weak Veblen 函数：修订间差异

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2025年8月26日 (二) 18:30的最新版本

weak Veblen 函数（又称“弱 φ”、“弱 Veblen”等），是 Veblen 函数的变体。

定义

现在有两种 Veblen 函数的变体可被称为 weak Veblen。

如果称 Veblen 函数的末位 + 1 等价于跳到下一个 ε 点（不动点进制），定义 Veblen 函数为 $F P φ$ ，那么可以直观地定义下面两种weak Veblen：

若 weak Veblen 函数的末位 + 1 对应于其序数值 $\times ω$ ，则称其为 $\times ω ϕ$ 。

若 weak Veblen 函数的末位 + 1 对应于其序数值 $+ 1$ ，则称其为 $+ 1 φ$ 。

以下是两种 weak Veblen 直到序元级别的定义：

$+ 1 ϕ$

$\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[0]=0$。
$\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)。[n+1]=\varphi(\#,\alpha\text{@}(\beta+1),\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[n]\text{@}\beta)$。
对于极限序数$\alpha$，$\varphi(\#,\alpha\text{@}\beta)[n]=\varphi(\#,\alpha[n]\text{@}\beta)$。
对于极限序数$\beta$，$\varphi(\#,(\alpha+1)\text{@}\beta,0\text{@}0)。[n]=\varphi(\#,\alpha\text{@}\beta,1\text{@}\beta[n])$。
$\varphi(\#,(\gamma+1)\text{@}0)=\varphi(\#,\gamma\text{@}0)+1$。

$\times ω ϕ$

$\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[0]=0$。
$\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)。[n+1]=\varphi(\#,\alpha\text{@}(\beta+1),\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[n]\text{@}\beta)$。
对于极限序数$\alpha$，$\varphi(\#,\alpha\text{@}\beta)[n]=\varphi(\#,\alpha[n]\text{@}\beta)$。
对于极限序数$\beta$，$\varphi(\#,(\alpha+1)\text{@}\beta,0\text{@}0)[n]=\varphi(\#,\alpha\text{@}\beta,1\text{@}\beta[n])$。
$\varphi(\#,(\gamma+1)\text{@}0)=\varphi(\#,\gamma\text{@}0)\cdot\varphi(1)$。

分析

以下分析中，左为 $F P ϕ$ ，中为 $\times ω ϕ$ ，右为 $+ 1 ϕ$ 。

$\varphi(0)=\varphi(0)=\varphi(0)$

$\varphi(0)\cdot2=\varphi(0)\cdot2=\varphi(\varphi(0))$

$\varphi(0)\cdot3=\varphi(0)\cdot3=\varphi(\varphi(\varphi(0)))$

$\varphi(1)=\varphi(1)=\varphi(1,0)=$ ω

$\varphi(1)+1=\varphi(1)+1=\varphi(1,1)$

$\varphi(1)+2=\varphi(1)+2=\varphi(1,2)$

$\varphi(1)\cdot2=\varphi(1)\cdot2=\varphi(1,\varphi(1,0))$

$\varphi(1)\cdot2+1=\varphi(1)\cdot2+1=\varphi(1,\varphi(1,1))$

$\varphi(1)\cdot2+2=\varphi(1)\cdot2+2=\varphi(1,\varphi(1,2))$

$\varphi(1)\cdot3=\varphi(1)\cdot3=\varphi(1,\varphi(1,\varphi(1,0)))$

$\varphi(2)=\varphi(2)=\varphi(2,0)$

$\varphi(2)+1=\varphi(2)+1=\varphi(2,1)$

$\varphi(2)+2=\varphi(2)+2=\varphi(2,2)$

$\varphi(2)+\varphi(1)=\varphi(2)+\varphi(1)=\varphi(2,\varphi(1,0))$

$\varphi(2)\cdot2=\varphi(2)\cdot2=\varphi(2,\varphi(2,0))$

$\varphi(2)\cdot3=\varphi(2)\cdot3=\varphi(2,\varphi(2,\varphi(2,0)))$

$\varphi(3)=\varphi(3)=\varphi(3,0)$

$\varphi(3)+\varphi(2)=\varphi(3)+\varphi(2)=\varphi(3,\varphi(2,0))$

$\varphi(4)=\varphi(4)=\varphi(4,0)$

$\varphi(\varphi(1))=\varphi(\varphi(1))=\varphi(\varphi(1,0),0)$

$\varphi(\varphi(1))+1=\varphi(\varphi(1))+1=\varphi(\varphi(1,0),1)$

$\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1,0),\varphi(1,0))$

$\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1,0),\varphi(2,0))$

$\varphi(\varphi(1))\cdot2=\varphi(\varphi(1))\cdot2=\varphi(\varphi(1,0),\varphi(\varphi(1,0),0))$

$\varphi(\varphi(1)+1)=\varphi(\varphi(1)+1)=\varphi(\varphi(1,1),0)$

$\varphi(\varphi(1)+2)=\varphi(\varphi(1)+2)=\varphi(\varphi(1,2),0)$

$\varphi(\varphi(1)+3)=\varphi(\varphi(1)+3)=\varphi(\varphi(1,3),0)$

$\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1,\varphi(1,0)),0)$

$\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1,\varphi(1,1)),0)$

$\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1,\varphi(1,\varphi(1,0))),0)$

$\varphi(\varphi(2))=\varphi(\varphi(2))=\varphi(\varphi(2,0),0)$

$\varphi(\varphi(2)+1)=\varphi(\varphi(2)+1)=\varphi(\varphi(2,1),0)$

$\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2,\varphi(1,0)),0)$

$\varphi(\varphi(3))=\varphi(\varphi(3))=\varphi(\varphi(3,0),0)$

$\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1,0),0),0)$

$\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1,0),1),0)$

$\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1,0),\varphi(1,0)),0)$

$\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1,0),\varphi(\varphi(1,0),0)),0)$

$\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1,1),0),0)$

$\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2,0),0),0)$

$\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3,0),0),0)$

$\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1,0),0),0),0)$

$\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1,0),0),0),0),0)$

$\varphi(1,0)=\varphi(1,0)=\varphi(1,0,0)=$ ε0

$\varphi(1,0)+1=\varphi(1,0)+1=\varphi(1,0,1)$

$\varphi(1,0)+\varphi(1)=\varphi(1,0)+\varphi(1)=\varphi(1,0,\varphi(1,0))$

$\varphi(1,0)\cdot2=\varphi(1,0)\cdot2=\varphi(1,0,\varphi(1,0,0))$

$\varphi(1,0)\cdot3=\varphi(1,0)\cdot3=\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))$

$\varphi(1,0)\cdot\varphi(1)=\varphi(1,1)=\varphi(1,1,0)$

$\varphi(1,0)\cdot\varphi(1)\cdot2=\varphi(1,1)\cdot2=\varphi(1,1,\varphi(1,1,0))$

$\varphi(1,0)\cdot\varphi(2)=\varphi(1,2)=\varphi(1,2,0)$

$\varphi(1,0)\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1))=\varphi(1,\varphi(1,0),0)$

$\varphi(1,0)\cdot\varphi(\varphi(1)+1)=\varphi(1,\varphi(1)+1)=\varphi(1,\varphi(1,1),0)$

$\varphi(1,0)\cdot\varphi(\varphi(2))=\varphi(1,\varphi(2))=\varphi(1,\varphi(2,0),0)$

$\varphi(1,0)\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1,0),0),0)$

$\varphi(1,0)^{2}=\varphi(1,\varphi(1,0))=\varphi(1,\varphi(1,0,0),0)$

$\varphi(1,0)^{2}\cdot\varphi(1)=\varphi(1,\varphi(1,0)+1)=\varphi(1,\varphi(\varphi(1,0,0)),0)$

$\varphi(1,0)^{2}\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(1,0)+\varphi(\varphi(1)))=\varphi(1,\varphi(1,0,\varphi(\varphi(1,0),0)),0)$

$\varphi(1,0)^{3}=\varphi(1,\varphi(1,0)\cdot2)=\varphi(1,\varphi(1,0,\varphi(1,0,0),0),0)$

$\varphi(1,0)^{\varphi(1)}=\varphi(1,\varphi(1,1))=\varphi(1,\varphi(1,1,0),0)$

$\varphi(1,0)^{\varphi(1)}\cdot\varphi(1)=\varphi(1,\varphi(1,1)+1)=\varphi(1,\varphi(\varphi(1,1,0)),0)$

$\varphi(1,0)^{\varphi(1)}\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1,1)+2)=\varphi(1,\varphi(\varphi(\varphi(1,1,0))),0)$

$\varphi(1,0)^{(\varphi(1)+1)}=\varphi(1,\varphi(1,1)+\varphi(1,0))=\varphi(1,\varphi(1,1,\varphi(1,0,0)),0)$

$\varphi(1,0)^{(\varphi(1)\cdot2)}=\varphi(1,\varphi(1,1)\cdot2)=\varphi(1,\varphi(1,1,\varphi(1,1,0)),0)$

$\varphi(1,0)^{\varphi(1)^{2}}=\varphi(1,\varphi(1,2))=\varphi(1,\varphi(1,2,0),0)$

$\varphi(1,0)^{\varphi(1)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1)))=\varphi(1,\varphi(1,\varphi(1,0),0),0)$

$\varphi(1,0)^{\varphi(1,0)}=\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(1,\varphi(1,\varphi(1,0,0),0),0)$

$\varphi(1,0)^{(\varphi(1,0)+1)}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,0))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,0,0)),0)$

$\varphi(1,0)^{(\varphi(1,0)\cdot\varphi(1))}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,1))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,1,0)),0)$

$\varphi(1,0)^{\varphi(1,0)^{2}}=\varphi(1,\varphi(1,\varphi(1,2)))=\varphi(1,\varphi(1,\varphi(1,2,0),0),0)$

$\varphi(1,0)^{\varphi(1,0)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0),0),0),0)$

$\varphi(1,0)^{\varphi(1,0)^{\varphi(1,0)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)$

$\varphi(1,1)=\varphi(2,0)=\varphi(2,0,0)$

$\varphi(1,1)\cdot\varphi(1,0)=\varphi(2,\varphi(1,0))=\varphi(2,\varphi(1,0,0),0)$

$\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1)}=\varphi(2,\varphi(1,\varphi(1,1)))=\varphi(2,\varphi(1,\varphi(1,1,0),0),0)$

$\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1,0)}=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)$

$\varphi(1,1)^{2}=\varphi(2,\varphi(2,0))=\varphi(2,\varphi(2,0,0),0)$

$\varphi(1,1)^{\varphi(1)}=\varphi(2,\varphi(2,1))=\varphi(2,\varphi(2,\varphi(1,0),0),0)$

$\varphi(1,1)^{\varphi(1,0)}=\varphi(2,\varphi(2,\varphi(1,0)))=\varphi(2,\varphi(2,\varphi(1,0,0),0),0)$

$\varphi(1,1)^{\varphi(1,1)}=\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(2,\varphi(2,\varphi(2,0,0),0),0)$

$\varphi(1,2)=\varphi(3,0)=\varphi(3,0,0)$

$\varphi(1,\varphi(1))=\varphi(\varphi(1),0)=\varphi(\varphi(1,0),0,0)$

$\varphi(1,\varphi(1))\cdot\varphi(1,0)=\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0),\varphi(1,0,0),0)$

$\varphi(1,\varphi(1))\cdot\varphi(1,1)=\varphi(\varphi(1),\varphi(1,1))=\varphi(\varphi(1,0),\varphi(1,1,0),0)$

$\varphi(1,\varphi(1))^{2}=\varphi(\varphi(1),\varphi(1,\varphi(1)))=\varphi(\varphi(1,0),\varphi(1,\varphi(1,0),0))$

$\varphi(1,\varphi(1)+1)=\varphi(\varphi(1)+1,0)=\varphi(\varphi(1,1),0,0)$

$\varphi(1,\varphi(1)+2)=\varphi(\varphi(1)+2,0)=\varphi(\varphi(1,2),0,0)$

$\varphi(1,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0)=\varphi(\varphi(1,\varphi(1,0)),0,0)$

