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+ω法序数超运算分析

2026-04-21T21:47:58Z

量子杰克：

+ω法序数超运算是一种超运算型记号，作者是量子杰克，与+1法超运算类似，但区别是遇到不动点会+ω而不是+1，原因是根据分析，这样能给出更整的基本列。序数超运算可以进行许多更高级别的拓展。

== 定义 ==
一个序数超运算的格式必须是<math>\beta\{\lambda\}\alpha</math>，其中β,λ,α都是正序数。

底数β只能是正整数或超限基数。若底数是超限基数，只能是<math>\Omega_x</math>，其中x是序数。定义：<math>\Omega_0=\omega</math>，对于正序数x，<math>\Omega_x</math>为第x个不可数基数（不是可数非递归序数，因为在超运算型记号中，不可数与可数非递归的效果存在本质区别）。

若底数有限，则指数α必须有限。若底数为超限基数，则指数的势必须小于等于底数的势。例如，若底数为ω，指数必须可数。若底数为<math>\Omega_x</math>，指数必须小于<math>\Omega_{x+1}</math>。

算符序数λ的势在目前版本中最多允许比底数多不超过I（首个不可达基数）。.

计算规则：

若指数为1，则值为底数。<math>\beta\{\lambda\}1=\beta</math>

若算符序数与指数均为后继序数，使用带跳不动点函数的简单迭代规则。<math>\beta\{\lambda+1\}(\alpha+1)=if(\beta\{\lambda\}\beta\{\lambda+1\}\alpha=\beta\{\lambda+1\}\alpha)?\beta\{\lambda\}j(\beta\{\lambda+1\}\alpha)\ else\ \beta\{\lambda\}\beta\{\lambda+1\}\alpha</math>

跳跃函数j(x)取<math>j(x)=x+\omega</math>，虽然可以取其他值，但根据分析，取x+ω能使基本列更整。

若指数为极限序数，值为对指数取基本列时得到的值的极限。<math>\beta\{\lambda\}\alpha=lim(\beta\{\lambda\}\alpha[x])</math>。共尾性<math>\Omega_x</math>的极限序数的基本列长度为<math>\Omega_x</math>。

若指数为后继序数，算符序数为共尾性小于等于底数的极限序数：对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_y,\ y\leq x</math>，将指数分解为一个<math>\Omega_y</math>的倍数与一个小于<math>\Omega_y</math>的序数之和，<math>\alpha=\Omega_y\times c+d</math>。若α<<math>\Omega_y</math>即c=0，则<math>\beta\{\lambda\}\alpha=\beta\{\lambda\}d=\beta\{\lambda[d]\}\beta</math>。若α><math>\Omega_y</math>即c>0,<math>\beta\{\lambda\}\alpha=\beta\{\lambda\}(\Omega_y\times c+d)=\beta\{\lambda[d]\}j(\beta\{\lambda\}(\Omega_y\times c))</math> 。

若指数为后继序数，算符序数为共尾性为底数的下一个超限基数：对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_{x+1}</math>，展开为：<math>\beta\{\lambda\}(\alpha+1)=if(\beta\{\lambda[\beta\{\lambda\}\alpha]\}\beta=\beta\{\lambda\}\alpha)?\beta\{\lambda[j(\beta\{\lambda\}\alpha)]\}\beta\ else\ \beta\{\lambda[\beta\{\lambda\}\alpha]\}\beta</math>。

若指数是后继序数，算符序数的共尾性大于底数的下一个超限基数，底数必须是超限基数（不能为有限数）。对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_{x+y+1}</math>，y为正序数，将指数分为ω的倍数与自然数之和：<math>\alpha=\omega\times c+d</math>。

定义：<math>\lambda_1=\lambda</math>；<math>\lambda_n</math>（n为大于1的正整数）= 将<math>\lambda_{n-1}</math>的表达式中所有x>0的<math>\Omega_x</math>全部替换为<math>\Omega_{x+y}</math>。若指数有限，β{λ}α = Ω_x{λ[Ω_(x+y+1)]}d；β{λ}1 = Ω_x{λ[Ω_(x+y)]}ω；β{λ}(n+1) = 将β{λ}n表达式中的λ_n[Ω_(x+y*n)]替换为λ_n[Ω_(x+y*n)]{λ_(n+1)[Ω_(x+y*(n+1))]}ω。若指数为超限后继序数，β{λ}α = Ω_x{λ[Ω_(x+y+1)]}(ω*c+d)；β{λ}(ω*c+1) = Ω_x{λ[Ω_(x+y)]}j(β{λ}(ω*c))；β{λ}(ω*c+n+1) = 将β{λ}n表达式中的λ_n[Ω_(x+y*n)]替换为λ_n[Ω_(x+y*n)]{λ_(n+1)[Ω_(x+y*(n+1))]}ω；若指数为极限序数，β{λ}α = lim(β{λ}(α[n]))。

== 分析1 ==
{| class="wikitable"
|+分析1：SCO~HCO
!序数超运算
!BMS
|-
|<math>\omega\uparrow\uparrow\omega</math>
|(0)(1,1) = [[SCO]]
|-
|<math>\omega\uparrow\uparrow(\omega+1)=\omega^{\omega\uparrow\uparrow\omega+\omega}</math>
|(0)(1,1)(1)(2)
|-
|<math>\omega\uparrow\uparrow(\omega+2)=\omega^{\omega^{\omega\uparrow\uparrow\omega+\omega}}</math>
|(0)(1,1)(1)(2,1)(2)(3)
|-
|<math>\omega\uparrow\uparrow(\omega+3)</math>
|(0)(1,1)(1)(2,1)(2)(3,1)(3)(4)
|-
|<math>\omega\uparrow\uparrow\omega2</math>
|(0)(1,1)(1,1)
|-
|<math>\omega\uparrow\uparrow(\omega2+1)=\omega^{\omega\uparrow\uparrow\omega2+\omega}</math>
|(0)(1,1)(1,1)(1)(2)
|-
|<math>\omega\uparrow\uparrow\omega3</math>
|(0)(1,1)(1,1)(1,1)
|-
|<math>\omega\uparrow\uparrow\omega^2</math>
|(0)(1,1)(2)
|-
|<math>\omega\uparrow\uparrow(\omega^2+\omega)</math>
|(0)(1,1)(2)(1,1)
|-
|<math>\omega\uparrow\uparrow\omega^22</math>
|(0)(1,1)(2)(1,1)(2)
|-
|<math>\omega\uparrow\uparrow\omega^3</math>
|(0)(1,1)(2)(2)
|-
|<math>\omega\uparrow\uparrow\omega^\omega</math>
|(0)(1,1)(2)(3)
|-
|<math>\omega\uparrow\uparrow\uparrow3=\omega\uparrow\uparrow\omega\uparrow\uparrow\omega</math>
|(0)(1,1)(2)(3,1)
|-
|<math>\omega\uparrow\uparrow\uparrow4</math>
|(0)(1,1)(2)(3,1)(4)(5,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega</math>
|(0)(1,1)(2,1) = [[CO]]
|-
|<math>\omega\uparrow\uparrow\uparrow(\omega+1)=\omega\uparrow\uparrow(\omega\uparrow\uparrow\uparrow\omega+\omega)</math>
|(0)(1,1)(2,1)(1,1)
|-
|<math>\omega\uparrow\uparrow\uparrow(\omega+2)</math>
|(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega2</math>
|(0)(1,1)(2,1)(1,1)(2,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega^2</math>
|(0)(1,1)(2,1)(2)
|-
|<math>\omega\{4\}3</math>
|(0)(1,1)(2,1)(2)(3,1)(4,1)
|-
|<math>\omega\{4\}\omega</math>
|(0)(1,1)(2,1)(2,1) = [[LCO]]
|-
|<math>\omega\{4\}(\omega+1) = \omega\uparrow\uparrow\uparrow(\omega\{4\}\omega+\omega)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2,1)
|-
|<math>\omega\{4\}\omega2</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)
|-
|<math>\omega\{5\}\omega</math>
|(0)(1,1)(2,1)(2,1)(2,1)
|-
|<math>\omega\{\omega\}\omega</math>
|(0)(1,1)(2,1)(3) = [[HCO]]
|}

== 分析2 ==
{| class="wikitable"
|+分析2：HCO~FSO
!序数超运算
!BMS
|-
|<math>\omega\{\omega\}\omega</math>
|(0)(1,1)(2,1)(3) = [[HCO]]
|-
|<math>\omega\{\omega\}(\omega+1)=\omega^{\omega\{\omega\}\omega+\omega}</math>
|(0)(1,1)(2,1)(3)(1)(2)
|-
|<math>\omega\{\omega\}(\omega+2)=\omega\uparrow\uparrow(\omega\{\omega\}\omega+\omega)</math>
|(0)(1,1)(2,1)(3)(1,1)
|-
|<math>\omega\{\omega\}(\omega+3)=\omega\uparrow\uparrow\uparrow(\omega\{\omega\}\omega+\omega)</math>
|(0)(1,1)(2,1)(3)(1,1)(2,1)
|-
|<math>\omega\{\omega\}\omega2</math>
|(0)(1,1)(2,1)(3)(1,1)(2,1)(3)
|-
|<math>\omega\{\omega\}\omega^2</math>
|(0)(1,1)(2,1)(3)(2)
|-
|<math>\omega\{\omega+1\}3</math>
|(0)(1,1)(2,1)(3)(2)(3,1)(4,1)(5)
|-
|<math>\omega\{\omega+1\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)
|-
|<math>\omega\{\omega+2\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)(2,1)
|-
|<math>\omega\{\omega2\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)(3)
|-
|<math>\omega\{\omega^2\}\omega</math>
|(0)(1,1)(2,1)(3)(3)
|-
|<math>\omega\{\omega^\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4)
|-
|<math>\omega\{\omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4,1)
|-
|<math>\omega\{\Omega\}3=\omega\{\omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4,1)(5,1)(6)
|-
|<math>\omega\{\Omega\}4</math>
|(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1)(9)
|-
|<math>\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1) = [[FSO]]
|}

