2025-05-16) [2025-07-02]: 35-36.</ref>的超运算记号。

若定义后继运算的运算等级为<math>0</math>，那么 <math>n</math> 个高德纳箭头的运算等级为 <math>n+2</math>

==== 历史 ====

高德纳箭头是由 <math>\mathrm{Donald\ Ervin\ Knuth}</math> 在1976年发明的大数记号<ref>Donald E. Knuth, ''Mathematics and Computer Science: Coping with Finiteness'', Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations, Science 194, pp. 1235--1242, 1976.https://cse-robotics.engr.tamu.edu/dshell/cs625/finiteness.pdf</ref>，曾在1977年被 <math> \mathrm{Martin\ Gardner} </math> 用于递归地定义[[葛立恒数]]。<ref>GARDNER M. Mathematical games[J]. Scientific American, 1977, 237(3): 28-38.https://raw.githubusercontent.com/AllenDowney/ModSimPy/master/papers/scientific_american_nov_77.pdf</ref>

而<math>\mathrm{Ronald\ Graham}</math>本人并未在论文中使用高德纳箭头或超运算来估计 [[葛立恒数#葛立恒问题|Graham问题]] 的上界，而是使用了类似 [[ACKERMANN函数]] 的递归函数 <math>F(m,n)</math>，和分别近似为 <math>2 \uparrow\uparrow n,2 \uparrow\uparrow\uparrow n</math> 的函数<math>TOWER(n),WOW(n)</math>。<ref>GRAHAM R L, ROTHSCHILD B L. Ramsey’s theorem for ff-parameter sets[J]. Transactions of the American Mathematical Society, 1971, 159: 257-292. https://www.ams.org/journals/tran/1971-159-00/S0002-9947-1971-0284352-8/S0002-9947-1971-0284352-8.pdf
</ref><ref>GRAHAM R L, ROTHSCHILD B L, SPENCER J H. Ramsey theory: Vol. 20[M]. John
Wiley & Sons, 1991.
https://people.dm.unipi.it/dinasso/ULTRABIBLIO/Graham_Rothschild_Spencer%20-%20Ramsey%20Theory%20(2nd%20edition).pdf</ref>

'''直观理解'''

高德纳箭头本质上是一种高级运算“折叠”低级运算的记号。

后继是最基础的运算，表现为 <math>n+1</math>.

在<math>n+m</math>中，运算 <math>+</math> 折叠了对 <math>n</math> 的 <math>m</math> 次后继运算，即

<math>\ n+m=n+\underbrace{1+1+\cdots+1}_{\text{m个1}}</math>.

在<math>n\times m</math>中，运算 <math>\times</math> 折叠了对 <math>n</math> 的 <math>m</math> 次 <math>+</math> 运算，即

<math>\ n\times m=\underbrace{n+n+\cdots+n}_{\text{m个n}}</math>.

在 <math>n^{m}</math> ( <math>n\uparrow m</math> ) 中，运算<math>\uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\times</math> 运算，即

<math>\ n^{m}=\underbrace{n\times n\times \cdots \times n}_{\text{m个n}}</math>.

在 <math>{}^m\!n</math> ( <math>n\uparrow \uparrow m</math> ) 中，运算<math>\uparrow \uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow</math> 运算，即

<math>\ n\uparrow \uparrow m=\underbrace{n\uparrow n\uparrow \cdots \uparrow n}_{\text{m个n}}=\underbrace{n^{n^{n^{\cdots}}}}_{\text{m个n}}</math>. （注意是右结合）

以此类推。最终，我们得到了高德纳箭头的形式化定义：

在<math>n \uparrow^{k} m</math>中，运算<math>\uparrow^{k}</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow^{k-1}</math> 运算，即

<math>\ n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>
==== 计算示例 ====

<math>\begin{align} 3 \uparrow\uparrow\uparrow 3\ & = 3\uparrow\uparrow3\uparrow\uparrow 3 \\ & = 3\uparrow\uparrow (3\uparrow 3\uparrow 3) \\ & = 3\uparrow\uparrow(3\uparrow27) \\ & = 3\uparrow\uparrow 7625597484987 \\ & = \underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \end{align}</math>

<math>\begin{align} 3 \uparrow\uparrow\uparrow\uparrow 4\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3 \\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \\ & = 3\uparrow\uparrow\uparrow(\ \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}\ ) \\ & = \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}}\end{align}</math>
==== 参考资料 ====
<references />
[[分类:记号]]
[[分类:入门]]