$\varphi(1,\varphi(1)^{2})=\varphi(\varphi(2),0)=\varphi(\varphi(2,0),0,0)$

$\varphi(1,\varphi(1)^{3})=\varphi(\varphi(3),0)=\varphi(\varphi(3,0),0,0)$

$\varphi(1,\varphi(1)^{\varphi(1)})=\varphi(\varphi(\varphi(1)),0)=\varphi(\varphi(\varphi(1,0),0),0,0)$

$\varphi(1,\varphi(1,0))=\varphi(\varphi(1,0),0)=\varphi(\varphi(1,0,0),0,0)$

$\varphi(1,\varphi(1,0)+1)=\varphi(\varphi(1,0)+1,0)=\varphi(\varphi(1,0,1),0,0)$

$\varphi(1,\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,1),0)=\varphi(\varphi(1,1,0),0,0)$

$\varphi(1,\varphi(1,0)^{\varphi(1)})=\varphi(\varphi(1,\varphi(1,1)),0)=\varphi(\varphi(1,\varphi(1,1,0),0),0,0)$

$\varphi(1,\varphi(1,1))=\varphi(\varphi(2,0),0)=\varphi(\varphi(2,0,0),0,0)$

$\varphi(1,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0)=\varphi(\varphi(\varphi(1,0),0,0),0,0)$

$\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(\varphi(\varphi(1,0),0),0)=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)$

$\varphi(2,0)=\varphi(1,0,0)=\varphi(1,0,0,0)=$ ζ0

$\varphi(1,\varphi(2,0)+1)=\varphi(\varphi(1,0,0),0)=\varphi(\varphi(1,0,0,0),0,0)$

$\varphi(1,\varphi(2,0)+\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(\varphi(1,0,0,\varphi(1,0)),0,0)$

$\varphi(1,\varphi(2,0)+\varphi(1,0))=\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(\varphi(1,0,0,\varphi(1,0,0)),0,0)$

$\varphi(1,\varphi(2,0)+\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1,1),0)=\varphi(\varphi(1,0,0,\varphi(1,1,0)),0,0)$

$\varphi(1,\varphi(2,0)+\varphi(1,1))=\varphi(\varphi(1,0,0)+\varphi(2,0),0)=\varphi(\varphi(1,0,0,\varphi(2,0,0)),0,0)$

$\varphi(1,\varphi(2,0)+\varphi(1,\varphi(1,0)))=\varphi(\varphi(1,0,0)+\varphi(\varphi(1,0),0),0)=\varphi(\varphi(1,0,0,\varphi(\varphi(1,0,0),0,0)),0,0))$

$\varphi(1,\varphi(2,0)\cdot2)=\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(\varphi(1,0,1,0),0,0)$

$\varphi(1,\varphi(2,0)\cdot\varphi(1))=\varphi(\varphi(1,1,0),0)=\varphi(\varphi(1,0,\varphi(1,0),0),0,0)$

$\varphi(1,\varphi(2,0)^{2})=\varphi(\varphi(1,0,\varphi(1,0,0)),0)=\varphi(\varphi(1,0,\varphi(1,0,0,0),0),0,0)$

$\varphi(1,\varphi(1,\varphi(2,0)+1))=\varphi(\varphi(\varphi(1,0,0),0),0)=\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0)$

$\varphi(1,\varphi(1,\varphi(1,\varphi(2,0)+1)))=\varphi(\varphi(\varphi(\varphi(1,0,0),0),0),0)=\varphi(\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0),0,0)$

$\varphi(2,1)=\varphi(2,0,0)=\varphi(2,0,0,0)$

$\varphi(1,\varphi(2,1)+1)=\varphi(\varphi(2,0,0),0)=\varphi(\varphi(2,0,0,0),0,0)$

$\varphi(1,\varphi(1,\varphi(2,1)+1))=\varphi(\varphi(\varphi(2,0,0),0),0)=\varphi(\varphi(\varphi(2,0,0,0),0,0),0,0)$

$\varphi(2,2)=\varphi(3,0,0)=\varphi(3,0,0,0)$

$\varphi(2,\varphi(1))=\varphi(\varphi(1),0,0)=\varphi(\varphi(1,0),0,0,0)$

$\varphi(2,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0,0)=\varphi(\varphi(1,\varphi(1,0)),0,0,0)$

$\varphi(2,\varphi(2))=\varphi(\varphi(2),0,0)=\varphi(\varphi(2,0),0,0,0)$

$\varphi(2,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0)$

$\varphi(2,\varphi(1,0))=\varphi(\varphi(1,0),0,0)=\varphi(\varphi(1,0,0),0,0,0)$

$\varphi(2,\varphi(1,1))=\varphi(\varphi(2,0),0,0)=\varphi(\varphi(2,0,0),0,0,0)$

$\varphi(2,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0,0)=\varphi(\varphi(\varphi(1,0),0,0),0,0,0)$

$\varphi(2,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0)=\varphi(\varphi(1,0,0,0),0,0,0)$

$\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)=\varphi(\varphi(\varphi(1,0,0,0),0,0,0),0,0,0)$

$\varphi(3,0)=\varphi(1,0,0,0)=\varphi(1,0,0,0,0)=$ η0

$\varphi(1,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0)=\varphi(\varphi(1,0,0,0,0),0,0)$

$\varphi(2,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0)$

$\varphi(3,1)=\varphi(2,0,0,0)=\varphi(2,0,0,0,0)$

$\varphi(3,\varphi(1))=\varphi(\varphi(1),0,0,0)=\varphi(\varphi(1,0),0,0,0,0)$

$\varphi(3,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0,0)$

$\varphi(3,\varphi(1,0))=\varphi(\varphi(1,0),0,0,0)=\varphi(\varphi(1,0,0),0,0,0,0)$

$\varphi(3,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0,0)=\varphi(\varphi(1,0,0,0),0,0,0,0)$

$\varphi(3,\varphi(3,0))=\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0,0)$

$\varphi(4,0)=\varphi(1,0,0,0,0)=\varphi(1,0,0,0,0,0)$

$\varphi(\varphi(1),0)=\varphi(1@\varphi(1))=\varphi(1@\varphi(1,0))=$ HCO

$\varphi(1,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0)=\varphi(\varphi(1@\varphi(1,0)),0,0)$

$\varphi(2,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0,0)=\varphi(\varphi(1@\varphi(1,0)),0,0,0)$

$\varphi(3,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0,0,0)=\varphi(\varphi(1@\varphi(1,0)),0,0,0,0)$

$\varphi(\varphi(1),1)=\varphi(2@\varphi(1))=\varphi(2@\varphi(1,0))$

$\varphi(1,\varphi(\varphi(1),1)+1)=\varphi(\varphi(2@\varphi(1)),0)=\varphi(\varphi(2@\varphi(1,0)),0,0)$

$\varphi(\varphi(1),2)=\varphi(3@\varphi(1))=\varphi(3@\varphi(1,0))$

$\varphi(\varphi(1),\varphi(1))=\varphi(\varphi(1)@\varphi(1))=\varphi(\varphi(1,0)@\varphi(1,0))$

$\varphi(\varphi(1),\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2@\varphi(1))=\varphi(\varphi(1,0)\cdot2@\varphi(1,0))$

$\varphi(\varphi(1),\varphi(2))=\varphi(\varphi(2)@\varphi(1))=\varphi(\varphi(2,0)@\varphi(1,0))$

$\varphi(\varphi(1),\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1))@\varphi(1))=\varphi(\varphi(\varphi(1,0),0)@\varphi(1,0))$

$\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0)@\varphi(1))=\varphi(\varphi(1,0,0)@\varphi(1,0))$

$\varphi(\varphi(1),\varphi(2,0))=\varphi(\varphi(1,0,0)@\varphi(1))=\varphi(\varphi(1,0,0,0)@\varphi(1,0))$

$\varphi(\varphi(1),\varphi(3,0))=\varphi(\varphi(1,0,0,0)@\varphi(1))=\varphi(\varphi(1,0,0,0,0)@\varphi(1,0))$

$\varphi(\varphi(1),\varphi(\varphi(1),0))=\varphi(\varphi(1@\varphi(1))@\varphi(1))=\varphi(\varphi(1@\varphi(1,0))@\varphi(1,0))$

$\varphi(\varphi(1)+1,0)=\varphi(1@\varphi(1)+1)=\varphi(1@\varphi(1,1))$

$\varphi(\varphi(1)+1,\varphi(\varphi(1)+1,0))=\varphi(\varphi(1@\varphi(1)+1)@\varphi(1)+1)=\varphi(\varphi(1@\varphi(1,1))@\varphi(1,1))$

$\varphi(\varphi(1)+2,0)=\varphi(1@\varphi(1)+2)=\varphi(1@\varphi(1,2))$

$\varphi(\varphi(1)\cdot2,0)=\varphi(1@\varphi(1)\cdot2)=\varphi(1@\varphi(1,\varphi(1)))$

$\varphi(\varphi(1)\cdot3,0)=\varphi(1@\varphi(1)\cdot3)=\varphi(1@\varphi(1,\varphi(1,\varphi(1,0))))$

$\varphi(\varphi(2),0)=\varphi(1@\varphi(2))=\varphi(1@\varphi(2,0))$

$\varphi(\varphi(\varphi(1)),0)=\varphi(1@\varphi(\varphi(1)))=\varphi(1@\varphi(\varphi(1,0),0))$

$\varphi(\varphi(1,0),0)=\varphi(1@\varphi(1,0))=\varphi(1@\varphi(1,0,0))$

$\varphi(\varphi(2,0),0)=\varphi(1@\varphi(1,0,0))=\varphi(1@\varphi(1,0,0,0))$

$\varphi(\varphi(3,0),0)=\varphi(1@\varphi(1,0,0,0))=\varphi(1@\varphi(1,0,0,0,0))$

$\varphi(\varphi(\varphi(1),0),0)=\varphi(1@\varphi(1@\varphi(1)))=\varphi(1@\varphi(1@\varphi(1,0)))$

$\varphi(\varphi(\varphi(1,0),0),0)=\varphi(1@\varphi(1@\varphi(1,0)))=\varphi(1@\varphi(1@\varphi(1,0,0)))$

$\varphi(\varphi(\varphi(2,0),0),0)=\varphi(1@\varphi(1@\varphi(1,0,0)))=\varphi(1@\varphi(1@\varphi(1,0,0,0)))$

$\varphi(1,0,0)=\varphi(1@(1,0))=\varphi(1@(1,0))=$ FSO

$\varphi(1,\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0)),0)=\varphi(\varphi(1@(1,0)),0,0)$

$\varphi(2,\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0)),0,0)=\varphi(\varphi(1@(1,0)),0,0,0)$

$\varphi(\varphi(1),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1))=\varphi(\varphi(1@(1,0))@\varphi(1,0))$

$\varphi(\varphi(1,0),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1,0))=\varphi(\varphi(1@(1,0))@\varphi(1,0,0))$

$\varphi(\varphi(2,0),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@\varphi(1,0,0,0))$

$\varphi(\varphi(1,0,0),1)=\varphi(2@\varphi(1@(1,0)))=\varphi(2@\varphi(1@(1,0)))$

$\varphi(\varphi(1,0,0),2)=\varphi(3@\varphi(1@(1,0)))=\varphi(3@\varphi(1@(1,0)))$

$\varphi(\varphi(1,0,0),\varphi(1))=\varphi(\varphi(1)@\varphi(1@(1,0)))=\varphi(\varphi(1,0)@\varphi(1@(1,0)))$

$\varphi(\varphi(1,0,0),\varphi(1,0))=\varphi(\varphi(1,0)@\varphi(1@(1,0)))=\varphi(\varphi(1,0,0)@\varphi(1@(1,0)))$

$\varphi(\varphi(1,0,0),\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@\varphi(1@(1,0)))=\varphi(\varphi(1@(1,0))@\varphi(1@(1,0)))$

$\varphi(\varphi(1,0,0)+1,0)=\varphi(1@\varphi(1@(1,0))+1)=\varphi(1@\varphi(1@(1,0),1@0))$

$\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(1@\varphi(1@(1,0))+\varphi(1))=\varphi(1@\varphi(1@(1,0),\varphi(1,0)@0))$

$\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(1@\varphi(1@(1,0))+\varphi(1,0))=\varphi(1@\varphi(1@(1,0),\varphi(1,0,0)@0))$