== 分析3 ==
{| class="wikitable"
|+分析3：FSO~BHO
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1) = [[FSO]]
|-
|<math>\omega\{\omega\{\Omega\}\omega\}(\omega+1)=\omega\{\omega\}\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)
|-
|<math>\omega\{\omega\{\Omega\}\omega\}(\omega+2)=\omega\{\omega\{\omega\}\omega\}\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6)
|-
|<math>\omega\{\omega\{\Omega\}\omega\}\omega2</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)
|-
|<math>\omega\{\omega\{\Omega\}\omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)
|-
|<math>\omega\{\Omega\}(\omega+1)=\omega\{\omega\{\Omega\}\omega+\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)(3)
|-
|<math>\omega\{\Omega\}(\omega+2)=\omega\{\omega\{\omega\{\Omega\}\omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)(5,1)(6)(7,1)(8,1)(9,1)(5,1)(6)
|-
|<math>\omega\{\Omega\}\omega2</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\}\omega^2</math>
|(0)(1,1)(2,1)(3,1)(2)
|-
|<math>\omega\{\Omega+1\}3</math>
|(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)
|-
|<math>\omega\{\Omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)
|-
|<math>\omega\{\Omega+2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(2,1)
|-
|<math>\omega\{\Omega+\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3)
|-
|<math>\omega\{\Omega2\}3=\omega\{\Omega+\omega\{\Omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1)(5,1)(4,1)(5)
|-
|<math>\omega\{\Omega2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3)
|-
|<math>\omega\{\Omega^2\}3=\omega\{\Omega\times\omega\{\Omega\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3)(4,1)(5,1)(5)
|-
|<math>\omega\{\Omega^2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1) = [[ACO]]
|-
|<math>\omega\{\Omega^2+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)
|-
|<math>\omega\{\Omega^22\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1)
|-
|<math>\omega\{\Omega^2\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(3)
|-
|<math>\omega\{\Omega^3\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(3,1)
|-
|<math>\omega\{\Omega^\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4) = [[SVO]]
|-
|<math>\omega\{\Omega^\Omega\}3=\omega\{\Omega^{\omega\{\Omega^\omega\}\omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8)
|-
|<math>\omega\{\Omega^\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1) = [[LVO]]
|-
|<math>\omega\{\Omega^\Omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(2,1)
|-
|<math>\omega\{\Omega^{\Omega+1}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)
|-
|<math>\omega\{\Omega^{\Omega2}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1)
|-
|<math>\omega\{\Omega^{\Omega^2}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(4,1)
|-
|<math>\omega\{\Omega^{\Omega^\omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}3=\omega\{\Omega^{\Omega^\Omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}4=\omega\{\Omega\uparrow\uparrow4\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2) = [[BHO]]
|}

== 分析4 ==
{| class="wikitable"
|+分析4：BHO~(0)(1,1)(2,2)(3,2)(4)
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2) = [[BHO]]
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}(\omega+1)=\omega\{\Omega\}(\omega\{\Omega\uparrow\uparrow\omega\}\omega+\omega)</math>
|(0)(1,1)(2,2)(1,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}(\omega+1)=\omega\{\Omega^\Omega\}(\omega\{\Omega\uparrow\uparrow\omega\}\omega+\omega)</math>
|(0)(1,1)(2,2)(1,1)(2,1)(3,1)(4,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega2</math>
|(0)(1,1)(2,2)(1,1)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega^2</math>
|(0)(1,1)(2,2)(2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+1\}3</math>
|(0)(1,1)(2,2)(2)(3,1)(4,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+1\}\omega</math>
|(0)(1,1)(2,2)(2,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+\Omega\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\times2\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\times\omega\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow(\omega+1)\}\omega=\omega\{\omega^{\Omega\uparrow\uparrow\omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega2\}\omega</math>
|(0)(1,1)(2,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega3\}\omega</math>
|(0)(1,1)(2,2)(2,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\}3=\omega\{\Omega\uparrow\uparrow\omega\{\Omega\uparrow\uparrow\Omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3)(4,1)(5,2)(6)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow(\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,1)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega2\}\omega</math>
|(0)(1,1)(2,2)(3,1)(2,2)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2)(3,1)(4,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow3\}\omega</math>
|(0)(1,1)(2,2)(3,1)(4,2)(5,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow(\omega+1)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\uparrow\uparrow\uparrow\omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(2,2)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,1)
|-
|<math>\omega\{\Omega\{4\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)
|-
|<math>\omega\{\Omega\{\omega\}4\}\omega=\omega\{\Omega\{4\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)(3,1)
|-
|<math>\omega\{\Omega\{5\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)(3,2)
|-
|<math>\omega\{\Omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)
|}

== 分析5 ==
{| class="wikitable"
|+分析：(0)(1,1)(2,2)(3,2)(4)~(0)(1,1)(2,2)(3,2)(4,2)
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega\}(\omega+1)\}\omega=\omega\{\Omega^{\Omega\{\omega\}\omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\{\omega\}(\omega+2)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\{\omega\}\omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,2)
|-
|<math>\omega\{\Omega\{\omega\}\omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega\}\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3)
|-
|<math>\omega\{\Omega\{\Omega\}\omega\}\omega=\omega\{\Omega\{\omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,1)
|-
|<math>\omega\{\Omega\{\omega+1\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,2)
|-
|<math>\omega\{\Omega\{\omega2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega^2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(4)
|-
|<math>\omega\{\Omega\{\omega\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(5,1)
|-
|<math>\omega\{\Omega\{\omega\{\Omega\uparrow\uparrow\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(5,1)(6,2)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+1)\}\omega=\omega\{\Omega^{\Omega\{\Omega\}\Omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+2)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\{\Omega\}\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+\omega)\}\omega=\omega\{\Omega\{\omega\}(\Omega\{\Omega\}\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,1)
|-
|<math>\omega\{\Omega\{\Omega+1\}3\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,1)(4,2)(5,3)(6,1)
|-
|<math>\omega\{\Omega\{\Omega+1\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)
|-
|<math>\omega\{\Omega\{\Omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)(4)
|-
|<math>\omega\{\Omega\{\Omega2\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(4)
|-
|<math>\omega\{\Omega\{\Omega^2\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,1)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)
|-
|<math>\omega\{\Omega\{\Omega\{\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}3=\omega\{\Omega\{\Omega_2\}3\}\omega=\omega\{\Omega\{\Omega\{\Omega\}\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}4=\omega\{\Omega\{\Omega_2\}4\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)(8,2)(9,2)(10,1)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,2)
|}

== 预期强度 ==
{| class="wikitable"
|+预期强度（已进行部分分析，待填写）
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\{\Omega_2\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)
|-
|<math>\omega\{\Omega\{\Omega_2\uparrow\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)
|-
|<math>\omega\{\Omega\{\Omega_2\{\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)(5)
|-
|<math>\omega\{\Omega\{\Omega_2\{\Omega_3\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)(5,3)
|-
|<math>\omega\{\Omega\{\Omega_2\{\Omega_3\uparrow\uparrow\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,4)
|-
|<math>\omega\{\Omega_2\}\omega</math>
|(0)(1,1,1) = [[BO]]
|-
|<math>\omega\{\Omega_22\}\omega</math>
|(0)(1,1,1)(1,1,1)
|-
|<math>\omega\{\Omega_2\Omega\}\omega</math>
|(0)(1,1,1)(2,1)
|-
|<math>\omega\{\Omega_2\times(\Omega+1)\}\omega</math>
|(0)(1,1,1)(2,1)(1,1,1)
|-
|<math>\omega\{\Omega_2\times(\Omega\uparrow\uparrow\omega)\}\omega</math>
|(0)(1,1,1)(2,1)(3,2) = [[TFBO]]
|-
|<math>\omega\{\Omega_2\times(\Omega\{\Omega_3\}\omega)\}\omega</math>
|(0)(1,1,1)(2,1)(3,2,1)
|-
|<math>\omega\{\Omega_2^2\}\omega</math>
|(0)(1,1,1)(2,1,1)
|-
|<math>\omega\{\Omega_2^\Omega\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1) = [[BIO]]
|-
|<math>\omega\{\Omega_2^\Omega\omega\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1)(2) = [[EBO]]
|-
|<math>\omega\{\Omega_2^{\Omega\uparrow\uparrow\omega}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1)(4,2) = [[JO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1) = [[SIO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2^2}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1) = [[SMO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2^{\Omega_2}}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) = [[SKO]]
|-
|<math>\omega\{\Omega_2\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1,1)(2,2) = [[SSO]]
|-
|<math>\omega\{\Omega_2\{\Omega\}\Omega\times\omega\}\omega</math>
|(0)(1,1,1)(2,2)(3,2)(4,1)(2) = [[LSO]]
|-
|<math>\omega\{\Omega_2\{\Omega_2\}\Omega_2\}\omega</math>
|(0)(1,1,1)(2,2)(3,2)(4,1,1) = [[APO]]
|-
|<math>\omega\{\Omega_3\}\omega</math>
|(0)(1,1,1)(2,2,1) = [[BGO]]
|-
|<math>\omega\{\Omega_4\}\omega</math>
|(0)(1,1,1)(2,2,1)(2,2,1)
|-
|<math>\omega\{\Omega_\omega\}\omega</math>
|(0)(1,1,1)(2,2,1)(3) = [[SDO]]
|-
|<math>\omega\{\Omega_\Omega\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,1)
|-
|<math>\omega\{\Omega_{\Omega_2}\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,1,1)
|-
|<math>\omega\{\Phi(1,0)\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,2) = [[LDO]]
|-
|<math>psd.\omega\{I\}\omega=\omega\{\psi_I(X)\}\omega\ ct.\ \psi(X)</math>
|(0)(1,1,1)(2,2,1)(3,2,1)
|-
|<math>psd. \omega\{X\}\omega\ ct.\ \psi(X)</math>
|(0)(1,1,1)(2,2,2) = [[LRO|pfec LRO]]
|}
[[分类:分析]]
[[分类:序数超运算]]