序数表

2025-07-03T14:05:25Z

Optimism：/* 序数表 */

讨论:首页

2025-07-03T13:54:15Z

Optimism：/* 果糕糕 */ 新章节

讨论:首页

2025-07-03T13:53:58Z

Optimism：

高德纳箭头

2025-07-03T13:33:27Z

Optimism：/* 增长率 */

'''高德纳箭头(Knuth's arrow notation, 亦称"上箭头记号")'''，一种满足'''右结合律'''的二元运算。其定义如下：

* <math>a \uparrow b = a^{b}</math>
* <math>a \uparrow^{c} 1 = a</math>
* <math>a \uparrow^{c+1} b+1 = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

其中，<math>a,c</math>均为'''正整数，'''<math>b</math>为'''自然数'''，<math>a \uparrow^{c} b = a\ \underbrace{ \uparrow\uparrow\cdots\uparrow }_{c}\ b</math>.

==== 性质 ====

高德纳箭头有如下性质：

===== 右结合律 =====

<math>a \uparrow^{c} b\uparrow^{c} c=a \uparrow^{c} (b\uparrow^{c} c) \neq (a \uparrow^{c} b)\uparrow^{c} c </math>

若将高德纳箭头的右结合律更替为左结合律，其余定义不变，将得到[[下箭号表示法|下箭头记号]]。

===== 恒等律 =====

<math>2 \uparrow^{c+1} 2 = 2 \uparrow^{c} 2 = 4</math>

<math>1 \uparrow^{c+1} b = 1 \uparrow^{c} b = 1</math>

<math>a \uparrow^{c+1} 0 = a \uparrow^{c} 0 = 1</math>

===== 增长率 =====

高德纳箭头的[[FGH]]增长率为<math>\omega</math>，特别地，

<math>a \uparrow^{c} b \approx f_{c+1}(b) </math>，

该推论可通过审视以下三组式子得到：

<math>a\uparrow b\approx f_2(b)=2^b\times b</math>

<math>a \uparrow^{c+1} (b+1) = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

<math>f_{c+1}(b+1)=f_{c}^{b+1}(b+1)=f_c(f_c^b(b+1))\approx f_c(f_c^b(b))=f_c(f_{c+1}(b))</math>

===== 超运算 =====

高德纳箭头是目前已被广泛认可、基本采用<ref>曹知秋. 大数理论: Vol.1[EB/OL]. (2025-05-16) [2025-07-02]: 35-36.</ref>的超运算记号。

若定义后继运算的运算等级为<math>0</math>，那么 <math>n</math> 个高德纳箭头的运算等级为 <math>n+2</math>

==== 历史 ====

高德纳箭头是由 <math>\mathrm{Donald\ Ervin\ Knuth}</math> 在1976年发明的大数记号<ref>Donald E. Knuth, ''Mathematics and Computer Science: Coping with Finiteness'', Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations, Science 194, pp. 1235--1242, 1976.https://cse-robotics.engr.tamu.edu/dshell/cs625/finiteness.pdf</ref>，曾在1977年被 <math> \mathrm{Martin\ Gardner} </math> 用于递归地定义[[葛立恒数]]。<ref>GARDNER M. Mathematical games[J]. Scientific American, 1977, 237(3): 28-38.https://raw.githubusercontent.com/AllenDowney/ModSimPy/master/papers/scientific_american_nov_77.pdf</ref>

而<math>\mathrm{Ronald\ Graham}</math>本人并未在论文中使用高德纳箭头或超运算来估计 [[葛立恒数#葛立恒问题|Graham问题]] 的上界，而是使用了类似 [[ACKERMANN函数]] 的递归函数 <math>F(m,n)</math>，和分别近似为 <math>2 \uparrow\uparrow n,2 \uparrow\uparrow\uparrow n</math> 的函数<math>TOWER(n),WOW(n)</math>。<ref>GRAHAM R L, ROTHSCHILD B L. Ramsey’s theorem for ff-parameter sets[J]. Transactions of the American Mathematical Society, 1971, 159: 257-292. https://www.ams.org/journals/tran/1971-159-00/S0002-9947-1971-0284352-8/S0002-9947-1971-0284352-8.pdf
</ref><ref>GRAHAM R L, ROTHSCHILD B L, SPENCER J H. Ramsey theory: Vol. 20[M]. John
Wiley & Sons, 1991.
https://people.dm.unipi.it/dinasso/ULTRABIBLIO/Graham_Rothschild_Spencer%20-%20Ramsey%20Theory%20(2nd%20edition).pdf</ref>
==== 形式化定义 ====

<math>n \uparrow m = n^{m}</math>

<math>n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>

该定义可通过以下分析与推理得到：

高德纳箭头本质上是一种高级运算“折叠”低级运算的记号。

后继是最基础的运算，表现为 <math>n+1</math>.