$\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(1@\varphi(1@(1,0))\cdot2)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))$

$\varphi(\varphi(1,0,0)\cdot\varphi(1),0)=\varphi(1@\varphi(1@(1,0),1@0))=\varphi(1@\varphi(1@(1,0),1@1))$

$\varphi(\varphi(1,0,0)^{2},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))$

$\varphi(\varphi(1,0,0)^{\varphi(1)},0)=\varphi(1@\varphi(1,\varphi(1@(1,0),1@0),0))=\varphi(1@\varphi(1,\varphi(1@(1,0),1@1),0))$

$\varphi(\varphi(1,0,0)^{\varphi(1,0)},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1,0)@0)@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1,0)@0)@0))$

$\varphi(\varphi(1,0,0)^{\varphi(1,0,0)},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1@(1,0))@0)@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1@(1,0))@0)@0))$

$\varphi(\varphi(1,\varphi(1,0,0)+1),0)=\varphi(1@\varphi(\varphi(1@(1,0)),0))=\varphi(1@\varphi(\varphi(1@(1,0)),0,0))$

$\varphi(\varphi(1,\varphi(1,0,0)+\varphi(1)),0)=\varphi(1@\varphi(\varphi(1@(1,0))+\varphi(1),0))=\varphi(1@\varphi(\varphi(1@(1,0))+\varphi(1,0),0))$

$\varphi(\varphi(\varphi(1),\varphi(1,0,0)+1),0)=\varphi(1@\varphi(\varphi(1@(1,0))@\varphi(1)))=\varphi(1@\varphi(\varphi(1@(1,0))@\varphi(1,0)))$

$\varphi(1,0,1)=\varphi(1@(1,1))=\varphi(1@(1,1))$

$\varphi(1,0,2)=\varphi(2@(1,1))=\varphi(2@(1,1))$

$\varphi(1,0,\varphi(1))=\varphi(\varphi(1)@(1,1))=\varphi(\varphi(1,0)@(1,1))$

$\varphi(1,0,\varphi(1,0))=\varphi(\varphi(1,0)@(1,1))=\varphi(\varphi(1,0,0)@(1,1))$

$\varphi(1,0,\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@(1,1))=\varphi(\varphi(1@(1,0))@(1,1))$

$\varphi(1,1,0)=\varphi(1@(1,1))=\varphi(1@(1,1))$

$\varphi(1,1,\varphi(1,1,0))=\varphi(\varphi(1@(1,1))@(1,1))=\varphi(\varphi(1@(1,1))@(1,1))$

$\varphi(1,2,0)=\varphi(1@(1,2))=\varphi(1@(1,2))$

$\varphi(1,\varphi(1),0)=\varphi(1@(1,\varphi(1)))=\varphi(1@(1,\varphi(1,0)))$

$\varphi(1,\varphi(1,0),0)=\varphi(1@(1,\varphi(1,0)))=\varphi(1@(1,\varphi(1,0,0)))$

$\varphi(1,\varphi(1,0,0),0)=\varphi(1@(1,\varphi(1@(1,0))))=\varphi(1@(1,\varphi(1@(1,0))))$

$\varphi(2,0,0)=\varphi(1@(2,0))=\varphi(1@(2,0))$

$\varphi(2,0,1)=\varphi(2@(2,0))=\varphi(2@(2,0))$

$\varphi(2,1,0)=\varphi(1@(2,1))=\varphi(1@(2,1))$

$\varphi(2,\varphi(2,0,0),0)=\varphi(1@(2,\varphi(1@(2,0))))=\varphi(1@(2,\varphi(1@(2,0))))$

$\varphi(3,0,0)=\varphi(1@(3,0))=\varphi(1@(3,0))$

$\varphi(\varphi(1),0,0)=\varphi(1@(\varphi(1),0))=\varphi(1@(\varphi(1,0),0))$

$\varphi(\varphi(1,0),0,0)=\varphi(1@(\varphi(1,0),0))=\varphi(1@(\varphi(1,0,0),0))$

$\varphi(\varphi(1,0,0),0,0)=\varphi(1@(\varphi(1@(1,0)),0))=\varphi(1@(\varphi(1@(1,0)),0))$

$\varphi(1,0,0,0)=\varphi(1@(1,0,0))=\varphi(1@(1,0,0))=$ ACO

$\varphi(1,0,0,1)=\varphi(2@(1,0,0))=\varphi(2@(1,0,0))$

$\varphi(1,0,1,0)=\varphi(1@(1,0,1))=\varphi(1@(1,0,1))$

$\varphi(1,1,0,0)=\varphi(1@(1,1,0))=\varphi(1@(1,1,0))$

$\varphi(2,0,0,0)=\varphi(1@(2,0,0))=\varphi(1@(2,0,0))$

$\varphi(\varphi(1),0,0,0)=\varphi(1@(\varphi(1),0,0))=\varphi(1@(\varphi(1,0),0,0))$

$\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(1@(\varphi(1@(1,0,0)),0,0))=\varphi(1@(\varphi(1@(1,0,0)),0,0))$

$\varphi(1,0,0,0,0)=\varphi(1@(1,0,0,0))=\varphi(1@(1,0,0,0))$

$\varphi(1,0,0,0,0,0)=\varphi(1@(1,0,0,0,0))=\varphi(1@(1,0,0,0,0))$

$\varphi(1,1,4,5,1,4)=\varphi(5@(1,1,4,5,1))=\varphi(5@(1,1,4,5,1))$

$\varphi(1@\varphi(1))=\varphi(1@(1@\varphi(1)))=\varphi(1@(1@\varphi(1,0)))=$ SVO

$\varphi(1,\varphi(1@\varphi(1))+1)=\varphi(1@(1@\varphi(1)),1@1)=\varphi(1@(1@\varphi(1,0)),1@2)$

$\varphi(1@\varphi(1),1@0)=\varphi(2@(1@\varphi(1)))=\varphi(2@(1@\varphi(1,0)))$

$\varphi(1@\varphi(1),1@1)=\varphi(1@(1@\varphi(1),1@0))=\varphi(1@(1@\varphi(1,0),1@0))$

$\varphi(1@\varphi(1),1@2)=\varphi(1@(1@\varphi(1),1@1))=\varphi(1@(1@\varphi(1,0),1@1))$

$\varphi(2@\varphi(1))=\varphi(1@(2@\varphi(1)))=\varphi(1@(2@\varphi(1,0)))$

$\varphi(\varphi(1)@\varphi(1))=\varphi(1@(\varphi(1)@\varphi(1)))=\varphi(1@(\varphi(1,0)@\varphi(1,0)))$

$\varphi(1@\varphi(1)+1)=\varphi(1@(1@\varphi(1)+1))=\varphi(1@(1@\varphi(1,1)))$

$\varphi(1@\varphi(1)\cdot2)=\varphi(1@(1@\varphi(1)\cdot2))=\varphi(1@(1@\varphi(1,\varphi(1,0))))$

$\varphi(1@\varphi(2))=\varphi(1@(1@\varphi(2)))=\varphi(1@(1@\varphi(2,0)))$

$\varphi(1@\varphi(1,0))=\varphi(1@(1@\varphi(1,0)))=\varphi(1@(1@\varphi(1,0,0)))$

$\varphi(1@\varphi(2,0))=\varphi(1@(1@\varphi(1,0,0)))=\varphi(1@(1@\varphi(1,0,0,0)))$

$\varphi(1@\varphi(1,0,0))=\varphi(1@(1@\varphi(1@(1,0))))=\varphi(1@(1@\varphi(1@(1,0))))$

$\varphi(1@\varphi(1@\varphi(1)))=\varphi(1@(1@\varphi(1@(1@\varphi(1)))))=\varphi(1@(1@\varphi(1@(1@\varphi(1,0)))))=$ LVO

历史

据信在 2024 年下旬及以前的 Weak Veblen 都是指的 $\times ω φ$ ，但在这之后 $\times ω φ$ 的生态位快速被 $+ 1 φ$ 所代替了，所以出现了一些较为混乱的局面。