+ω法序数超运算分析

2026-04-21T20:30:50Z

量子杰克：分析

+ω法序数超运算是一种超运算型记号，作者是量子杰克，与+1法超运算类似，但区别是遇到不动点会+ω而不是+1，原因是根据分析，这样能给出更整的基本列。序数超运算可以进行许多更高级别的拓展。

== 定义 ==
一个序数超运算的格式必须是<math>\beta\{\lambda\}\alpha</math>，其中β,λ,α都是正序数。

底数β只能是正整数或超限基数。若底数是超限基数，只能是<math>\Omega_x</math>，其中x是序数。定义：<math>\Omega_0=\omega</math>，对于正序数x，<math>\Omega_x</math>为第x个不可数基数（不是可数非递归序数，因为在超运算型记号中，不可数与可数非递归的效果存在本质区别）。

若底数有限，则指数α必须有限。若底数为超限基数，则指数的势必须小于等于底数的势。例如，若底数为ω，指数必须可数。若底数为<math>\Omega_x</math>，指数必须小于<math>\Omega_{x+1}</math>。

算符序数λ的势在目前版本中最多允许比底数多不超过I（首个不可达基数）。.

计算规则：

若指数为1，则值为底数。<math>\beta\{\lambda\}1=\beta</math>

若算符序数与指数均为后继序数，使用带跳不动点函数的简单迭代规则。<math>\beta\{\lambda+1\}(\alpha+1)=if(\beta\{\lambda\}\beta\{\lambda+1\}\alpha=\beta\{\lambda+1\}\alpha)?\beta\{\lambda\}j(\beta\{\lambda+1\}\alpha)\ else\ \beta\{\lambda\}\beta\{\lambda+1\}\alpha</math>

跳跃函数j(x)取<math>j(x)=x+\omega</math>，虽然可以取其他值，但根据分析，取x+ω能使基本列更整。

若指数为极限序数，值为对指数取基本列时得到的值的极限。<math>\beta\{\lambda\}\alpha=lim(\beta\{\lambda\}\alpha[x])</math>。共尾性<math>\Omega_x</math>的极限序数的基本列长度为<math>\Omega_x</math>。

若指数为后继序数，算符序数为共尾性小于等于底数的极限序数：对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_y,\ y\leq x</math>，将指数分解为一个<math>\Omega_y</math>的倍数与一个小于<math>\Omega_y</math>的序数之和，<math>\alpha=\Omega_y\times c+d</math>。若α<<math>\Omega_y</math>即c=0，则<math>\beta\{\lambda\}\alpha=\beta\{\lambda\}d=\beta\{\lambda[d]\}\beta</math>。若α><math>\Omega_y</math>即c>0,<math>\beta\{\lambda\}\alpha=\beta\{\lambda\}(\Omega_y\times c+d)=\beta\{\lambda[d]\}j(\beta\{\lambda\}(\Omega_y\times c))</math> 。

若指数为后继序数，算符序数为共尾性为底数的下一个超限基数：对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_{x+1}</math>，展开为：<math>\beta\{\lambda\}(\alpha+1)=if(\beta\{\lambda[\beta\{\lambda\}\alpha]\}\beta=\beta\{\lambda\}\alpha)?\beta\{\lambda[j(\beta\{\lambda\}\alpha)]\}\beta\ else\ \beta\{\lambda[\beta\{\lambda\}\alpha]\}\beta</math>。

若指数是后继序数，算符序数的共尾性大于底数的下一个超限基数，底数必须是超限基数（不能为有限数）。对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_{x+y+1}</math>，y为正序数，将指数分为ω的倍数与自然数之和：<math>\alpha=\omega\times c+d</math>。

定义：<math>\lambda_1=\lambda</math>；<math>\lambda_n</math>（n为大于1的正整数）= 将<math>\lambda_{n-1}</math>的表达式中所有x>0的<math>\Omega_x</math>全部替换为<math>\Omega_{x+y}</math>。若指数有限，β{λ}α = Ω_x{λ[Ω_(x+y+1)]}d；β{λ}1 = Ω_x{λ[Ω_(x+y)]}ω；β{λ}(n+1) = 将β{λ}n表达式中的λ_n[Ω_(x+y*n)]替换为λ_n[Ω_(x+y*n)]{λ_(n+1)[Ω_(x+y*(n+1))]}ω。若指数为超限后继序数，β{λ}α = Ω_x{λ[Ω_(x+y+1)]}(ω*c+d)；β{λ}(ω*c+1) = Ω_x{λ[Ω_(x+y)]}j(β{λ}(ω*c))；β{λ}(ω*c+n+1) = 将β{λ}n表达式中的λ_n[Ω_(x+y*n)]替换为λ_n[Ω_(x+y*n)]{λ_(n+1)[Ω_(x+y*(n+1))]}ω；若指数为极限序数，β{λ}α = lim(β{λ}(α[n]))。

== 分析1 ==
{| class="wikitable"
|+分析1：SCO~HCO
!序数超运算
!BMS
|-
|<math>\omega\uparrow\uparrow\omega</math>
|(0)(1,1) = [[SCO]]
|-
|<math>\omega\uparrow\uparrow(\omega+1)=\omega^{\omega\uparrow\uparrow\omega+\omega}</math>
|(0)(1,1)(1)(2)
|-
|<math>\omega\uparrow\uparrow(\omega+2)=\omega^{\omega^{\omega\uparrow\uparrow\omega+\omega}}</math>
|(0)(1,1)(1)(2,1)(2)(3)
|-
|<math>\omega\uparrow\uparrow(\omega+3)</math>
|(0)(1,1)(1)(2,1)(2)(3,1)(3)(4)
|-
|<math>\omega\uparrow\uparrow\omega2</math>
|(0)(1,1)(1,1)
|-
|<math>\omega\uparrow\uparrow(\omega2+1)=\omega^{\omega\uparrow\uparrow\omega2+\omega}</math>
|(0)(1,1)(1,1)(1)(2)
|-
|<math>\omega\uparrow\uparrow\omega3</math>
|(0)(1,1)(1,1)(1,1)
|-
|<math>\omega\uparrow\uparrow\omega^2</math>
|(0)(1,1)(2)
|-
|<math>\omega\uparrow\uparrow(\omega^2+\omega)</math>
|(0)(1,1)(2)(1,1)
|-
|<math>\omega\uparrow\uparrow\omega^22</math>
|(0)(1,1)(2)(1,1)(2)
|-
|<math>\omega\uparrow\uparrow\omega^3</math>
|(0)(1,1)(2)(2)
|-
|<math>\omega\uparrow\uparrow\omega^\omega</math>
|(0)(1,1)(2)(3)
|-
|<math>\omega\uparrow\uparrow\uparrow3=\omega\uparrow\uparrow\omega\uparrow\uparrow\omega</math>
|(0)(1,1)(2)(3,1)
|-
|<math>\omega\uparrow\uparrow\uparrow4</math>
|(0)(1,1)(2)(3,1)(4)(5,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega</math>
|(0)(1,1)(2,1) = [[CO]]
|-
|<math>\omega\uparrow\uparrow\uparrow(\omega+1)=\omega\uparrow\uparrow(\omega\uparrow\uparrow\uparrow\omega+\omega)</math>
|(0)(1,1)(2,1)(1,1)
|-
|<math>\omega\uparrow\uparrow\uparrow(\omega+2)</math>
|(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega2</math>
|(0)(1,1)(2,1)(1,1)(2,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega^2</math>
|(0)(1,1)(2,1)(2)
|-
|<math>\omega\{4\}3</math>
|(0)(1,1)(2,1)(2)(3,1)(4,1)
|-
|<math>\omega\{4\}\omega</math>
|(0)(1,1)(2,1)(2,1) = [[LCO]]
|-
|<math>\omega\{4\}(\omega+1) = \omega\uparrow\uparrow\uparrow(\omega\{4\}\omega+\omega)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2,1)
|-
|<math>\omega\{4\}\omega2</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)
|-
|<math>\omega\{5\}\omega</math>
|(0)(1,1)(2,1)(2,1)(2,1)
|-
|<math>\omega\{\omega\}\omega</math>
|(0)(1,1)(2,1)(3) = [[HCO]]
|}

