+ω法序数超运算分析

2026-04-21T21:47:58Z

←上一版本		2026年4月22日 (三) 05:47的版本
（未显示同一用户的1个中间版本）
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	\|(0)(1,1,1)(2,2,1)(3,2,1)		\|(0)(1,1,1)(2,2,1)(3,2,1)
	\|-		\|-
	\|<math>psd. \omega\{~~\psi(~~X)\}\omega\ ct.\ \psi(X)</math>		\|<math>psd. \omega\{X\}\omega\ ct.\ \psi(X)</math>
	\|(0)(1,1,1)(2,2,2) = [[LRO\|pfec LRO]]		\|(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

分类

←上一版本		2026年4月22日 (三) 04:30的版本
（未显示同一用户的1个中间版本）
第120行：		第120行：
	第三中试图解决问题的方案是+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。关于~~+1法序数超运算，我们有：		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。		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。
第199行：		第201行：

	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。		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 ==		== 总结 ==
第204行：		第208行：
	[[分类:记号]]		[[分类:记号]]
	[[分类:入门]]		[[分类:入门]]
			[[分类:序数超运算]]

+ω法序数超运算分析

2026-04-21T19:31:09Z

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…”

新页面

+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]]
|}

Googology Wiki - 最近更改 [zh-cn]

+ω法序数超运算分析

序数超运算

+ω法序数超运算分析

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