@@ 第1行： / 第1行： @@
-'''Weak Veblen 函数（又称“弱 φ”、“弱 Veblen”等）'''，是 [[Veblen函数|Veblen 函数]]的变体。
+'''weak Veblen 函数（又称“弱 φ”、“弱 Veblen”等）'''，是 [[Veblen 函数|Veblen 函数]]的变体。
 === 定义 ===
-现在有两种 Veblen 函数的变体可被称为 Weak Veblen。
+现在有两种 Veblen 函数的变体可被称为 weak Veblen。
-如果称 Veblen 函数的末位+1等价于跳到下一个 ε 点（[[不动点]]进制），定义 Veblen 函数为 <math>\mathrm{FP}\ \phi</math>，那么可以直观地定义下面两种Weak Veblen：
+如果称 Veblen 函数的末位 + 1 等价于跳到下一个 ε 点（[[不动点]]进制），定义 Veblen 函数为 <math>\mathrm{FP}\ \varphi</math>，那么可以直观地定义下面两种weak Veblen：
-* 若 Weak Veblen 函数的末位 <math>+1</math> 对应于其序数值<math> \times\omega</math>，则称其为 <math>\times\omega\ \phi</math>。
+* 若 weak Veblen 函数的末位 + 1 对应于其序数值 <math> \times\omega</math>，则称其为 <math>\times\omega\ \phi</math>。
-* 若 Weak Veblen 函数的末位 <math>+1</math> 对应于其序数值 <math>+1</math>，则称其为 <math>+1\ \phi</math>。
+* 若 weak Veblen 函数的末位 + 1 对应于其序数值 <math>+1</math>，则称其为 <math>+1\ \varphi</math>。
-以下是两种Weak Veblen直到序元级别的定义：
+以下是两种 weak Veblen 直到序元级别的定义：
 <big>'''<math>+1\ \phi</math>'''</big>
@@ 第30行： / 第30行： @@
 以下分析中，左为 <math>\mathrm{FP}\ \phi</math>，中为 <math>\times\omega\ \phi</math>，右为 <math>+1\ \phi</math>。
-\(φ(0)=φ(0)=φ(0)\)
+\(\varphi(0)=\varphi(0)=\varphi(0)\)
-\(φ(0)\cdot2=φ(0)\cdot2=φ(φ(0))\)
+\(\varphi(0)\cdot2=\varphi(0)\cdot2=\varphi(\varphi(0))\)
-\(φ(0)\cdot3=φ(0)\cdot3=φ(φ(φ(0)))\)
+\(\varphi(0)\cdot3=\varphi(0)\cdot3=\varphi(\varphi(\varphi(0)))\)
-\(φ(1)=φ(1)=φ(1,0)=\) '''[[FTO|ω]]'''
+\(\varphi(1)=\varphi(1)=\varphi(1,0)=\) [[FTO|ω]]
-\(φ(1)+1=φ(1)+1=φ(1,1)\)
+\(\varphi(1)+1=\varphi(1)+1=\varphi(1,1)\)
-\(φ(1)+2=φ(1)+2=φ(1,2)\)
+\(\varphi(1)+2=\varphi(1)+2=\varphi(1,2)\)
-\(φ(1)\cdot2=φ(1)\cdot2=φ(1,φ(1,0))\)
+\(\varphi(1)\cdot2=\varphi(1)\cdot2=\varphi(1,\varphi(1,0))\)
-\(φ(1)\cdot2+1=φ(1)\cdot2+1=φ(1,φ(1,1))\)
+\(\varphi(1)\cdot2+1=\varphi(1)\cdot2+1=\varphi(1,\varphi(1,1))\)
-\(φ(1)\cdot2+2=φ(1)\cdot2+2=φ(1,φ(1,2))\)
+\(\varphi(1)\cdot2+2=\varphi(1)\cdot2+2=\varphi(1,\varphi(1,2))\)
-\(φ(1)\cdot3=φ(1)\cdot3=φ(1,φ(1,φ(1,0)))\)
+\(\varphi(1)\cdot3=\varphi(1)\cdot3=\varphi(1,\varphi(1,\varphi(1,0)))\)
-\(φ(2)=φ(2)=φ(2,0)\)
+\(\varphi(2)=\varphi(2)=\varphi(2,0)\)
-\(φ(2)+1=φ(2)+1=φ(2,1)\)
+\(\varphi(2)+1=\varphi(2)+1=\varphi(2,1)\)
-\(φ(2)+2=φ(2)+2=φ(2,2)\)
+\(\varphi(2)+2=\varphi(2)+2=\varphi(2,2)\)
-\(φ(2)+φ(1)=φ(2)+φ(1)=φ(2,φ(1,0))\)
+\(\varphi(2)+\varphi(1)=\varphi(2)+\varphi(1)=\varphi(2,\varphi(1,0))\)
-\(φ(2)\cdot2=φ(2)\cdot2=φ(2,φ(2,0))\)
+\(\varphi(2)\cdot2=\varphi(2)\cdot2=\varphi(2,\varphi(2,0))\)
-\(φ(2)\cdot3=φ(2)\cdot3=φ(2,φ(2,φ(2,0)))\)
+\(\varphi(2)\cdot3=\varphi(2)\cdot3=\varphi(2,\varphi(2,\varphi(2,0)))\)
-\(φ(3)=φ(3)=φ(3,0)\)
+\(\varphi(3)=\varphi(3)=\varphi(3,0)\)
-\(φ(3)+φ(2)=φ(3)+φ(2)=φ(3,φ(2,0))\)
+\(\varphi(3)+\varphi(2)=\varphi(3)+\varphi(2)=\varphi(3,\varphi(2,0))\)
-\(φ(4)=φ(4)=φ(4,0)\)
+\(\varphi(4)=\varphi(4)=\varphi(4,0)\)
-\(φ(φ(1))=φ(φ(1))=φ(φ(1,0),0)\)
+\(\varphi(\varphi(1))=\varphi(\varphi(1))=\varphi(\varphi(1,0),0)\)
-\(φ(φ(1))+1=φ(φ(1))+1=φ(φ(1,0),1)\)
+\(\varphi(\varphi(1))+1=\varphi(\varphi(1))+1=\varphi(\varphi(1,0),1)\)
-\(φ(φ(1))+φ(1)=φ(φ(1))+φ(1)=φ(φ(1,0),φ(1,0))\)
+\(\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1,0),\varphi(1,0))\)
-\(φ(φ(1))+φ(2)=φ(φ(1))+φ(2)=φ(φ(1,0),φ(2,0))\)
+\(\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1,0),\varphi(2,0))\)
-\(φ(φ(1))\cdot2=φ(φ(1))\cdot2=φ(φ(1,0),φ(φ(1,0),0))\)
+\(\varphi(\varphi(1))\cdot2=\varphi(\varphi(1))\cdot2=\varphi(\varphi(1,0),\varphi(\varphi(1,0),0))\)
-\(φ(φ(1)+1)=φ(φ(1)+1)=φ(φ(1,1),0)\)
+\(\varphi(\varphi(1)+1)=\varphi(\varphi(1)+1)=\varphi(\varphi(1,1),0)\)
-\(φ(φ(1)+2)=φ(φ(1)+2)=φ(φ(1,2),0)\)
+\(\varphi(\varphi(1)+2)=\varphi(\varphi(1)+2)=\varphi(\varphi(1,2),0)\)
-\(φ(φ(1)+3)=φ(φ(1)+3)=φ(φ(1,3),0)\)
+\(\varphi(\varphi(1)+3)=\varphi(\varphi(1)+3)=\varphi(\varphi(1,3),0)\)
-\(φ(φ(1)\cdot2)=φ(φ(1)\cdot2)=φ(φ(1,φ(1,0)),0)\)
+\(\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1,\varphi(1,0)),0)\)
-\(φ(φ(1)\cdot2+1)=φ(φ(1)\cdot2+1)=φ(φ(1,φ(1,1)),0)\)
+\(\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1,\varphi(1,1)),0)\)
-\(φ(φ(1)\cdot3)=φ(φ(1)\cdot3)=φ(φ(1,φ(1,φ(1,0))),0)\)
+\(\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1,\varphi(1,\varphi(1,0))),0)\)
-\(φ(φ(2))=φ(φ(2))=φ(φ(2,0),0)\)
+\(\varphi(\varphi(2))=\varphi(\varphi(2))=\varphi(\varphi(2,0),0)\)
-\(φ(φ(2)+1)=φ(φ(2)+1)=φ(φ(2,1),0)\)
+\(\varphi(\varphi(2)+1)=\varphi(\varphi(2)+1)=\varphi(\varphi(2,1),0)\)
-\(φ(φ(2)+φ(1))=φ(φ(2)+φ(1))=φ(φ(2,φ(1,0)),0)\)
+\(\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2,\varphi(1,0)),0)\)
-\(φ(φ(3))=φ(φ(3))=φ(φ(3,0),0)\)
+\(\varphi(\varphi(3))=\varphi(\varphi(3))=\varphi(\varphi(3,0),0)\)
-\(φ(φ(φ(1)))=φ(φ(φ(1)))=φ(φ(φ(1,0),0),0)\)
+\(\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1,0),0),0)\)
-\(φ(φ(φ(1))+1)=φ(φ(φ(1))+1)=φ(φ(φ(1,0),1),0)\)
+\(\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1,0),1),0)\)
-\(φ(φ(φ(1))+φ(1))=φ(φ(φ(1))+φ(1))=φ(φ(φ(1,0),φ(1,0)),0)\)
+\(\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1,0),\varphi(1,0)),0)\)
-\(φ(φ(φ(1))\cdot2)=φ(φ(φ(1))\cdot2)=φ(φ(φ(1,0),φ(φ(1,0),0)),0)\)
+\(\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1,0),\varphi(\varphi(1,0),0)),0)\)
-\(φ(φ(φ(1)+1))=φ(φ(φ(1)+1))=φ(φ(φ(1,1),0),0)\)
+\(\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1,1),0),0)\)
-\(φ(φ(φ(2)))=φ(φ(φ(2)))=φ(φ(φ(2,0),0),0)\)
+\(\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2,0),0),0)\)
-\(φ(φ(φ(3)))=φ(φ(φ(3)))=φ(φ(φ(3,0),0),0)\)
+\(\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3,0),0),0)\)
-\(φ(φ(φ(φ(1))))=φ(φ(φ(φ(1))))=φ(φ(φ(φ(1,0),0),0),0)\)
+\(\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1,0),0),0),0)\)
-\(φ(φ(φ(φ(φ(1)))))=φ(φ(φ(φ(φ(1)))))=φ(φ(φ(φ(φ(1,0),0),0),0),0)\)
+\(\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1,0),0),0),0),0)\)
-\(φ(1,0)=φ(1,0)=φ(1,0,0)=\) '''[[SCO|ε<sub>0</sub>]]'''
+\(\varphi(1,0)=\varphi(1,0)=\varphi(1,0,0)=\) [[SCO|ε0]]
-\(φ(1,0)+1=φ(1,0)+1=φ(1,0,1)\)
+\(\varphi(1,0)+1=\varphi(1,0)+1=\varphi(1,0,1)\)
-\(φ(1,0)+φ(1)=φ(1,0)+φ(1)=φ(1,0,φ(1,0))\)
+\(\varphi(1,0)+\varphi(1)=\varphi(1,0)+\varphi(1)=\varphi(1,0,\varphi(1,0))\)
-\(φ(1,0)\cdot2=φ(1,0)\cdot2=φ(1,0,φ(1,0,0))\)
+\(\varphi(1,0)\cdot2=\varphi(1,0)\cdot2=\varphi(1,0,\varphi(1,0,0))\)
-\(φ(1,0)\cdot3=φ(1,0)\cdot3=φ(1,0,φ(1,0,φ(1,0,0)))\)
+\(\varphi(1,0)\cdot3=\varphi(1,0)\cdot3=\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))\)
-\(φ(1,0)\cdotφ(1)=φ(1,1)=φ(1,1,0)\)
+\(\varphi(1,0)\cdot\varphi(1)=\varphi(1,1)=\varphi(1,1,0)\)
-\(φ(1,0)\cdotφ(1)\cdot2=φ(1,1)\cdot2=φ(1,1,φ(1,1,0))\)
+\(\varphi(1,0)\cdot\varphi(1)\cdot2=\varphi(1,1)\cdot2=\varphi(1,1,\varphi(1,1,0))\)
-\(φ(1,0)\cdotφ(2)=φ(1,2)=φ(1,2,0)\)
+\(\varphi(1,0)\cdot\varphi(2)=\varphi(1,2)=\varphi(1,2,0)\)
-\(φ(1,0)\cdotφ(φ(1))=φ(1,φ(1))=φ(1,φ(1,0),0)\)
+\(\varphi(1,0)\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1))=\varphi(1,\varphi(1,0),0)\)
-\(φ(1,0)\cdotφ(φ(1)+1)=φ(1,φ(1)+1)=φ(1,φ(1,1),0)\)