在<math>n+m</math>中，运算 <math>+</math> 折叠了对 <math>n</math> 的 <math>m</math> 次后继运算，即

<math>\ n+m=n+\underbrace{1+1+\cdots+1}_{\text{m个1}}</math>.

在<math>n\times m</math>中，运算 <math>\times</math> 折叠了对 <math>n</math> 的 <math>m</math> 次 <math>+</math> 运算，即

<math>\ n\times m=\underbrace{n+n+\cdots+n}_{\text{m个n}}</math>.

在 <math>n^{m}</math> ( <math>n\uparrow m</math> ) 中，运算<math>\uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\times</math> 运算，即

<math>\ n^{m}=\underbrace{n\times n\times \cdots \times n}_{\text{m个n}}</math>.

在 <math>{}^m\!n</math> ( <math>n\uparrow \uparrow m</math> ) 中，运算<math>\uparrow \uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow</math> 运算，即

<math>\ n\uparrow \uparrow m=\underbrace{n\uparrow n\uparrow \cdots \uparrow n}_{\text{m个n}}=\underbrace{n^{n^{n^{\cdots}}}}_{\text{m个n}}</math>. （注意是右结合）

以此类推。最终，我们得到了高德纳箭头的形式化定义：

在<math>n \uparrow^{k} m</math>中，运算<math>\uparrow^{k}</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow^{k-1}</math> 运算，即

<math>\ n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>
==== 计算示例 ====

<math>\begin{align} 3 \uparrow\uparrow\uparrow 3\ & = 3\uparrow\uparrow3\uparrow\uparrow 3 \\ & = 3\uparrow\uparrow (3\uparrow 3\uparrow 3) \\ & = 3\uparrow\uparrow(3\uparrow27) \\ & = 3\uparrow\uparrow 7625597484987 \\ & = \underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \end{align}</math>

<math>\begin{align} 3 \uparrow\uparrow\uparrow\uparrow 4\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3 \\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \\ & = 3\uparrow\uparrow\uparrow(\ \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}\ ) \\ & = \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}}\end{align}</math>
==== 参考资料 ====
<references />
[[分类:记号]]
[[分类:入门]]

高德纳箭头

2025-07-03T13:23:31Z

Optimism：/* 增长率 */

'''高德纳箭头(Knuth's arrow notation, 亦称"上箭头记号")'''，一种满足'''右结合律'''的二元运算。其定义如下：

* <math>a \uparrow b = a^{b}</math>
* <math>a \uparrow^{c} 1 = a</math>
* <math>a \uparrow^{c+1} b+1 = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

其中，<math>a,c</math>均为'''正整数，'''<math>b</math>为'''自然数'''，<math>a \uparrow^{c} b = a\ \underbrace{ \uparrow\uparrow\cdots\uparrow }_{c}\ b</math>.

==== 性质 ====

高德纳箭头有如下性质：

===== 右结合律 =====

<math>a \uparrow^{c} b\uparrow^{c} c=a \uparrow^{c} (b\uparrow^{c} c) \neq (a \uparrow^{c} b)\uparrow^{c} c </math>

若将高德纳箭头的右结合律更替为左结合律，其余定义不变，将得到[[下箭号表示法|下箭头记号]]。

===== 恒等律 =====

<math>2 \uparrow^{c+1} 2 = 2 \uparrow^{c} 2 = 4</math>

<math>1 \uparrow^{c+1} b = 1 \uparrow^{c} b = 1</math>

<math>a \uparrow^{c+1} 0 = a \uparrow^{c} 0 = 1</math>

===== 增长率 =====

高德纳箭头的[[FGH]]增长率为<math>\omega</math>，特别地，

<math>a \uparrow^{c} b \approx f_{c+1}(b) </math>，

该推论可通过审视以下两组等式得到：

<math>a \uparrow^{c+1} (b+1) = a \uparrow^{c} ( a \uparrow^{c+1} b)</math>

<math>f_{c+1}(b+1)=f_{c}^{b+1}(b+1)</math>

===== 超运算 =====

高德纳箭头是目前已被广泛认可、基本采用<ref>曹知秋. 大数理论: Vol.1[EB/OL]. (2025-05-16) [2025-07-02]: 35-36.</ref>的超运算记号。