== 分析2 ==
{| class="wikitable"
|+分析2：HCO~FSO
!序数超运算
!BMS
|-
|<math>\omega\{\omega\}\omega</math>
|(0)(1,1)(2,1)(3) = [[HCO]]
|-
|<math>\omega\{\omega\}(\omega+1)=\omega^{\omega\{\omega\}\omega+\omega}</math>
|(0)(1,1)(2,1)(3)(1)(2)
|-
|<math>\omega\{\omega\}(\omega+2)=\omega\uparrow\uparrow(\omega\{\omega\}\omega+\omega)</math>
|(0)(1,1)(2,1)(3)(1,1)
|-
|<math>\omega\{\omega\}(\omega+3)=\omega\uparrow\uparrow\uparrow(\omega\{\omega\}\omega+\omega)</math>
|(0)(1,1)(2,1)(3)(1,1)(2,1)
|-
|<math>\omega\{\omega\}\omega2</math>
|(0)(1,1)(2,1)(3)(1,1)(2,1)(3)
|-
|<math>\omega\{\omega\}\omega^2</math>
|(0)(1,1)(2,1)(3)(2)
|-
|<math>\omega\{\omega+1\}3</math>
|(0)(1,1)(2,1)(3)(2)(3,1)(4,1)(5)
|-
|<math>\omega\{\omega+1\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)
|-
|<math>\omega\{\omega+2\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)(2,1)
|-
|<math>\omega\{\omega2\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)(3)
|-
|<math>\omega\{\omega^2\}\omega</math>
|(0)(1,1)(2,1)(3)(3)
|-
|<math>\omega\{\omega^\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4)
|-
|<math>\omega\{\omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4,1)
|-
|<math>\omega\{\Omega\}3=\omega\{\omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4,1)(5,1)(6)
|-
|<math>\omega\{\Omega\}4</math>
|(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1)(9)
|-
|<math>\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1) = [[FSO]]
|}

== 分析3 ==
{| class="wikitable"
|+分析3：FSO~BHO
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1) = [[FSO]]
|-
|<math>\omega\{\omega\{\Omega\}\omega\}(\omega+1)=\omega\{\omega\}\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)
|-
|<math>\omega\{\omega\{\Omega\}\omega\}(\omega+2)=\omega\{\omega\{\omega\}\omega\}\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6)
|-
|<math>\omega\{\omega\{\Omega\}\omega\}\omega2</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)
|-
|<math>\omega\{\omega\{\Omega\}\omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)
|-
|<math>\omega\{\Omega\}(\omega+1)=\omega\{\omega\{\Omega\}\omega+\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)(3)
|-
|<math>\omega\{\Omega\}(\omega+2)=\omega\{\omega\{\omega\{\Omega\}\omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)(5,1)(6)(7,1)(8,1)(9,1)(5,1)(6)
|-
|<math>\omega\{\Omega\}\omega2</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\}\omega^2</math>
|(0)(1,1)(2,1)(3,1)(2)
|-
|<math>\omega\{\Omega+1\}3</math>
|(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)
|-
|<math>\omega\{\Omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)
|-
|<math>\omega\{\Omega+2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(2,1)
|-
|<math>\omega\{\Omega+\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3)
|-
|<math>\omega\{\Omega2\}3=\omega\{\Omega+\omega\{\Omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1)(5,1)(4,1)(5)
|-
|<math>\omega\{\Omega2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3)
|-
|<math>\omega\{\Omega^2\}3=\omega\{\Omega\times\omega\{\Omega\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3)(4,1)(5,1)(5)
|-
|<math>\omega\{\Omega^2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1) = [[ACO]]
|-
|<math>\omega\{\Omega^2+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)
|-
|<math>\omega\{\Omega^22\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1)
|-
|<math>\omega\{\Omega^2\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(3)
|-
|<math>\omega\{\Omega^3\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(3,1)
|-
|<math>\omega\{\Omega^\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4) = [[SVO]]
|-
|<math>\omega\{\Omega^\Omega\}3=\omega\{\Omega^{\omega\{\Omega^\omega\}\omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8)
|-
|<math>\omega\{\Omega^\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1) = [[LVO]]
|-
|<math>\omega\{\Omega^\Omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(2,1)
|-
|<math>\omega\{\Omega^{\Omega+1}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)
|-
|<math>\omega\{\Omega^{\Omega2}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1)
|-
|<math>\omega\{\Omega^{\Omega^2}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(4,1)
|-
|<math>\omega\{\Omega^{\Omega^\omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}3=\omega\{\Omega^{\Omega^\Omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}4=\omega\{\Omega\uparrow\uparrow4\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2) = [[BHO]]
|}

== 分析4 ==
{| class="wikitable"
|+分析4：BHO~(0)(1,1)(2,2)(3,2)(4)
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2) = [[BHO]]
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}(\omega+1)=\omega\{\Omega\}(\omega\{\Omega\uparrow\uparrow\omega\}\omega+\omega)</math>
|(0)(1,1)(2,2)(1,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}(\omega+1)=\omega\{\Omega^\Omega\}(\omega\{\Omega\uparrow\uparrow\omega\}\omega+\omega)</math>
|(0)(1,1)(2,2)(1,1)(2,1)(3,1)(4,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega2</math>
|(0)(1,1)(2,2)(1,1)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega^2</math>
|(0)(1,1)(2,2)(2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+1\}3</math>
|(0)(1,1)(2,2)(2)(3,1)(4,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+1\}\omega</math>
|(0)(1,1)(2,2)(2,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+\Omega\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\times2\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\times\omega\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow(\omega+1)\}\omega=\omega\{\omega^{\Omega\uparrow\uparrow\omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega2\}\omega</math>
|(0)(1,1)(2,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega3\}\omega</math>
|(0)(1,1)(2,2)(2,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\}3=\omega\{\Omega\uparrow\uparrow\omega\{\Omega\uparrow\uparrow\Omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3)(4,1)(5,2)(6)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow(\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,1)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega2\}\omega</math>
|(0)(1,1)(2,2)(3,1)(2,2)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2)(3,1)(4,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow3\}\omega</math>
|(0)(1,1)(2,2)(3,1)(4,2)(5,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow(\omega+1)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\uparrow\uparrow\uparrow\omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(2,2)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,1)
|-
|<math>\omega\{\Omega\{4\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)
|-
|<math>\omega\{\Omega\{\omega\}4\}\omega=\omega\{\Omega\{4\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)(3,1)
|-
|<math>\omega\{\Omega\{5\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)(3,2)
|-
|<math>\omega\{\Omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)
|}

== 分析5 ==
{| class="wikitable"
|+分析：(0)(1,1)(2,2)(3,2)(4)~(0)(1,1)(2,2)(3,2)(4,2)
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega\}(\omega+1)\}\omega=\omega\{\Omega^{\Omega\{\omega\}\omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\{\omega\}(\omega+2)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\{\omega\}\omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,2)
|-
|<math>\omega\{\Omega\{\omega\}\omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega\}\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3)
|-
|<math>\omega\{\Omega\{\Omega\}\omega\}\omega=\omega\{\Omega\{\omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,1)
|-
|<math>\omega\{\Omega\{\omega+1\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,2)
|-
|<math>\omega\{\Omega\{\omega2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega^2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(4)
|-
|<math>\omega\{\Omega\{\omega\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(5,1)
|-
|<math>\omega\{\Omega\{\omega\{\Omega\uparrow\uparrow\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(5,1)(6,2)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+1)\}\omega=\omega\{\Omega^{\Omega\{\Omega\}\Omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+2)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\{\Omega\}\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+\omega)\}\omega=\omega\{\Omega\{\omega\}(\Omega\{\Omega\}\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,1)
|-
|<math>\omega\{\Omega\{\Omega+1\}3\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,1)(4,2)(5,3)(6,1)
|-
|<math>\omega\{\Omega\{\Omega+1\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)
|-
|<math>\omega\{\Omega\{\Omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)(4)
|-
|<math>\omega\{\Omega\{\Omega2\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(4)
|-
|<math>\omega\{\Omega\{\Omega^2\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,1)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)
|-
|<math>\omega\{\Omega\{\Omega\{\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}3=\omega\{\Omega\{\Omega_2\}3\}\omega=\omega\{\Omega\{\Omega\{\Omega\}\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}4=\omega\{\Omega\{\Omega_2\}4\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)(8,2)(9,2)(10,1)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,2)
|}