+\(\varphi(1,0)\cdot\varphi(\varphi(1)+1)=\varphi(1,\varphi(1)+1)=\varphi(1,\varphi(1,1),0)\)
-\(φ(1,0)\cdotφ(φ(2))=φ(1,φ(2))=φ(1,φ(2,0),0)\)
+\(\varphi(1,0)\cdot\varphi(\varphi(2))=\varphi(1,\varphi(2))=\varphi(1,\varphi(2,0),0)\)
-\(φ(1,0)\cdotφ(φ(φ(1)))=φ(1,φ(φ(1)))=φ(1,φ(φ(1,0),0),0)\)
+\(\varphi(1,0)\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1,0),0),0)\)
-\(φ(1,0)^{2}=φ(1,φ(1,0))=φ(1,φ(1,0,0),0)\)
+\(\varphi(1,0)^{2}=\varphi(1,\varphi(1,0))=\varphi(1,\varphi(1,0,0),0)\)
-\(φ(1,0)^{2}\cdotφ(1)=φ(1,φ(1,0)+1)=φ(1,φ(φ(1,0,0)),0)\)
+\(\varphi(1,0)^{2}\cdot\varphi(1)=\varphi(1,\varphi(1,0)+1)=\varphi(1,\varphi(\varphi(1,0,0)),0)\)
-\(φ(1,0)^{2}\cdotφ(φ(φ(1)))=φ(1,φ(1,0)+φ(φ(1)))=φ(1,φ(1,0,φ(φ(1,0),0)),0)\)
+\(\varphi(1,0)^{2}\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(1,0)+\varphi(\varphi(1)))=\varphi(1,\varphi(1,0,\varphi(\varphi(1,0),0)),0)\)
-\(φ(1,0)^{3}=φ(1,φ(1,0)\cdot2)=φ(1,φ(1,0,φ(1,0,0),0),0)\)
+\(\varphi(1,0)^{3}=\varphi(1,\varphi(1,0)\cdot2)=\varphi(1,\varphi(1,0,\varphi(1,0,0),0),0)\)
-\(φ(1,0)^{φ(1)}=φ(1,φ(1,1))=φ(1,φ(1,1,0),0)\)
+\(\varphi(1,0)^{\varphi(1)}=\varphi(1,\varphi(1,1))=\varphi(1,\varphi(1,1,0),0)\)
-\(φ(1,0)^{φ(1)}\cdotφ(1)=φ(1,φ(1,1)+1)=φ(1,φ(φ(1,1,0)),0)\)
+\(\varphi(1,0)^{\varphi(1)}\cdot\varphi(1)=\varphi(1,\varphi(1,1)+1)=\varphi(1,\varphi(\varphi(1,1,0)),0)\)
-\(φ(1,0)^{φ(1)}\cdotφ(φ(1))=φ(1,φ(1,1)+2)=φ(1,φ(φ(φ(1,1,0))),0)\)
+\(\varphi(1,0)^{\varphi(1)}\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1,1)+2)=\varphi(1,\varphi(\varphi(\varphi(1,1,0))),0)\)
-\(φ(1,0)^{(φ(1)+1)}=φ(1,φ(1,1)+φ(1,0))=φ(1,φ(1,1,φ(1,0,0)),0)\)
+\(\varphi(1,0)^{(\varphi(1)+1)}=\varphi(1,\varphi(1,1)+\varphi(1,0))=\varphi(1,\varphi(1,1,\varphi(1,0,0)),0)\)
-\(φ(1,0)^{(φ(1)\cdot2)}=φ(1,φ(1,1)\cdot2)=φ(1,φ(1,1,φ(1,1,0)),0)\)
+\(\varphi(1,0)^{(\varphi(1)\cdot2)}=\varphi(1,\varphi(1,1)\cdot2)=\varphi(1,\varphi(1,1,\varphi(1,1,0)),0)\)
-\(φ(1,0)^{φ(1)^{2}}=φ(1,φ(1,2))=φ(1,φ(1,2,0),0)\)
+<nowiki>\(\varphi(1,0)^{\varphi(1)^{2}}=\varphi(1,\varphi(1,2))=\varphi(1,\varphi(1,2,0),0)\)</nowiki>
-\(φ(1,0)^{φ(1)^{φ(1)}}=φ(1,φ(1,φ(1)))=φ(1,φ(1,φ(1,0),0),0)\)
+<nowiki>\(\varphi(1,0)^{\varphi(1)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1)))=\varphi(1,\varphi(1,\varphi(1,0),0),0)\)</nowiki>
-\(φ(1,0)^{φ(1,0)}=φ(1,φ(1,φ(1,0)))=φ(1,φ(1,φ(1,0,0),0),0)\)
+\(\varphi(1,0)^{\varphi(1,0)}=\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(1,\varphi(1,\varphi(1,0,0),0),0)\)
-\(φ(1,0)^{(φ(1,0)+1)}=φ(1,φ(1,φ(1,0))+φ(1,0))=φ(1,φ(1,φ(1,0,0),φ(1,0,0)),0)\)
+\(\varphi(1,0)^{(\varphi(1,0)+1)}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,0))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,0,0)),0)\)
-\(φ(1,0)^{(φ(1,0)\cdotφ(1))}=φ(1,φ(1,φ(1,0))+φ(1,1))=φ(1,φ(1,φ(1,0,0),φ(1,1,0)),0)\)
+\(\varphi(1,0)^{(\varphi(1,0)\cdot\varphi(1))}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,1))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,1,0)),0)\)
-\(φ(1,0)^{φ(1,0)^{2}}=φ(1,φ(1,φ(1,2)))=φ(1,φ(1,φ(1,2,0),0),0)\)
+<nowiki>\(\varphi(1,0)^{\varphi(1,0)^{2}}=\varphi(1,\varphi(1,\varphi(1,2)))=\varphi(1,\varphi(1,\varphi(1,2,0),0),0)\)</nowiki>
-\(φ(1,0)^{φ(1,0)^{φ(1)}}=φ(1,φ(1,φ(1,φ(1))))=φ(1,φ(1,φ(1,φ(1,0),0),0),0)\)
+<nowiki>\(\varphi(1,0)^{\varphi(1,0)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0),0),0),0)\)</nowiki>
-\(φ(1,0)^{φ(1,0)^{φ(1,0)}}=φ(1,φ(1,φ(1,φ(1,0))))=φ(1,φ(1,φ(1,φ(1,0,0),0),0),0)\)
+<nowiki>\(\varphi(1,0)^{\varphi(1,0)^{\varphi(1,0)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)\)</nowiki>
-\(φ(1,1)=φ(2,0)=φ(2,0,0)\)
+\(\varphi(1,1)=\varphi(2,0)=\varphi(2,0,0)\)
-\(φ(1,1)\cdotφ(1,0)=φ(2,φ(1,0))=φ(2,φ(1,0,0),0)\)
+\(\varphi(1,1)\cdot\varphi(1,0)=\varphi(2,\varphi(1,0))=\varphi(2,\varphi(1,0,0),0)\)
-\(φ(1,1)\cdotφ(1,0)^{φ(1)}=φ(2,φ(1,φ(1,1)))=φ(2,φ(1,φ(1,1,0),0),0)\)
+\(\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1)}=\varphi(2,\varphi(1,\varphi(1,1)))=\varphi(2,\varphi(1,\varphi(1,1,0),0),0)\)
-\(φ(1,1)\cdotφ(1,0)^{φ(1,0)}=φ(2,φ(1,φ(1,φ(1,0))))=φ(2,φ(1,φ(1,φ(1,0,0),0),0),0)\)
+\(\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1,0)}=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)\)
-\(φ(1,1)^{2}=φ(2,φ(2,0))=φ(2,φ(2,0,0),0)\)
+\(\varphi(1,1)^{2}=\varphi(2,\varphi(2,0))=\varphi(2,\varphi(2,0,0),0)\)
-\(φ(1,1)^{φ(1)}=φ(2,φ(2,1))=φ(2,φ(2,φ(1,0),0),0)\)
+\(\varphi(1,1)^{\varphi(1)}=\varphi(2,\varphi(2,1))=\varphi(2,\varphi(2,\varphi(1,0),0),0)\)
-\(φ(1,1)^{φ(1,0)}=φ(2,φ(2,φ(1,0)))=φ(2,φ(2,φ(1,0,0),0),0)\)
+\(\varphi(1,1)^{\varphi(1,0)}=\varphi(2,\varphi(2,\varphi(1,0)))=\varphi(2,\varphi(2,\varphi(1,0,0),0),0)\)
-\(φ(1,1)^{φ(1,1)}=φ(2,φ(2,φ(2,0)))=φ(2,φ(2,φ(2,0,0),0),0)\)
+\(\varphi(1,1)^{\varphi(1,1)}=\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(2,\varphi(2,\varphi(2,0,0),0),0)\)
-\(φ(1,2)=φ(3,0)=φ(3,0,0)\)
+\(\varphi(1,2)=\varphi(3,0)=\varphi(3,0,0)\)
-\(φ(1,φ(1))=φ(φ(1),0)=φ(φ(1,0),0,0)\)
+\(\varphi(1,\varphi(1))=\varphi(\varphi(1),0)=\varphi(\varphi(1,0),0,0)\)
-\(φ(1,φ(1))\cdotφ(1,0)=φ(φ(1),φ(1,0))=φ(φ(1,0),φ(1,0,0),0)\)
+\(\varphi(1,\varphi(1))\cdot\varphi(1,0)=\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0),\varphi(1,0,0),0)\)
-\(φ(1,φ(1))\cdotφ(1,1)=φ(φ(1),φ(1,1))=φ(φ(1,0),φ(1,1,0),0)\)
+\(\varphi(1,\varphi(1))\cdot\varphi(1,1)=\varphi(\varphi(1),\varphi(1,1))=\varphi(\varphi(1,0),\varphi(1,1,0),0)\)
-\(φ(1,φ(1))^{2}=φ(φ(1),φ(1,φ(1)))=φ(φ(1,0),φ(1,φ(1,0),0))\)
+\(\varphi(1,\varphi(1))^{2}=\varphi(\varphi(1),\varphi(1,\varphi(1)))=\varphi(\varphi(1,0),\varphi(1,\varphi(1,0),0))\)
-\(φ(1,φ(1)+1)=φ(φ(1)+1,0)=φ(φ(1,1),0,0)\)
+\(\varphi(1,\varphi(1)+1)=\varphi(\varphi(1)+1,0)=\varphi(\varphi(1,1),0,0)\)
-\(φ(1,φ(1)+2)=φ(φ(1)+2,0)=φ(φ(1,2),0,0)\)
+\(\varphi(1,\varphi(1)+2)=\varphi(\varphi(1)+2,0)=\varphi(\varphi(1,2),0,0)\)
-\(φ(1,φ(1)\cdot2)=φ(φ(1)\cdot2,0)=φ(φ(1,φ(1,0)),0,0)\)
+\(\varphi(1,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0)=\varphi(\varphi(1,\varphi(1,0)),0,0)\)
-\(φ(1,φ(1)^{2})=φ(φ(2),0)=φ(φ(2,0),0,0)\)
+\(\varphi(1,\varphi(1)^{2})=\varphi(\varphi(2),0)=\varphi(\varphi(2,0),0,0)\)
-\(φ(1,φ(1)^{3})=φ(φ(3),0)=φ(φ(3,0),0,0)\)
+\(\varphi(1,\varphi(1)^{3})=\varphi(\varphi(3),0)=\varphi(\varphi(3,0),0,0)\)
-\(φ(1,φ(1)^{φ(1)})=φ(φ(φ(1)),0)=φ(φ(φ(1,0),0),0,0)\)
+\(\varphi(1,\varphi(1)^{\varphi(1)})=\varphi(\varphi(\varphi(1)),0)=\varphi(\varphi(\varphi(1,0),0),0,0)\)
-\(φ(1,φ(1,0))=φ(φ(1,0),0)=φ(φ(1,0,0),0,0)\)
+\(\varphi(1,\varphi(1,0))=\varphi(\varphi(1,0),0)=\varphi(\varphi(1,0,0),0,0)\)
-\(φ(1,φ(1,0)+1)=φ(φ(1,0)+1,0)=φ(φ(1,0,1),0,0)\)
+\(\varphi(1,\varphi(1,0)+1)=\varphi(\varphi(1,0)+1,0)=\varphi(\varphi(1,0,1),0,0)\)
-\(φ(1,φ(1,0)\cdotφ(1))=φ(φ(1,1),0)=φ(φ(1,1,0),0,0)\)
+\(\varphi(1,\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,1),0)=\varphi(\varphi(1,1,0),0,0)\)
-\(φ(1,φ(1,0)^{φ(1)})=φ(φ(1,φ(1,1)),0)=φ(φ(1,φ(1,1,0),0),0,0)\)
+\(\varphi(1,\varphi(1,0)^{\varphi(1)})=\varphi(\varphi(1,\varphi(1,1)),0)=\varphi(\varphi(1,\varphi(1,1,0),0),0,0)\)