若定义后继运算的运算等级为<math>0</math>，那么 <math>n</math> 个高德纳箭头的运算等级为 <math>n+2</math>

==== 历史 ====

高德纳箭头是由 <math>\mathrm{Donald\ Ervin\ Knuth}</math> 在1976年发明的大数记号<ref>Donald E. Knuth, ''Mathematics and Computer Science: Coping with Finiteness'', Advances in Our Ability to Compute are Bringing Us Substantially Closer to Ultimate Limitations, Science 194, pp. 1235--1242, 1976.https://cse-robotics.engr.tamu.edu/dshell/cs625/finiteness.pdf</ref>，曾在1977年被 <math> \mathrm{Martin\ Gardner} </math> 用于递归地定义[[葛立恒数]]。<ref>GARDNER M. Mathematical games[J]. Scientific American, 1977, 237(3): 28-38.https://raw.githubusercontent.com/AllenDowney/ModSimPy/master/papers/scientific_american_nov_77.pdf</ref>

而<math>\mathrm{Ronald\ Graham}</math>本人并未在论文中使用高德纳箭头或超运算来估计 [[葛立恒数#葛立恒问题|Graham问题]] 的上界，而是使用了类似 [[ACKERMANN函数]] 的递归函数 <math>F(m,n)</math>，和分别近似为 <math>2 \uparrow\uparrow n,2 \uparrow\uparrow\uparrow n</math> 的函数<math>TOWER(n),WOW(n)</math>。<ref>GRAHAM R L, ROTHSCHILD B L. Ramsey’s theorem for ff-parameter sets[J]. Transactions of the American Mathematical Society, 1971, 159: 257-292. https://www.ams.org/journals/tran/1971-159-00/S0002-9947-1971-0284352-8/S0002-9947-1971-0284352-8.pdf
</ref><ref>GRAHAM R L, ROTHSCHILD B L, SPENCER J H. Ramsey theory: Vol. 20[M]. John
Wiley & Sons, 1991.
https://people.dm.unipi.it/dinasso/ULTRABIBLIO/Graham_Rothschild_Spencer%20-%20Ramsey%20Theory%20(2nd%20edition).pdf</ref>
==== 形式化定义 ====

<math>n \uparrow m = n^{m}</math>

<math>n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>

该定义可通过以下分析与推理得到：

高德纳箭头本质上是一种高级运算“折叠”低级运算的记号。

后继是最基础的运算，表现为 <math>n+1</math>.

在<math>n+m</math>中，运算 <math>+</math> 折叠了对 <math>n</math> 的 <math>m</math> 次后继运算，即

<math>\ n+m=n+\underbrace{1+1+\cdots+1}_{\text{m个1}}</math>.

在<math>n\times m</math>中，运算 <math>\times</math> 折叠了对 <math>n</math> 的 <math>m</math> 次 <math>+</math> 运算，即

<math>\ n\times m=\underbrace{n+n+\cdots+n}_{\text{m个n}}</math>.

在 <math>n^{m}</math> ( <math>n\uparrow m</math> ) 中，运算<math>\uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\times</math> 运算，即

<math>\ n^{m}=\underbrace{n\times n\times \cdots \times n}_{\text{m个n}}</math>.

在 <math>{}^m\!n</math> ( <math>n\uparrow \uparrow m</math> ) 中，运算<math>\uparrow \uparrow</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow</math> 运算，即

<math>\ n\uparrow \uparrow m=\underbrace{n\uparrow n\uparrow \cdots \uparrow n}_{\text{m个n}}=\underbrace{n^{n^{n^{\cdots}}}}_{\text{m个n}}</math>. （注意是右结合）

以此类推。最终，我们得到了高德纳箭头的形式化定义：

在<math>n \uparrow^{k} m</math>中，运算<math>\uparrow^{k}</math>折叠了对 <math>n</math> 的 <math>m</math> 次 <math>\uparrow^{k-1}</math> 运算，即

<math>\ n \uparrow^{k} m = \underbrace{n \uparrow^{k-1} n \uparrow^{k-1} \cdots \uparrow^{k-1} n }_{\text{m个n}}</math>
==== 计算示例 ====

<math>\begin{align} 3 \uparrow\uparrow\uparrow 3\ & = 3\uparrow\uparrow3\uparrow\uparrow 3 \\ & = 3\uparrow\uparrow (3\uparrow 3\uparrow 3) \\ & = 3\uparrow\uparrow(3\uparrow27) \\ & = 3\uparrow\uparrow 7625597484987 \\ & = \underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \end{align}</math>

<math>\begin{align} 3 \uparrow\uparrow\uparrow\uparrow 4\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow3 \\ & = 3\uparrow\uparrow\uparrow3\uparrow\uparrow\uparrow\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987} \\ & = 3\uparrow\uparrow\uparrow(\ \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}\ ) \\ & = \underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3\uparrow\uparrow3\uparrow\uparrow\cdots\uparrow\uparrow3}_{\underbrace{3^{3^{3^{\cdots}}}}_{7625597484987}}}\end{align}</math>
==== 参考资料 ====
<references />
[[分类:记号]]
[[分类:入门]]