== 预期强度 ==
{| class="wikitable"
|+预期强度（已进行部分分析，待填写）
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\{\Omega_2\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)
|-
|<math>\omega\{\Omega\{\Omega_2\uparrow\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)
|-
|<math>\omega\{\Omega\{\Omega_2\{\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)(5)
|-
|<math>\omega\{\Omega\{\Omega_2\{\Omega_3\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)(5,3)
|-
|<math>\omega\{\Omega\{\Omega_2\{\Omega_3\uparrow\uparrow\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,4)
|-
|<math>\omega\{\Omega_2\}\omega</math>
|(0)(1,1,1) = [[BO]]
|-
|<math>\omega\{\Omega_22\}\omega</math>
|(0)(1,1,1)(1,1,1)
|-
|<math>\omega\{\Omega_2\Omega\}\omega</math>
|(0)(1,1,1)(2,1)
|-
|<math>\omega\{\Omega_2\times(\Omega+1)\}\omega</math>
|(0)(1,1,1)(2,1)(1,1,1)
|-
|<math>\omega\{\Omega_2\times(\Omega\uparrow\uparrow\omega)\}\omega</math>
|(0)(1,1,1)(2,1)(3,2) = [[TFBO]]
|-
|<math>\omega\{\Omega_2\times(\Omega\{\Omega_3\}\omega)\}\omega</math>
|(0)(1,1,1)(2,1)(3,2,1)
|-
|<math>\omega\{\Omega_2^2\}\omega</math>
|(0)(1,1,1)(2,1,1)
|-
|<math>\omega\{\Omega_2^\Omega\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1) = [[BIO]]
|-
|<math>\omega\{\Omega_2^\Omega\omega\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1)(2) = [[EBO]]
|-
|<math>\omega\{\Omega_2^{\Omega\uparrow\uparrow\omega}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1)(4,2) = [[JO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1) = [[SIO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2^2}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1) = [[SMO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2^{\Omega_2}}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) = [[SKO]]
|-
|<math>\omega\{\Omega_2\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1,1)(2,2) = [[SSO]]
|-
|<math>\omega\{\Omega_2\{\Omega\}\Omega\times\omega\}\omega</math>
|(0)(1,1,1)(2,2)(3,2)(4,1)(2) = [[LSO]]
|-
|<math>\omega\{\Omega_2\{\Omega_2\}\Omega_2\}\omega</math>
|(0)(1,1,1)(2,2)(3,2)(4,1,1) = [[APO]]
|-
|<math>\omega\{\Omega_3\}\omega</math>
|(0)(1,1,1)(2,2,1) = [[BGO]]
|-
|<math>\omega\{\Omega_4\}\omega</math>
|(0)(1,1,1)(2,2,1)(2,2,1)
|-
|<math>\omega\{\Omega_\omega\}\omega</math>
|(0)(1,1,1)(2,2,1)(3) = [[SDO]]
|-
|<math>\omega\{\Omega_\Omega\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,1)
|-
|<math>\omega\{\Omega_{\Omega_2}\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,1,1)
|-
|<math>\omega\{\Phi(1,0)\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,2) = [[LDO]]
|-
|<math>psd.\omega\{I\}\omega=\omega\{\psi_I(X)\}\omega\ ct.\ \psi(X)</math>
|(0)(1,1,1)(2,2,1)(3,2,1)
|-
|<math>psd. \omega\{\psi(X)\}\omega\ ct.\ \psi(X)</math>
|(0)(1,1,1)(2,2,2) = [[LRO|pfec LRO]]
|}
[[分类:分析]]
[[分类:序数超运算]]

序数超运算

2026-04-21T20:30:11Z

量子杰克：分类

序数超运算是对序数使用[[超运算序列|超运算]]的尝试。它虽然是最直观、新人最容易想到的模式，但已经被长期的[[googology]]实践所证明是低效、难以扩展的。

== 原定义 ==
首先，我们仿照[[高德纳箭头]]在自然数上的定义和[[序数#序数的运算|序数运算]]的定义，给出序数使用高德纳箭头的定义：

# <math>\alpha\uparrow^1\beta=\alpha^{\beta}</math>
# <math>\alpha\uparrow^c1=\alpha</math>
# <math>\alpha\uparrow^{c+1}(\beta+1)=\alpha\uparrow^c(\alpha\uparrow^{c+1}\beta)</math>
# <math>\alpha\uparrow^c\lambda=\sup\{\alpha\uparrow^c\beta|\beta<\lambda\}</math>

其中<math>\alpha,\beta</math>是任意序数，c是自然数，<math>\lambda</math>是非0极限序数。

这样一来，就有<math>\omega\uparrow^2(\omega+1)=\omega^{\omega\uparrow^2\omega}=\sup\{\omega^{\omega\uparrow^2n}|n<\omega\}=\sup\{\omega\uparrow^2n|n<\omega\}=\omega\uparrow^2\omega</math>.进一步，对任意<math>\beta\geq\omega</math>，都有<math>\omega\uparrow^2\beta=\omega\uparrow^2\omega</math>.再进一步，对任意的<math>c\geq2,\beta\geq\omega</math>,都有<math>\omega\uparrow^c\beta=\omega\uparrow^2\omega</math>.

这显然不是我们所期待的。

== 左结合法 ==
第一种试图解决问题的方案是左结合法。它借鉴了[[下箭号表示法|下箭头表示法]]，给出序数使用下箭头的定义：

# <math>\alpha\downarrow^1\beta=\alpha^{\beta}</math>
# <math>\alpha\downarrow^c1=\alpha</math>
# <math>\alpha\downarrow^{c+1}(\beta+1)=(\alpha\downarrow^{c+1}\beta)\downarrow\alpha</math>
# <math>\alpha\downarrow^c\lambda=\sup\{\alpha\downarrow^c\beta|\beta<\lambda\}</math>

其中<math>\alpha,\beta</math>是任意序数，c是自然数，<math>\lambda</math>是非0极限序数。

于是有：

<math>\omega\downarrow^22=\omega^\omega
</math>

<math>\omega\downarrow^23=\omega^{\omega^2}</math>

<math>\omega\downarrow^2\omega=\omega^{\omega^\omega}</math>

<math>(\omega\downarrow^2\omega)\downarrow^22=\omega^{\omega^{\omega^\omega}}</math>

<math>\omega\downarrow^33=\omega^{\omega^{\omega^{\omega^\omega}}}</math>

<math>\omega\downarrow^34=\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}</math>

<math>\omega\downarrow^3\omega=\varepsilon_0</math>

<math>(\omega\downarrow^3\omega)\downarrow^22=\varepsilon_0^{\varepsilon_0}</math>

<math>(\omega\downarrow^3\omega)\downarrow^2\omega=\varepsilon_0^{\varepsilon_0^\omega}</math>

<math>(\omega\downarrow^3\omega)\downarrow^32=\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}</math>

<math>(\omega\downarrow^3\omega)\downarrow^33=\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}}</math>

<math>\omega\downarrow^43=\varepsilon_1</math>

<math>\omega\downarrow^44=\varepsilon_2</math>

<math>\omega\downarrow^4\omega=\varepsilon_\omega</math>

<math>(\omega\downarrow^4\omega)\downarrow^3\omega=\varepsilon_{\omega+1}</math>

<math>(\omega\downarrow^4\omega)\downarrow^42=\varepsilon_{\varepsilon_\omega}</math>

<math>(\omega\downarrow^4\omega)\downarrow^43=\varepsilon_{\varepsilon_\omega2}</math>

<math>\omega\downarrow^53=\varepsilon_{\varepsilon_\omega\omega}</math>

<math>\omega\downarrow^54=\varepsilon_{\varepsilon_{\varepsilon_\omega\omega}\omega}</math>

<math>\omega\downarrow^5\omega=\zeta_0</math>

<math>\omega\downarrow^6\omega=\zeta_\omega</math>

<math>\omega\downarrow^7\omega=\varphi(3,0)</math>

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

把下箭头用到序数上，其极限为<math>\varphi(\omega,0)</math>，也符合箭头运算的强度。

但是，左结合的下箭头行为和高德纳箭头差异还是不小，而且两个箭头对应一个<math>\varphi(n,0)</math>还是不太符合我们的预期。

== 攀爬法 ==
第二种试图解决问题的方案是攀爬法。攀爬法提供了一种更强的推广。我们知道，<math>\varepsilon_1</math>的基本列是<math>\{\varepsilon_0+1,\omega^{\varepsilon_0+1},\omega^{\omega^{\varepsilon_0+1}},\omega^{\omega^{\omega^{\varepsilon_0+1}}},\cdots\}</math>，我们可以将其表示为<math>\{\omega^{\omega^{\omega^{\cdots}}}+1,\omega^{\omega^{\omega^{\cdots}}+1},\omega^{\omega^{\omega^{\cdots}+1}},\cdots\}</math>.在这里我们把<math>\omega</math>的指数塔固定在<math>\omega</math>。在这样的基本列中，+1像在指数塔攀爬一样，攀爬法也因此而得名。在基本列的尽头，+1攀爬到了指数塔的顶端，与原来在顶端的1相加变为2。因此我们得到<math>\varepsilon_1=\underbrace{\omega^{\omega^{\cdots^{2}}}}_{\omega+1 \text{ layers}}</math>.进一步的，按照攀爬法我们有<math>\varepsilon_{\omega}=\underbrace{\omega^{\omega^{\cdots^{\omega}}}}_{\omega+1 \text{ layers}}</math>,我们将其记为<math>\omega\uparrow\uparrow(\omega+1)=\varepsilon_{\omega}</math>.

进一步有：

<math>\omega\uparrow\uparrow(\omega+2)=\varepsilon_{\omega^{\omega}}</math>

<math>\omega\uparrow\uparrow(\omega\times2)=\varepsilon_{\varepsilon_0}</math>

<math>\omega\uparrow\uparrow(\omega^2)=\zeta_0</math>

<math>\omega\uparrow\uparrow(\omega^3)=\eta_0</math>

<math>\omega\uparrow\uparrow(\omega^{\omega})=\varphi(\omega,0)</math>

<math>\omega\uparrow\uparrow\omega\uparrow\uparrow\omega=\varphi(\varphi(1,0),0)</math>

<math>\omega\uparrow^34=\varphi(\varphi(\varphi(1,0),0),0)</math>

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

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

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

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

<math>\omega\uparrow^44=\varphi(1,\varphi(1,\varphi(1,0,0),0),0)</math>

<math>\omega\uparrow^4\omega=\varphi(2,0,0)</math>

<math>\omega\uparrow^5\omega=\varphi(3,0,0)</math>

<math>\omega\uparrow^{\omega}\omega=\varphi(\omega,0,0)</math>

由此我们得到了攀爬法序数超运算的极限是<math>\varphi(\omega,0,0)</math>.