-\(φ(1,φ(1,1))=φ(φ(2,0),0)=φ(φ(2,0,0),0,0)\)
+\(\varphi(1,\varphi(1,1))=\varphi(\varphi(2,0),0)=\varphi(\varphi(2,0,0),0,0)\)
-\(φ(1,φ(1,φ(1)))=φ(φ(φ(1),0),0)=φ(φ(φ(1,0),0,0),0,0)\)
+\(\varphi(1,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0)=\varphi(\varphi(\varphi(1,0),0,0),0,0)\)
-\(φ(1,φ(1,φ(1,0)))=φ(φ(φ(1,0),0),0)=φ(φ(φ(1,0,0),0,0),0,0)\)
+\(\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(\varphi(\varphi(1,0),0),0)=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)\)
-\(φ(2,0)=φ(1,0,0)=φ(1,0,0,0)=\) '''[[CO|ζ<sub>0</sub>]]'''
+\(\varphi(2,0)=\varphi(1,0,0)=\varphi(1,0,0,0)=\) [[CO|ζ0]]
-\(φ(1,φ(2,0)+1)=φ(φ(1,0,0),0)=φ(φ(1,0,0,0),0,0)\)
+\(\varphi(1,\varphi(2,0)+1)=\varphi(\varphi(1,0,0),0)=\varphi(\varphi(1,0,0,0),0,0)\)
-\(φ(1,φ(2,0)+φ(1))=φ(φ(1,0,0)+φ(1),0)=φ(φ(1,0,0,φ(1,0)),0,0)\)
+\(\varphi(1,\varphi(2,0)+\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(\varphi(1,0,0,\varphi(1,0)),0,0)\)
-\(φ(1,φ(2,0)+φ(1,0))=φ(φ(1,0,0)+φ(1,0),0)=φ(φ(1,0,0,φ(1,0,0)),0,0)\)
+\(\varphi(1,\varphi(2,0)+\varphi(1,0))=\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(\varphi(1,0,0,\varphi(1,0,0)),0,0)\)
-\(φ(1,φ(2,0)+φ(1,0)\cdotφ(1))=φ(φ(1,0,0)+φ(1,1),0)=φ(φ(1,0,0,φ(1,1,0)),0,0)\)
+\(\varphi(1,\varphi(2,0)+\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1,1),0)=\varphi(\varphi(1,0,0,\varphi(1,1,0)),0,0)\)
-\(φ(1,φ(2,0)+φ(1,1))=φ(φ(1,0,0)+φ(2,0),0)=φ(φ(1,0,0,φ(2,0,0)),0,0)\)
+\(\varphi(1,\varphi(2,0)+\varphi(1,1))=\varphi(\varphi(1,0,0)+\varphi(2,0),0)=\varphi(\varphi(1,0,0,\varphi(2,0,0)),0,0)\)
-\(φ(1,φ(2,0)+φ(1,φ(1,0)))=φ(φ(1,0,0)+φ(φ(1,0),0),0)=φ(φ(1,0,0,φ(φ(1,0,0),0,0)),0,0))\)
+\(\varphi(1,\varphi(2,0)+\varphi(1,\varphi(1,0)))=\varphi(\varphi(1,0,0)+\varphi(\varphi(1,0),0),0)=\varphi(\varphi(1,0,0,\varphi(\varphi(1,0,0),0,0)),0,0))\)
-\(φ(1,φ(2,0)\cdot2)=φ(φ(1,0,0)\cdot2,0)=φ(φ(1,0,1,0),0,0)\)
+\(\varphi(1,\varphi(2,0)\cdot2)=\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(\varphi(1,0,1,0),0,0)\)
-\(φ(1,φ(2,0)\cdotφ(1))=φ(φ(1,1,0),0)=φ(φ(1,0,φ(1,0),0),0,0)\)
+\(\varphi(1,\varphi(2,0)\cdot\varphi(1))=\varphi(\varphi(1,1,0),0)=\varphi(\varphi(1,0,\varphi(1,0),0),0,0)\)
-\(φ(1,φ(2,0)^{2})=φ(φ(1,0,φ(1,0,0)),0)=φ(φ(1,0,φ(1,0,0,0),0),0,0)\)
+\(\varphi(1,\varphi(2,0)^{2})=\varphi(\varphi(1,0,\varphi(1,0,0)),0)=\varphi(\varphi(1,0,\varphi(1,0,0,0),0),0,0)\)
-\(φ(1,φ(1,φ(2,0)+1))=φ(φ(φ(1,0,0),0),0)=φ(φ(φ(1,0,0,0),0,0),0,0)\)
+\(\varphi(1,\varphi(1,\varphi(2,0)+1))=\varphi(\varphi(\varphi(1,0,0),0),0)=\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0)\)
-\(φ(1,φ(1,φ(1,φ(2,0)+1)))=φ(φ(φ(φ(1,0,0),0),0),0)=φ(φ(φ(φ(1,0,0,0),0,0),0,0),0,0)\)
+\(\varphi(1,\varphi(1,\varphi(1,\varphi(2,0)+1)))=\varphi(\varphi(\varphi(\varphi(1,0,0),0),0),0)=\varphi(\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0),0,0)\)
-\(φ(2,1)=φ(2,0,0)=φ(2,0,0,0)\)
+\(\varphi(2,1)=\varphi(2,0,0)=\varphi(2,0,0,0)\)
-\(φ(1,φ(2,1)+1)=φ(φ(2,0,0),0)=φ(φ(2,0,0,0),0,0)\)
+\(\varphi(1,\varphi(2,1)+1)=\varphi(\varphi(2,0,0),0)=\varphi(\varphi(2,0,0,0),0,0)\)
-\(φ(1,φ(1,φ(2,1)+1))=φ(φ(φ(2,0,0),0),0)=φ(φ(φ(2,0,0,0),0,0),0,0)\)
+\(\varphi(1,\varphi(1,\varphi(2,1)+1))=\varphi(\varphi(\varphi(2,0,0),0),0)=\varphi(\varphi(\varphi(2,0,0,0),0,0),0,0)\)
-\(φ(2,2)=φ(3,0,0)=φ(3,0,0,0)\)
+\(\varphi(2,2)=\varphi(3,0,0)=\varphi(3,0,0,0)\)
-\(φ(2,φ(1))=φ(φ(1),0,0)=φ(φ(1,0),0,0,0)\)
+\(\varphi(2,\varphi(1))=\varphi(\varphi(1),0,0)=\varphi(\varphi(1,0),0,0,0)\)
-\(φ(2,φ(1)\cdot2)=φ(φ(1)\cdot2,0,0)=φ(φ(1,φ(1,0)),0,0,0)\)
+\(\varphi(2,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0,0)=\varphi(\varphi(1,\varphi(1,0)),0,0,0)\)
-\(φ(2,φ(2))=φ(φ(2),0,0)=φ(φ(2,0),0,0,0)\)
+\(\varphi(2,\varphi(2))=\varphi(\varphi(2),0,0)=\varphi(\varphi(2,0),0,0,0)\)
-\(φ(2,φ(φ(1)))=φ(φ(φ(1)),0,0)=φ(φ(φ(1,0),0),0,0,0)\)
+\(\varphi(2,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0)\)
-\(φ(2,φ(1,0))=φ(φ(1,0),0,0)=φ(φ(1,0,0),0,0,0)\)
+\(\varphi(2,\varphi(1,0))=\varphi(\varphi(1,0),0,0)=\varphi(\varphi(1,0,0),0,0,0)\)
-\(φ(2,φ(1,1))=φ(φ(2,0),0,0)=φ(φ(2,0,0),0,0,0)\)
+\(\varphi(2,\varphi(1,1))=\varphi(\varphi(2,0),0,0)=\varphi(\varphi(2,0,0),0,0,0)\)
-\(φ(2,φ(1,φ(1)))=φ(φ(φ(1),0),0,0)=φ(φ(φ(1,0),0,0),0,0,0)\)
+\(\varphi(2,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0,0)=\varphi(\varphi(\varphi(1,0),0,0),0,0,0)\)
-\(φ(2,φ(2,0))=φ(φ(1,0,0),0,0)=φ(φ(1,0,0,0),0,0,0)\)
+\(\varphi(2,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0)=\varphi(\varphi(1,0,0,0),0,0,0)\)
-\(φ(2,φ(2,φ(2,0)))=φ(φ(φ(1,0,0),0,0),0,0)=φ(φ(φ(1,0,0,0),0,0,0),0,0,0)\)
+\(\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)=\varphi(\varphi(\varphi(1,0,0,0),0,0,0),0,0,0)\)
-\(φ(3,0)=φ(1,0,0,0)=φ(1,0,0,0,0)=\) '''[[LCO|η<sub>0</sub>]]'''
+\(\varphi(3,0)=\varphi(1,0,0,0)=\varphi(1,0,0,0,0)=\) [[LCO|η0]]
-\(φ(1,φ(3,0)+1)=φ(φ(1,0,0,0),0)=φ(φ(1,0,0,0,0),0,0)\)
+\(\varphi(1,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0)=\varphi(\varphi(1,0,0,0,0),0,0)\)
-\(φ(2,φ(3,0)+1)=φ(φ(1,0,0,0),0,0)=φ(φ(1,0,0,0,0),0,0,0)\)
+\(\varphi(2,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0)\)
-\(φ(3,1)=φ(2,0,0,0)=φ(2,0,0,0,0)\)
+\(\varphi(3,1)=\varphi(2,0,0,0)=\varphi(2,0,0,0,0)\)
-\(φ(3,φ(1))=φ(φ(1),0,0,0)=φ(φ(1,0),0,0,0,0)\)
+\(\varphi(3,\varphi(1))=\varphi(\varphi(1),0,0,0)=\varphi(\varphi(1,0),0,0,0,0)\)
-\(φ(3,φ(φ(1)))=φ(φ(φ(1)),0,0,0)=φ(φ(φ(1,0),0),0,0,0,0)\)
+\(\varphi(3,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0,0)\)
-\(φ(3,φ(1,0))=φ(φ(1,0),0,0,0)=φ(φ(1,0,0),0,0,0,0)\)
+\(\varphi(3,\varphi(1,0))=\varphi(\varphi(1,0),0,0,0)=\varphi(\varphi(1,0,0),0,0,0,0)\)
-\(φ(3,φ(2,0))=φ(φ(1,0,0),0,0,0)=φ(φ(1,0,0,0),0,0,0,0)\)
+\(\varphi(3,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0,0)=\varphi(\varphi(1,0,0,0),0,0,0,0)\)
-\(φ(3,φ(3,0))=φ(φ(1,0,0,0),0,0,0)=φ(φ(1,0,0,0,0),0,0,0,0)\)
+\(\varphi(3,\varphi(3,0))=\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0,0)\)
-\(φ(4,0)=φ(1,0,0,0,0)=φ(1,0,0,0,0,0)\)
+\(\varphi(4,0)=\varphi(1,0,0,0,0)=\varphi(1,0,0,0,0,0)\)
-\(φ(φ(1),0)=φ(1@φ(1))=φ(1@φ(1,0))=\) '''[[HCO]]'''
+\(\varphi(\varphi(1),0)=\varphi(1@\varphi(1))=\varphi(1@\varphi(1,0))=\) [[HCO]]
-\(φ(1,φ(φ(1),0)+1)=φ(φ(1@φ(1)),0)=φ(φ(1@φ(1,0)),0,0)\)
+\(\varphi(1,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0)=\varphi(\varphi(1@\varphi(1,0)),0,0)\)
-\(φ(2,φ(φ(1),0)+1)=φ(φ(1@φ(1)),0,0)=φ(φ(1@φ(1,0)),0,0,0)\)
+\(\varphi(2,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0,0)=\varphi(\varphi(1@\varphi(1,0)),0,0,0)\)
-\(φ(3,φ(φ(1),0)+1)=φ(φ(1@φ(1)),0,0,0)=φ(φ(1@φ(1,0)),0,0,0,0)\)
+\(\varphi(3,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0,0,0)=\varphi(\varphi(1@\varphi(1,0)),0,0,0,0)\)
-\(φ(φ(1),1)=φ(2@φ(1))=φ(2@φ(1,0))\)
+\(\varphi(\varphi(1),1)=\varphi(2@\varphi(1))=\varphi(2@\varphi(1,0))\)
-\(φ(1,φ(φ(1),1)+1)=φ(φ(2@φ(1)),0)=φ(φ(2@φ(1,0)),0,0)\)
+\(\varphi(1,\varphi(\varphi(1),1)+1)=\varphi(\varphi(2@\varphi(1)),0)=\varphi(\varphi(2@\varphi(1,0)),0,0)\)