但是现在已经证明了，攀爬法是非良序的。因此这一做法得到的推广是不可靠的。

== +1法 ==
第三中试图解决问题的方案是+1法。

它基于一种特别朴素的想法，即：如果<math>\omega\uparrow^c\alpha=\alpha</math>,则修改其值为<math>\omega\uparrow^c(\alpha+1)</math>.显然，这一改变真正起到效果的是指数上的变化。当然，类似+1法，还可以用其他函数跳过不动点，例如+ω法，×2法，×ω法等，但不能达到下一个ε点。

关于+1法序数超运算，我们有：

例如：ω^^(ω+1)如果展开为ω^ω^^ω就会遇到不动点，因此触发上述的+1规则。

<math>\omega\uparrow\uparrow(\omega+1)=\omega^{\omega\uparrow\uparrow\omega+1}=\omega^{\varepsilon_0+1}</math>

然后，以此类推：

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

<math>\omega\uparrow\uparrow(\omega2)=\varepsilon_1</math>

<math>\omega\uparrow\uparrow(\omega3)=\varepsilon_2</math>

<math>\omega\uparrow\uparrow(\omega^2)=\varepsilon_{\omega}</math>

<math>\omega\uparrow^33=\omega\uparrow\uparrow\omega\uparrow\uparrow\omega=\varepsilon_{\varepsilon_0}</math>

<math>\omega\uparrow^3\omega=\zeta_0</math>

到ζ_0后，+1法的展开方式将会比较复杂，并产生一些奇特的基础序列。继续分析：

<math>\omega\uparrow^3(\omega+1)=\omega\uparrow\uparrow(\omega\uparrow^3\omega+1)
=\omega\uparrow(\omega\uparrow\uparrow(\omega\uparrow^3\omega)+1)
=\omega\uparrow(\omega\uparrow^3\omega+1)
=\omega^{\zeta_0+1}</math>

<math>\omega\uparrow\uparrow(\omega\uparrow^3\omega+\omega)=\varepsilon_{\zeta_0+1}</math>

<math>\omega\uparrow\uparrow(\omega\uparrow^3\omega*2)=\varepsilon_{\zeta_02}</math>

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

<math>\omega\uparrow^3(\omega+3)=\varepsilon_{\varepsilon_{\omega^{\zeta_0+1}}}</math>

<math>\omega\uparrow^3(\omega2)=\zeta_1</math>

<math>\omega\uparrow^43=\zeta_{\zeta_0}</math>

<math>\omega\uparrow^4\omega=\eta_0</math>

<math>\omega\uparrow^4(\omega+1)=\omega\uparrow^3(\omega\uparrow^4\omega+1)
=\omega\uparrow\omega\uparrow\uparrow\omega\uparrow^3(\omega\uparrow^4\omega+1)
=\omega\uparrow(\omega\uparrow^4\omega+1)
=\omega^{\eta_0+1}</math>

<math>\omega\uparrow^4(\omega+2)=\omega\uparrow^3(\omega\uparrow^4(\omega+1))=\zeta_{\omega^{\eta_0+1}}</math>

<math>\omega\uparrow^4(\omega2)=\eta_1</math>

<math>\omega\uparrow^5\omega=\varphi(4,0)</math>

<math>\omega\uparrow^{\omega}\omega=\varphi(\omega,0)</math>

于是我们得到其极限为<math>\varphi(\omega,0)</math>。它的优点是它和<math>\varphi</math>函数行为完全一致。但缺点也是这个。这导致了<math>\omega\uparrow\uparrow(\omega+1)=(\omega\uparrow\uparrow\omega)\times\omega</math>这种奇异的结果，某种意义上丢失了超运算自己的特性。因此可以说，+1法序数超运算被<math>\varphi</math>函数上位替代了。+1法序数超运算还可以继续拓展，在某些版本中，其增长率与类似的Veblen 函数及Feferman序数折叠函数类似，例如：

<math>\omega\{\omega\}(\omega2)=\varphi(\omega,1)</math>

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

<math>\omega\{\omega\uparrow\uparrow\omega\}\omega=\varphi(\varepsilon_0,0)</math>

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

<math>\omega\{\Omega\}\omega=\Gamma_0</math>

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

<math>\omega\{\Omega2\}\omega=\varphi(2,0,0)</math>

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

<math>\omega\{\Omega^\omega\}\omega=\psi(\Omega^{\Omega^\omega})</math>

<math>\omega\{\Omega^\Omega\}\omega=\psi(\Omega^{\Omega^\Omega})</math>

<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega=\psi(\Omega_2)</math>

但如此的拓展达到同样序数的难度要明显大于一般的[[序数坍缩函数]]。有些版本引入了更为强大的结构，但已经失去了超运算的特性，其强度主要为引入的更高级序数结构，建议使用以更高级核心的序数记号替代之。

类似+1法的+ω法序数超运算由于整度较大，容易分析，方便进行拓展，有如下分析：[[+ω法序数超运算分析]]

== 总结 ==
到目前为止，序数超运算不是不良定义，就是不理想的。我们不建议在任何地方使用高于<math>\varepsilon_0</math>的序数超运算和更高级别的运算。如果要使用，必须仅仅将它作为形式上的符号，并且明确地说明其具体含义。事实上，我们完全可以使用[[Veblen 函数]]这样的更加强大且清晰的[[序数记号]]来替代它。
[[分类:记号]]
[[分类:入门]]
[[分类:序数超运算]]

序数超运算

2026-04-21T20:28:57Z

量子杰克：+ω法序数超运算分析

+ω法序数超运算分析

2026-04-21T19:31:09Z

量子杰克：创建页面，内容为“+ω法序数超运算是一种超运算型记号，作者是量子杰克，与+1法超运算类似，但区别是遇到不动点会+ω而不是+1，原因是根据分析，这样能给出更整的基本列。序数超运算可以进行许多更高级别的拓展。 == 定义 == 一个序数超运算的格式必须是<math>\beta\{\lambda\}\alpha</math>，其中β,λ,α都是正序数。底数β只能是正整数或超限基数。若底数是超限基数，只…”

+ω法序数超运算是一种超运算型记号，作者是量子杰克，与+1法超运算类似，但区别是遇到不动点会+ω而不是+1，原因是根据分析，这样能给出更整的基本列。序数超运算可以进行许多更高级别的拓展。

== 定义 ==
一个序数超运算的格式必须是<math>\beta\{\lambda\}\alpha</math>，其中β,λ,α都是正序数。

底数β只能是正整数或超限基数。若底数是超限基数，只能是<math>\Omega_x</math>，其中x是序数。定义：<math>\Omega_0=\omega</math>，对于正序数x，<math>\Omega_x</math>为第x个不可数基数（不是可数非递归序数，因为在超运算型记号中，不可数与可数非递归的效果存在本质区别）。

若底数有限，则指数α必须有限。若底数为超限基数，则指数的势必须小于等于底数的势。例如，若底数为ω，指数必须可数。若底数为<math>\Omega_x</math>，指数必须小于<math>\Omega_{x+1}</math>。

算符序数λ的势在目前版本中最多允许比底数多不超过I（首个不可达基数）。.

计算规则：

若指数为1，则值为底数。<math>\beta\{\lambda\}1=\beta</math>

若算符序数与指数均为后继序数，使用带跳不动点函数的简单迭代规则。<math>\beta\{\lambda+1\}(\alpha+1)=if(\beta\{\lambda\}\beta\{\lambda+1\}\alpha=\beta\{\lambda+1\}\alpha)?\beta\{\lambda\}j(\beta\{\lambda+1\}\alpha)\ else\ \beta\{\lambda\}\beta\{\lambda+1\}\alpha</math>

跳跃函数j(x)取<math>j(x)=x+\omega</math>，虽然可以取其他值，但根据分析，取x+ω能使基本列更整。

若指数为极限序数，值为对指数取基本列时得到的值的极限。<math>\beta\{\lambda\}\alpha=lim(\beta\{\lambda\}\alpha[x])</math>。共尾性<math>\Omega_x</math>的极限序数的基本列长度为<math>\Omega_x</math>。

若指数为后继序数，算符序数为共尾性小于等于底数的极限序数：对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_y,\ y\leq x</math>，将指数分解为一个<math>\Omega_y</math>的倍数与一个小于<math>\Omega_y</math>的序数之和，<math>\alpha=\Omega_y\times c+d</math>。若α<<math>\Omega_y</math>即c=0，则<math>\beta\{\lambda\}\alpha=\beta\{\lambda\}d=\beta\{\lambda[d]\}\beta</math>。若α><math>\Omega_y</math>即c>0,<math>\beta\{\lambda\}\alpha=\beta\{\lambda\}(\Omega_y\times c+d)=\beta\{\lambda[d]\}j(\beta\{\lambda\}(\Omega_y\times c))</math> 。

若指数为后继序数，算符序数为共尾性为底数的下一个超限基数：对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_{x+1}</math>，展开为：<math>\beta\{\lambda\}(\alpha+1)=if(\beta\{\lambda[\beta\{\lambda\}\alpha]\}\beta=\beta\{\lambda\}\alpha)?\beta\{\lambda[j(\beta\{\lambda\}\alpha)]\}\beta\ else\ \beta\{\lambda[\beta\{\lambda\}\alpha]\}\beta</math>。

若指数是后继序数，算符序数的共尾性大于底数的下一个超限基数，底数必须是超限基数（不能为有限数）。对于<math>\beta=\Omega_x,\ cof(\lambda)=\Omega_{x+y+1}</math>，y为正序数，将指数分为ω的倍数与自然数之和：<math>\alpha=\omega\times c+d</math>。

定义：<math>\lambda_1=\lambda</math>；<math>\lambda_n</math>（n为大于1的正整数）= 将<math>\lambda_{n-1}</math>的表达式中所有x>0的<math>\Omega_x</math>全部替换为<math>\Omega_{x+y}</math>。若指数有限，β{λ}α = Ω_x{λ[Ω_(x+y+1)]}d；β{λ}1 = Ω_x{λ[Ω_(x+y)]}ω；β{λ}(n+1) = 将β{λ}n表达式中的λ_n[Ω_(x+y*n)]替换为λ_n[Ω_(x+y*n)]{λ_(n+1)[Ω_(x+y*(n+1))]}ω。若指数为超限后继序数，β{λ}α = Ω_x{λ[Ω_(x+y+1)]}(ω*c+d)；β{λ}(ω*c+1) = Ω_x{λ[Ω_(x+y)]}j(β{λ}(ω*c))；β{λ}(ω*c+n+1) = 将β{λ}n表达式中的λ_n[Ω_(x+y*n)]替换为λ_n[Ω_(x+y*n)]{λ_(n+1)[Ω_(x+y*(n+1))]}ω；若指数为极限序数，β{λ}α = lim(β{λ}(α[n]))。

== 分析1 ==
{| class="wikitable"
|+分析1：SCO~HCO
!序数超运算
!BMS
|-
|<math>\omega\uparrow\uparrow\omega</math>
|(0)(1,1) = [[SCO]]
|-
|<math>\omega\uparrow\uparrow(\omega+1)=\omega^{\omega\uparrow\uparrow\omega+\omega}</math>
|(0)(1,1)(1)(2)
|-
|<math>\omega\uparrow\uparrow(\omega+2)=\omega^{\omega^{\omega\uparrow\uparrow\omega+\omega}}</math>
|(0)(1,1)(1)(2,1)(2)(3)
|-
|<math>\omega\uparrow\uparrow(\omega+3)</math>
|(0)(1,1)(1)(2,1)(2)(3,1)(3)(4)
|-
|<math>\omega\uparrow\uparrow\omega2</math>
|(0)(1,1)(1,1)
|-
|<math>\omega\uparrow\uparrow(\omega2+1)=\omega^{\omega\uparrow\uparrow\omega2+\omega}</math>
|(0)(1,1)(1,1)(1)(2)
|-
|<math>\omega\uparrow\uparrow\omega3</math>
|(0)(1,1)(1,1)(1,1)
|-
|<math>\omega\uparrow\uparrow\omega^2</math>
|(0)(1,1)(2)
|-
|<math>\omega\uparrow\uparrow(\omega^2+\omega)</math>
|(0)(1,1)(2)(1,1)
|-
|<math>\omega\uparrow\uparrow\omega^22</math>
|(0)(1,1)(2)(1,1)(2)
|-
|<math>\omega\uparrow\uparrow\omega^3</math>
|(0)(1,1)(2)(2)
|-
|<math>\omega\uparrow\uparrow\omega^\omega</math>
|(0)(1,1)(2)(3)
|-
|<math>\omega\uparrow\uparrow\uparrow3=\omega\uparrow\uparrow\omega\uparrow\uparrow\omega</math>
|(0)(1,1)(2)(3,1)
|-
|<math>\omega\uparrow\uparrow\uparrow4</math>
|(0)(1,1)(2)(3,1)(4)(5,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega</math>
|(0)(1,1)(2,1) = [[CO]]
|-
|<math>\omega\uparrow\uparrow\uparrow(\omega+1)=\omega\uparrow\uparrow(\omega\uparrow\uparrow\uparrow\omega+\omega)</math>
|(0)(1,1)(2,1)(1,1)
|-
|<math>\omega\uparrow\uparrow\uparrow(\omega+2)</math>
|(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega2</math>
|(0)(1,1)(2,1)(1,1)(2,1)
|-
|<math>\omega\uparrow\uparrow\uparrow\omega^2</math>
|(0)(1,1)(2,1)(2)
|-
|<math>\omega\{4\}3</math>
|(0)(1,1)(2,1)(2)(3,1)(4,1)
|-
|<math>\omega\{4\}\omega</math>
|(0)(1,1)(2,1)(2,1) = [[LCO]]
|-
|<math>\omega\{4\}(\omega+1) = \omega\uparrow\uparrow\uparrow(\omega\{4\}\omega+\omega)</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2,1)
|-
|<math>\omega\{4\}\omega2</math>
|(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)
|-
|<math>\omega\{5\}\omega</math>
|(0)(1,1)(2,1)(2,1)(2,1)
|-
|<math>\omega\{\omega\}\omega</math>
|(0)(1,1)(2,1)(3) = [[HCO]]
|}

== 分析2 ==
{| class="wikitable"
|+分析2：HCO~FSO
!序数超运算
!BMS
|-
|<math>\omega\{\omega\}\omega</math>
|(0)(1,1)(2,1)(3) = [[HCO]]
|-
|<math>\omega\{\omega\}(\omega+1)=\omega^{\omega\{\omega\}\omega+\omega}</math>
|(0)(1,1)(2,1)(3)(1)(2)
|-
|<math>\omega\{\omega\}(\omega+2)=\omega\uparrow\uparrow(\omega\{\omega\}\omega+\omega)</math>
|(0)(1,1)(2,1)(3)(1,1)
|-
|<math>\omega\{\omega\}(\omega+3)=\omega\uparrow\uparrow\uparrow(\omega\{\omega\}\omega+\omega)</math>
|(0)(1,1)(2,1)(3)(1,1)(2,1)
|-
|<math>\omega\{\omega\}\omega2</math>
|(0)(1,1)(2,1)(3)(1,1)(2,1)(3)
|-
|<math>\omega\{\omega\}\omega^2</math>
|(0)(1,1)(2,1)(3)(2)
|-
|<math>\omega\{\omega+1\}3</math>
|(0)(1,1)(2,1)(3)(2)(3,1)(4,1)(5)
|-
|<math>\omega\{\omega+1\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)
|-
|<math>\omega\{\omega+2\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)(2,1)
|-
|<math>\omega\{\omega2\}\omega</math>
|(0)(1,1)(2,1)(3)(2,1)(3)
|-
|<math>\omega\{\omega^2\}\omega</math>
|(0)(1,1)(2,1)(3)(3)
|-
|<math>\omega\{\omega^\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4)
|-
|<math>\omega\{\omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4,1)
|-
|<math>\omega\{\Omega\}3=\omega\{\omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3)(4,1)(5,1)(6)
|-
|<math>\omega\{\Omega\}4</math>
|(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1)(9)
|-
|<math>\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1) = [[FSO]]
|}

== 分析3 ==
{| class="wikitable"
|+分析3：FSO~BHO
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1) = [[FSO]]
|-
|<math>\omega\{\omega\{\Omega\}\omega\}(\omega+1)=\omega\{\omega\}\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)
|-
|<math>\omega\{\omega\{\Omega\}\omega\}(\omega+2)=\omega\{\omega\{\omega\}\omega\}\omega\{\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6)
|-
|<math>\omega\{\omega\{\Omega\}\omega\}\omega2</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)
|-
|<math>\omega\{\omega\{\Omega\}\omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)
|-
|<math>\omega\{\Omega\}(\omega+1)=\omega\{\omega\{\Omega\}\omega+\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)(3)
|-
|<math>\omega\{\Omega\}(\omega+2)=\omega\{\omega\{\omega\{\Omega\}\omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)(5,1)(6)(7,1)(8,1)(9,1)(5,1)(6)
|-
|<math>\omega\{\Omega\}\omega2</math>
|(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\}\omega^2</math>
|(0)(1,1)(2,1)(3,1)(2)
|-
|<math>\omega\{\Omega+1\}3</math>
|(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)
|-
|<math>\omega\{\Omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)
|-
|<math>\omega\{\Omega+2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(2,1)
|-
|<math>\omega\{\Omega+\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3)
|-
|<math>\omega\{\Omega2\}3=\omega\{\Omega+\omega\{\Omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1)(5,1)(4,1)(5)
|-
|<math>\omega\{\Omega2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3)
|-
|<math>\omega\{\Omega^2\}3=\omega\{\Omega\times\omega\{\Omega\omega\}\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3)(4,1)(5,1)(5)
|-
|<math>\omega\{\Omega^2\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1) = [[ACO]]
|-
|<math>\omega\{\Omega^2+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)
|-
|<math>\omega\{\Omega^22\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1)
|-
|<math>\omega\{\Omega^2\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(3)
|-
|<math>\omega\{\Omega^3\}\omega</math>
|(0)(1,1)(2,1)(3,1)(3,1)(3,1)
|-
|<math>\omega\{\Omega^\omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4) = [[SVO]]
|-
|<math>\omega\{\Omega^\Omega\}3=\omega\{\Omega^{\omega\{\Omega^\omega\}\omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8)
|-
|<math>\omega\{\Omega^\Omega\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1) = [[LVO]]
|-
|<math>\omega\{\Omega^\Omega+1\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(2,1)
|-
|<math>\omega\{\Omega^{\Omega+1}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)
|-
|<math>\omega\{\Omega^{\Omega2}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1)
|-
|<math>\omega\{\Omega^{\Omega^2}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(4,1)
|-
|<math>\omega\{\Omega^{\Omega^\omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}3=\omega\{\Omega^{\Omega^\Omega}\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}4=\omega\{\Omega\uparrow\uparrow4\}\omega</math>
|(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2) = [[BHO]]
|}