-\(φ(φ(1),2)=φ(3@φ(1))=φ(3@φ(1,0))\)
+\(\varphi(\varphi(1),2)=\varphi(3@\varphi(1))=\varphi(3@\varphi(1,0))\)
-\(φ(φ(1),φ(1))=φ(φ(1)@φ(1))=φ(φ(1,0)@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(1))=\varphi(\varphi(1)@\varphi(1))=\varphi(\varphi(1,0)@\varphi(1,0))\)
-\(φ(φ(1),φ(1)\cdot2)=φ(φ(1)\cdot2@φ(1))=φ(φ(1,0)\cdot2@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2@\varphi(1))=\varphi(\varphi(1,0)\cdot2@\varphi(1,0))\)
-\(φ(φ(1),φ(2))=φ(φ(2)@φ(1))=φ(φ(2,0)@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(2))=\varphi(\varphi(2)@\varphi(1))=\varphi(\varphi(2,0)@\varphi(1,0))\)
-\(φ(φ(1),φ(φ(1)))=φ(φ(φ(1))@φ(1))=φ(φ(φ(1,0),0)@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1))@\varphi(1))=\varphi(\varphi(\varphi(1,0),0)@\varphi(1,0))\)
-\(φ(φ(1),φ(1,0))=φ(φ(1,0)@φ(1))=φ(φ(1,0,0)@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0)@\varphi(1))=\varphi(\varphi(1,0,0)@\varphi(1,0))\)
-\(φ(φ(1),φ(2,0))=φ(φ(1,0,0)@φ(1))=φ(φ(1,0,0,0)@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(2,0))=\varphi(\varphi(1,0,0)@\varphi(1))=\varphi(\varphi(1,0,0,0)@\varphi(1,0))\)
-\(φ(φ(1),φ(3,0))=φ(φ(1,0,0,0)@φ(1))=φ(φ(1,0,0,0,0)@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(3,0))=\varphi(\varphi(1,0,0,0)@\varphi(1))=\varphi(\varphi(1,0,0,0,0)@\varphi(1,0))\)
-\(φ(φ(1),φ(φ(1),0))=φ(φ(1@φ(1))@φ(1))=φ(φ(1@φ(1,0))@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(\varphi(1),0))=\varphi(\varphi(1@\varphi(1))@\varphi(1))=\varphi(\varphi(1@\varphi(1,0))@\varphi(1,0))\)
-\(φ(φ(1)+1,0)=φ(1@φ(1)+1)=φ(1@φ(1,1))\)
+\(\varphi(\varphi(1)+1,0)=\varphi(1@\varphi(1)+1)=\varphi(1@\varphi(1,1))\)
-\(φ(φ(1)+1,φ(φ(1)+1,0))=φ(φ(1@φ(1)+1)@φ(1)+1)=φ(φ(1@φ(1,1))@φ(1,1))\)
+\(\varphi(\varphi(1)+1,\varphi(\varphi(1)+1,0))=\varphi(\varphi(1@\varphi(1)+1)@\varphi(1)+1)=\varphi(\varphi(1@\varphi(1,1))@\varphi(1,1))\)
-\(φ(φ(1)+2,0)=φ(1@φ(1)+2)=φ(1@φ(1,2))\)
+\(\varphi(\varphi(1)+2,0)=\varphi(1@\varphi(1)+2)=\varphi(1@\varphi(1,2))\)
-\(φ(φ(1)\cdot2,0)=φ(1@φ(1)\cdot2)=φ(1@φ(1,φ(1)))\)
+\(\varphi(\varphi(1)\cdot2,0)=\varphi(1@\varphi(1)\cdot2)=\varphi(1@\varphi(1,\varphi(1)))\)
-\(φ(φ(1)\cdot3,0)=φ(1@φ(1)\cdot3)=φ(1@φ(1,φ(1,φ(1,0))))\)
+\(\varphi(\varphi(1)\cdot3,0)=\varphi(1@\varphi(1)\cdot3)=\varphi(1@\varphi(1,\varphi(1,\varphi(1,0))))\)
-\(φ(φ(2),0)=φ(1@φ(2))=φ(1@φ(2,0))\)
+\(\varphi(\varphi(2),0)=\varphi(1@\varphi(2))=\varphi(1@\varphi(2,0))\)
-\(φ(φ(φ(1)),0)=φ(1@φ(φ(1)))=φ(1@φ(φ(1,0),0))\)
+\(\varphi(\varphi(\varphi(1)),0)=\varphi(1@\varphi(\varphi(1)))=\varphi(1@\varphi(\varphi(1,0),0))\)
-\(φ(φ(1,0),0)=φ(1@φ(1,0))=φ(1@φ(1,0,0))\)
+\(\varphi(\varphi(1,0),0)=\varphi(1@\varphi(1,0))=\varphi(1@\varphi(1,0,0))\)
-\(φ(φ(2,0),0)=φ(1@φ(1,0,0))=φ(1@φ(1,0,0,0))\)
+\(\varphi(\varphi(2,0),0)=\varphi(1@\varphi(1,0,0))=\varphi(1@\varphi(1,0,0,0))\)
-\(φ(φ(3,0),0)=φ(1@φ(1,0,0,0))=φ(1@φ(1,0,0,0,0))\)
+\(\varphi(\varphi(3,0),0)=\varphi(1@\varphi(1,0,0,0))=\varphi(1@\varphi(1,0,0,0,0))\)
-\(φ(φ(φ(1),0),0)=φ(1@φ(1@φ(1)))=φ(1@φ(1@φ(1,0)))\)
+\(\varphi(\varphi(\varphi(1),0),0)=\varphi(1@\varphi(1@\varphi(1)))=\varphi(1@\varphi(1@\varphi(1,0)))\)
-\(φ(φ(φ(1,0),0),0)=φ(1@φ(1@φ(1,0)))=φ(1@φ(1@φ(1,0,0)))\)
+\(\varphi(\varphi(\varphi(1,0),0),0)=\varphi(1@\varphi(1@\varphi(1,0)))=\varphi(1@\varphi(1@\varphi(1,0,0)))\)
-\(φ(φ(φ(2,0),0),0)=φ(1@φ(1@φ(1,0,0)))=φ(1@φ(1@φ(1,0,0,0)))\)
+\(\varphi(\varphi(\varphi(2,0),0),0)=\varphi(1@\varphi(1@\varphi(1,0,0)))=\varphi(1@\varphi(1@\varphi(1,0,0,0)))\)
-\(φ(1,0,0)=φ(1@(1,0))=φ(1@(1,0))=\) '''[[FSO]]'''
+\(\varphi(1,0,0)=\varphi(1@(1,0))=\varphi(1@(1,0))=\) [[FSO]]
-\(φ(1,φ(1,0,0)+1)=φ(φ(1@(1,0)),0)=φ(φ(1@(1,0)),0,0)\)
+\(\varphi(1,\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0)),0)=\varphi(\varphi(1@(1,0)),0,0)\)
-\(φ(2,φ(1,0,0)+1)=φ(φ(1@(1,0)),0,0)=φ(φ(1@(1,0)),0,0,0)\)
+\(\varphi(2,\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0)),0,0)=\varphi(\varphi(1@(1,0)),0,0,0)\)
-\(φ(φ(1),φ(1,0,0)+1)=φ(φ(1@(1,0))@φ(1))=φ(φ(1@(1,0))@φ(1,0))\)
+\(\varphi(\varphi(1),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1))=\varphi(\varphi(1@(1,0))@\varphi(1,0))\)
-\(φ(φ(1,0),φ(1,0,0)+1)=φ(φ(1@(1,0))@φ(1,0))=φ(φ(1@(1,0))@φ(1,0,0))\)
+\(\varphi(\varphi(1,0),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1,0))=\varphi(\varphi(1@(1,0))@\varphi(1,0,0))\)
-\(φ(φ(2,0),φ(1,0,0)+1)=φ(φ(1@(1,0))@φ(1,0,0))=φ(φ(1@(1,0))@φ(1,0,0,0))\)
+\(\varphi(\varphi(2,0),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@\varphi(1,0,0,0))\)
-\(φ(φ(1,0,0),1)=φ(2@φ(1@(1,0)))=φ(2@φ(1@(1,0)))\)
+\(\varphi(\varphi(1,0,0),1)=\varphi(2@\varphi(1@(1,0)))=\varphi(2@\varphi(1@(1,0)))\)
-\(φ(φ(1,0,0),2)=φ(3@φ(1@(1,0)))=φ(3@φ(1@(1,0)))\)
+\(\varphi(\varphi(1,0,0),2)=\varphi(3@\varphi(1@(1,0)))=\varphi(3@\varphi(1@(1,0)))\)
-\(φ(φ(1,0,0),φ(1))=φ(φ(1)@φ(1@(1,0)))=φ(φ(1,0)@φ(1@(1,0)))\)
+\(\varphi(\varphi(1,0,0),\varphi(1))=\varphi(\varphi(1)@\varphi(1@(1,0)))=\varphi(\varphi(1,0)@\varphi(1@(1,0)))\)
-\(φ(φ(1,0,0),φ(1,0))=φ(φ(1,0)@φ(1@(1,0)))=φ(φ(1,0,0)@φ(1@(1,0)))\)
+\(\varphi(\varphi(1,0,0),\varphi(1,0))=\varphi(\varphi(1,0)@\varphi(1@(1,0)))=\varphi(\varphi(1,0,0)@\varphi(1@(1,0)))\)
-\(φ(φ(1,0,0),φ(1,0,0))=φ(φ(1@(1,0))@φ(1@(1,0)))=φ(φ(1@(1,0))@φ(1@(1,0)))\)
+\(\varphi(\varphi(1,0,0),\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@\varphi(1@(1,0)))=\varphi(\varphi(1@(1,0))@\varphi(1@(1,0)))\)
-\(φ(φ(1,0,0)+1,0)=φ(1@φ(1@(1,0))+1)=φ(1@φ(1@(1,0),1@0))\)
+\(\varphi(\varphi(1,0,0)+1,0)=\varphi(1@\varphi(1@(1,0))+1)=\varphi(1@\varphi(1@(1,0),1@0))\)
-\(φ(φ(1,0,0)+φ(1),0)=φ(1@φ(1@(1,0))+φ(1))=φ(1@φ(1@(1,0),φ(1,0)@0))\)
+\(\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(1@\varphi(1@(1,0))+\varphi(1))=\varphi(1@\varphi(1@(1,0),\varphi(1,0)@0))\)
-\(φ(φ(1,0,0)+φ(1,0),0)=φ(1@φ(1@(1,0))+φ(1,0))=φ(1@φ(1@(1,0),φ(1,0,0)@0))\)
+\(\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(1@\varphi(1@(1,0))+\varphi(1,0))=\varphi(1@\varphi(1@(1,0),\varphi(1,0,0)@0))\)
-\(φ(φ(1,0,0)\cdot2,0)=φ(1@φ(1@(1,0))\cdot2)=φ(1@φ(1@(1,0),φ(1@(1,0))@0))\)
+\(\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(1@\varphi(1@(1,0))\cdot2)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))\)
-\(φ(φ(1,0,0)\cdotφ(1),0)=φ(1@φ(1@(1,0),1@0))=φ(1@φ(1@(1,0),1@1))\)
+\(\varphi(\varphi(1,0,0)\cdot\varphi(1),0)=\varphi(1@\varphi(1@(1,0),1@0))=\varphi(1@\varphi(1@(1,0),1@1))\)
-\(φ(φ(1,0,0)^{2},0)=φ(1@φ(1@(1,0),φ(1@(1,0))@0))=φ(1@φ(1@(1,0),φ(1@(1,0))@0))\)
+\(\varphi(\varphi(1,0,0)^{2},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))\)
-\(φ(φ(1,0,0)^{φ(1)},0)=φ(1@φ(1,φ(1@(1,0),1@0),0))=φ(1@φ(1,φ(1@(1,0),1@1),0))\)
+\(\varphi(\varphi(1,0,0)^{\varphi(1)},0)=\varphi(1@\varphi(1,\varphi(1@(1,0),1@0),0))=\varphi(1@\varphi(1,\varphi(1@(1,0),1@1),0))\)
-\(φ(φ(1,0,0)^{φ(1,0)},0)=φ(1@φ(1@(1,0),φ(1@(1,0),φ(1,0)@0)@0))=φ(1@φ(1@(1,0),φ(1@(1,0),φ(1,0)@0)@0))\)
+\(\varphi(\varphi(1,0,0)^{\varphi(1,0)},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1,0)@0)@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1,0)@0)@0))\)