== 分析4 ==
{| class="wikitable"
|+分析4：BHO~(0)(1,1)(2,2)(3,2)(4)
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2) = [[BHO]]
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}(\omega+1)=\omega\{\Omega\}(\omega\{\Omega\uparrow\uparrow\omega\}\omega+\omega)</math>
|(0)(1,1)(2,2)(1,1)(2,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}(\omega+1)=\omega\{\Omega^\Omega\}(\omega\{\Omega\uparrow\uparrow\omega\}\omega+\omega)</math>
|(0)(1,1)(2,2)(1,1)(2,1)(3,1)(4,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega2</math>
|(0)(1,1)(2,2)(1,1)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\}\omega^2</math>
|(0)(1,1)(2,2)(2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+1\}3</math>
|(0)(1,1)(2,2)(2)(3,1)(4,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+1\}\omega</math>
|(0)(1,1)(2,2)(2,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega+\Omega\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\times2\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega\times\omega\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow(\omega+1)\}\omega=\omega\{\omega^{\Omega\uparrow\uparrow\omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega2\}\omega</math>
|(0)(1,1)(2,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega3\}\omega</math>
|(0)(1,1)(2,2)(2,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\}3=\omega\{\Omega\uparrow\uparrow\omega\{\Omega\uparrow\uparrow\Omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3)(4,1)(5,2)(6)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow(\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,1)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega2\}\omega</math>
|(0)(1,1)(2,2)(3,1)(2,2)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,1)(3,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\Omega\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2)(3,1)(4,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow3\}\omega</math>
|(0)(1,1)(2,2)(3,1)(4,2)(5,1)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow(\omega+1)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\uparrow\uparrow\uparrow\omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(2,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(2,2)(3,2)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3)
|-
|<math>\omega\{\Omega\uparrow\uparrow\uparrow\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,1)
|-
|<math>\omega\{\Omega\{4\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)
|-
|<math>\omega\{\Omega\{\omega\}4\}\omega=\omega\{\Omega\{4\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)(3,1)
|-
|<math>\omega\{\Omega\{5\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(3,2)(3,2)
|-
|<math>\omega\{\Omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)
|}

== 分析5 ==
{| class="wikitable"
|+分析：(0)(1,1)(2,2)(3,2)(4)~(0)(1,1)(2,2)(3,2)(4,2)
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\{\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega\}(\omega+1)\}\omega=\omega\{\Omega^{\Omega\{\omega\}\omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\{\omega\}(\omega+2)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\{\omega\}\omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,2)
|-
|<math>\omega\{\Omega\{\omega\}\omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega\}\omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3)
|-
|<math>\omega\{\Omega\{\Omega\}\omega\}\omega=\omega\{\Omega\{\omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,1)
|-
|<math>\omega\{\Omega\{\omega+1\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,2)
|-
|<math>\omega\{\Omega\{\omega2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(3,2)(4)
|-
|<math>\omega\{\Omega\{\omega^2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(4)
|-
|<math>\omega\{\Omega\{\omega\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(5,1)
|-
|<math>\omega\{\Omega\{\omega\{\Omega\uparrow\uparrow\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4)(5,1)(6,2)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+1)\}\omega=\omega\{\Omega^{\Omega\{\Omega\}\Omega+\omega}\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,1)(3,2)(3)(4)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+2)\}\omega=\omega\{\Omega\uparrow\uparrow(\Omega\{\Omega\}\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)
|-
|<math>\omega\{\Omega\{\Omega\}(\Omega+\omega)\}\omega=\omega\{\Omega\{\omega\}(\Omega\{\Omega\}\Omega+\omega)\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)(3,2)(4)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(2,2)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3)
|-
|<math>\omega\{\Omega\{\Omega\}\Omega^2\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,1)
|-
|<math>\omega\{\Omega\{\Omega+1\}3\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,1)(4,2)(5,3)(6,1)
|-
|<math>\omega\{\Omega\{\Omega+1\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)
|-
|<math>\omega\{\Omega\{\Omega+\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)(4)
|-
|<math>\omega\{\Omega\{\Omega2\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(3,2)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(4)
|-
|<math>\omega\{\Omega\{\Omega^2\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(4,1)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,1)
|-
|<math>\omega\{\Omega\{\Omega\uparrow\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)
|-
|<math>\omega\{\Omega\{\Omega\{\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}3=\omega\{\Omega\{\Omega_2\}3\}\omega=\omega\{\Omega\{\Omega\{\Omega\}\Omega\}\Omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}4=\omega\{\Omega\{\Omega_2\}4\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)(8,2)(9,2)(10,1)
|-
|<math>\omega\{\Omega\{\Omega_2\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,2)(4,2)
|}

== 预期强度 ==
{| class="wikitable"
|+预期强度（已进行部分分析，待填写）
!序数超运算
!BMS
|-
|<math>\omega\{\Omega\{\Omega_2\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)
|-
|<math>\omega\{\Omega\{\Omega_2\uparrow\uparrow\uparrow\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)
|-
|<math>\omega\{\Omega\{\Omega_2\{\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)(5)
|-
|<math>\omega\{\Omega\{\Omega_2\{\Omega_3\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,3)(5,3)
|-
|<math>\omega\{\Omega\{\Omega_2\{\Omega_3\uparrow\uparrow\omega\}\omega\}\omega\}\omega</math>
|(0)(1,1)(2,2)(3,3)(4,4)
|-
|<math>\omega\{\Omega_2\}\omega</math>
|(0)(1,1,1) = [[BO]]
|-
|<math>\omega\{\Omega_22\}\omega</math>
|(0)(1,1,1)(1,1,1)
|-
|<math>\omega\{\Omega_2\Omega\}\omega</math>
|(0)(1,1,1)(2,1)
|-
|<math>\omega\{\Omega_2\times(\Omega+1)\}\omega</math>
|(0)(1,1,1)(2,1)(1,1,1)
|-
|<math>\omega\{\Omega_2\times(\Omega\uparrow\uparrow\omega)\}\omega</math>
|(0)(1,1,1)(2,1)(3,2) = [[TFBO]]
|-
|<math>\omega\{\Omega_2\times(\Omega\{\Omega_3\}\omega)\}\omega</math>
|(0)(1,1,1)(2,1)(3,2,1)
|-
|<math>\omega\{\Omega_2^2\}\omega</math>
|(0)(1,1,1)(2,1,1)
|-
|<math>\omega\{\Omega_2^\Omega\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1) = [[BIO]]
|-
|<math>\omega\{\Omega_2^\Omega\omega\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1)(2) = [[EBO]]
|-
|<math>\omega\{\Omega_2^{\Omega\uparrow\uparrow\omega}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1)(4,2) = [[JO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1) = [[SIO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2^2}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1) = [[SMO]]
|-
|<math>\omega\{\Omega_2^{\Omega_2^{\Omega_2}}\}\omega</math>
|(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) = [[SKO]]
|-
|<math>\omega\{\Omega_2\uparrow\uparrow\omega\}\omega</math>
|(0)(1,1,1)(2,2) = [[SSO]]
|-
|<math>\omega\{\Omega_2\{\Omega\}\Omega\times\omega\}\omega</math>
|(0)(1,1,1)(2,2)(3,2)(4,1)(2) = [[LSO]]
|-
|<math>\omega\{\Omega_2\{\Omega_2\}\Omega_2\}\omega</math>
|(0)(1,1,1)(2,2)(3,2)(4,1,1) = [[APO]]
|-
|<math>\omega\{\Omega_3\}\omega</math>
|(0)(1,1,1)(2,2,1) = [[BGO]]
|-
|<math>\omega\{\Omega_4\}\omega</math>
|(0)(1,1,1)(2,2,1)(2,2,1)
|-
|<math>\omega\{\Omega_\omega\}\omega</math>
|(0)(1,1,1)(2,2,1)(3) = [[SDO]]
|-
|<math>\omega\{\Omega_\Omega\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,1)
|-
|<math>\omega\{\Omega_{\Omega_2}\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,1,1)
|-
|<math>\omega\{\Phi(1,0)\}\omega</math>
|(0)(1,1,1)(2,2,1)(3,2) = [[LDO]]
|-
|<math>psd.\omega\{I\}\omega=\omega\{\psi_I(X)\}\omega\ ct.\ \psi(X)</math>
|(0)(1,1,1)(2,2,1)(3,2,1)
|-
|<math>psd. \omega\{\psi(X)\}\omega\ ct.\ \psi(X)</math>
|(0)(1,1,1)(2,2,2) = [[LRO|pfec LRO]]
|}