-\(φ(φ(1,0,0)^{φ(1,0,0)},0)=φ(1@φ(1@(1,0),φ(1@(1,0),φ(1@(1,0))@0)@0))=φ(1@φ(1@(1,0),φ(1@(1,0),φ(1@(1,0))@0)@0))\)
+\(\varphi(\varphi(1,0,0)^{\varphi(1,0,0)},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1@(1,0))@0)@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1@(1,0))@0)@0))\)
-\(φ(φ(1,φ(1,0,0)+1),0)=φ(1@φ(φ(1@(1,0)),0))=φ(1@φ(φ(1@(1,0)),0,0))\)
+\(\varphi(\varphi(1,\varphi(1,0,0)+1),0)=\varphi(1@\varphi(\varphi(1@(1,0)),0))=\varphi(1@\varphi(\varphi(1@(1,0)),0,0))\)
-\(φ(φ(1,φ(1,0,0)+φ(1)),0)=φ(1@φ(φ(1@(1,0))+φ(1),0))=φ(1@φ(φ(1@(1,0))+φ(1,0),0))\)
+\(\varphi(\varphi(1,\varphi(1,0,0)+\varphi(1)),0)=\varphi(1@\varphi(\varphi(1@(1,0))+\varphi(1),0))=\varphi(1@\varphi(\varphi(1@(1,0))+\varphi(1,0),0))\)
-\(φ(φ(φ(1),φ(1,0,0)+1),0)=φ(1@φ(φ(1@(1,0))@φ(1)))=φ(1@φ(φ(1@(1,0))@φ(1,0)))\)
+\(\varphi(\varphi(\varphi(1),\varphi(1,0,0)+1),0)=\varphi(1@\varphi(\varphi(1@(1,0))@\varphi(1)))=\varphi(1@\varphi(\varphi(1@(1,0))@\varphi(1,0)))\)
-\(φ(1,0,1)=φ(1@(1,1))=φ(1@(1,1))\)
+\(\varphi(1,0,1)=\varphi(1@(1,1))=\varphi(1@(1,1))\)
-\(φ(1,0,2)=φ(2@(1,1))=φ(2@(1,1))\)
+\(\varphi(1,0,2)=\varphi(2@(1,1))=\varphi(2@(1,1))\)
-\(φ(1,0,φ(1))=φ(φ(1)@(1,1))=φ(φ(1,0)@(1,1))\)
+\(\varphi(1,0,\varphi(1))=\varphi(\varphi(1)@(1,1))=\varphi(\varphi(1,0)@(1,1))\)
-\(φ(1,0,φ(1,0))=φ(φ(1,0)@(1,1))=φ(φ(1,0,0)@(1,1))\)
+\(\varphi(1,0,\varphi(1,0))=\varphi(\varphi(1,0)@(1,1))=\varphi(\varphi(1,0,0)@(1,1))\)
-\(φ(1,0,φ(1,0,0))=φ(φ(1@(1,0))@(1,1))=φ(φ(1@(1,0))@(1,1))\)
+\(\varphi(1,0,\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@(1,1))=\varphi(\varphi(1@(1,0))@(1,1))\)
-\(φ(1,1,0)=φ(1@(1,1))=φ(1@(1,1))\)
+\(\varphi(1,1,0)=\varphi(1@(1,1))=\varphi(1@(1,1))\)
-\(φ(1,1,φ(1,1,0))=φ(φ(1@(1,1))@(1,1))=φ(φ(1@(1,1))@(1,1))\)
+\(\varphi(1,1,\varphi(1,1,0))=\varphi(\varphi(1@(1,1))@(1,1))=\varphi(\varphi(1@(1,1))@(1,1))\)
-\(φ(1,2,0)=φ(1@(1,2))=φ(1@(1,2))\)
+\(\varphi(1,2,0)=\varphi(1@(1,2))=\varphi(1@(1,2))\)
-\(φ(1,φ(1),0)=φ(1@(1,φ(1)))=φ(1@(1,φ(1,0)))\)
+\(\varphi(1,\varphi(1),0)=\varphi(1@(1,\varphi(1)))=\varphi(1@(1,\varphi(1,0)))\)
-\(φ(1,φ(1,0),0)=φ(1@(1,φ(1,0)))=φ(1@(1,φ(1,0,0)))\)
+\(\varphi(1,\varphi(1,0),0)=\varphi(1@(1,\varphi(1,0)))=\varphi(1@(1,\varphi(1,0,0)))\)
-\(φ(1,φ(1,0,0),0)=φ(1@(1,φ(1@(1,0))))=φ(1@(1,φ(1@(1,0))))\)
+\(\varphi(1,\varphi(1,0,0),0)=\varphi(1@(1,\varphi(1@(1,0))))=\varphi(1@(1,\varphi(1@(1,0))))\)
-\(φ(2,0,0)=φ(1@(2,0))=φ(1@(2,0))\)
+\(\varphi(2,0,0)=\varphi(1@(2,0))=\varphi(1@(2,0))\)
-\(φ(2,0,1)=φ(2@(2,0))=φ(2@(2,0))\)
+\(\varphi(2,0,1)=\varphi(2@(2,0))=\varphi(2@(2,0))\)
-\(φ(2,1,0)=φ(1@(2,1))=φ(1@(2,1))\)
+\(\varphi(2,1,0)=\varphi(1@(2,1))=\varphi(1@(2,1))\)
-\(φ(2,φ(2,0,0),0)=φ(1@(2,φ(1@(2,0))))=φ(1@(2,φ(1@(2,0))))\)
+\(\varphi(2,\varphi(2,0,0),0)=\varphi(1@(2,\varphi(1@(2,0))))=\varphi(1@(2,\varphi(1@(2,0))))\)
-\(φ(3,0,0)=φ(1@(3,0))=φ(1@(3,0))\)
+\(\varphi(3,0,0)=\varphi(1@(3,0))=\varphi(1@(3,0))\)
-\(φ(φ(1),0,0)=φ(1@(φ(1),0))=φ(1@(φ(1,0),0))\)
+\(\varphi(\varphi(1),0,0)=\varphi(1@(\varphi(1),0))=\varphi(1@(\varphi(1,0),0))\)
-\(φ(φ(1,0),0,0)=φ(1@(φ(1,0),0))=φ(1@(φ(1,0,0),0))\)
+\(\varphi(\varphi(1,0),0,0)=\varphi(1@(\varphi(1,0),0))=\varphi(1@(\varphi(1,0,0),0))\)
-\(φ(φ(1,0,0),0,0)=φ(1@(φ(1@(1,0)),0))=φ(1@(φ(1@(1,0)),0))\)
+\(\varphi(\varphi(1,0,0),0,0)=\varphi(1@(\varphi(1@(1,0)),0))=\varphi(1@(\varphi(1@(1,0)),0))\)
-\(φ(1,0,0,0)=φ(1@(1,0,0))=φ(1@(1,0,0))\)
+\(\varphi(1,0,0,0)=\varphi(1@(1,0,0))=\varphi(1@(1,0,0))=\) [[ACO]]
-\(φ(1,0,0,1)=φ(2@(1,0,0))=φ(2@(1,0,0))\)
+\(\varphi(1,0,0,1)=\varphi(2@(1,0,0))=\varphi(2@(1,0,0))\)
-\(φ(1,0,1,0)=φ(1@(1,0,1))=φ(1@(1,0,1))\)
+\(\varphi(1,0,1,0)=\varphi(1@(1,0,1))=\varphi(1@(1,0,1))\)
-\(φ(1,1,0,0)=φ(1@(1,1,0))=φ(1@(1,1,0))\)
+\(\varphi(1,1,0,0)=\varphi(1@(1,1,0))=\varphi(1@(1,1,0))\)
-\(φ(2,0,0,0)=φ(1@(2,0,0))=φ(1@(2,0,0))\)
+\(\varphi(2,0,0,0)=\varphi(1@(2,0,0))=\varphi(1@(2,0,0))\)
-\(φ(φ(1),0,0,0)=φ(1@(φ(1),0,0))=φ(1@(φ(1,0),0,0))\)
+\(\varphi(\varphi(1),0,0,0)=\varphi(1@(\varphi(1),0,0))=\varphi(1@(\varphi(1,0),0,0))\)
-\(φ(φ(1,0,0,0),0,0,0)=φ(1@(φ(1@(1,0,0)),0,0))=φ(1@(φ(1@(1,0,0)),0,0))\)
+\(\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(1@(\varphi(1@(1,0,0)),0,0))=\varphi(1@(\varphi(1@(1,0,0)),0,0))\)
-\(φ(1,0,0,0,0)=φ(1@(1,0,0,0))=φ(1@(1,0,0,0))\)
+\(\varphi(1,0,0,0,0)=\varphi(1@(1,0,0,0))=\varphi(1@(1,0,0,0))\)
-\(φ(1,0,0,0,0,0)=φ(1@(1,0,0,0,0))=φ(1@(1,0,0,0,0))\)
+\(\varphi(1,0,0,0,0,0)=\varphi(1@(1,0,0,0,0))=\varphi(1@(1,0,0,0,0))\)
-\(φ(1,1,4,5,1,4)=φ(5@(1,1,4,5,1))=φ(5@(1,1,4,5,1))\)
+\(\varphi(1,1,4,5,1,4)=\varphi(5@(1,1,4,5,1))=\varphi(5@(1,1,4,5,1))\)
-\(φ(1@φ(1))=φ(1@(1@φ(1)))=φ(1@(1@φ(1,0)))=\) '''[[SVO]]'''
+\(\varphi(1@\varphi(1))=\varphi(1@(1@\varphi(1)))=\varphi(1@(1@\varphi(1,0)))=\) [[SVO]]
-\(φ(1,φ(1@φ(1))+1)=φ(1@(1@φ(1)),1@1)=φ(1@(1@φ(1,0)),1@2)\)
+\(\varphi(1,\varphi(1@\varphi(1))+1)=\varphi(1@(1@\varphi(1)),1@1)=\varphi(1@(1@\varphi(1,0)),1@2)\)
-\(φ(1@φ(1),1@0)=φ(2@(1@φ(1)))=φ(2@(1@φ(1,0)))\)
+\(\varphi(1@\varphi(1),1@0)=\varphi(2@(1@\varphi(1)))=\varphi(2@(1@\varphi(1,0)))\)
-\(φ(1@φ(1),1@1)=φ(1@(1@φ(1),1@0))=φ(1@(1@φ(1,0),1@0))\)
+\(\varphi(1@\varphi(1),1@1)=\varphi(1@(1@\varphi(1),1@0))=\varphi(1@(1@\varphi(1,0),1@0))\)
-\(φ(1@φ(1),1@2)=φ(1@(1@φ(1),1@1))=φ(1@(1@φ(1,0),1@1))\)
+\(\varphi(1@\varphi(1),1@2)=\varphi(1@(1@\varphi(1),1@1))=\varphi(1@(1@\varphi(1,0),1@1))\)
-\(φ(2@φ(1))=φ(1@(2@φ(1)))=φ(1@(2@φ(1,0)))\)
+\(\varphi(2@\varphi(1))=\varphi(1@(2@\varphi(1)))=\varphi(1@(2@\varphi(1,0)))\)
-\(φ(φ(1)@φ(1))=φ(1@(φ(1)@φ(1)))=φ(1@(φ(1,0)@φ(1,0)))\)
+\(\varphi(\varphi(1)@\varphi(1))=\varphi(1@(\varphi(1)@\varphi(1)))=\varphi(1@(\varphi(1,0)@\varphi(1,0)))\)
-\(φ(1@φ(1)+1)=φ(1@(1@φ(1)+1))=φ(1@(1@φ(1,1)))\)
+\(\varphi(1@\varphi(1)+1)=\varphi(1@(1@\varphi(1)+1))=\varphi(1@(1@\varphi(1,1)))\)
-\(φ(1@φ(1)\cdot2)=φ(1@(1@φ(1)\cdot2))=φ(1@(1@φ(1,φ(1,0))))\)
+\(\varphi(1@\varphi(1)\cdot2)=\varphi(1@(1@\varphi(1)\cdot2))=\varphi(1@(1@\varphi(1,\varphi(1,0))))\)
-\(φ(1@φ(2))=φ(1@(1@φ(2)))=φ(1@(1@φ(2,0)))\)
+\(\varphi(1@\varphi(2))=\varphi(1@(1@\varphi(2)))=\varphi(1@(1@\varphi(2,0)))\)
-\(φ(1@φ(1,0))=φ(1@(1@φ(1,0)))=φ(1@(1@φ(1,0,0)))\)
+\(\varphi(1@\varphi(1,0))=\varphi(1@(1@\varphi(1,0)))=\varphi(1@(1@\varphi(1,0,0)))\)
-\(φ(1@φ(2,0))=φ(1@(1@φ(1,0,0)))=φ(1@(1@φ(1,0,0,0)))\)
+\(\varphi(1@\varphi(2,0))=\varphi(1@(1@\varphi(1,0,0)))=\varphi(1@(1@\varphi(1,0,0,0)))\)
-\(φ(1@φ(1,0,0))=φ(1@(1@φ(1@(1,0))))=φ(1@(1@φ(1@(1,0))))\)
+\(\varphi(1@\varphi(1,0,0))=\varphi(1@(1@\varphi(1@(1,0))))=\varphi(1@(1@\varphi(1@(1,0))))\)
-\(φ(1@φ(1@φ(1)))=φ(1@(1@φ(1@(1@φ(1)))))=φ(1@(1@φ(1@(1@φ(1,0)))))=\) '''[[LVO]]'''
+\(\varphi(1@\varphi(1@\varphi(1)))=\varphi(1@(1@\varphi(1@(1@\varphi(1)))))=\varphi(1@(1@\varphi(1@(1@\varphi(1,0)))))=\) [[LVO]]
 === 历史 ===
-据信在 2024 年下旬及以前的 Weak Veblen 都是指的 <math>\times\omega\ \phi</math>，但在这之后 <math>\times\omega\ \phi</math> 的生态位快速被 <math>+1\ \phi</math> 所代替了，所以出现了一些较为混乱的局面。
+据信在 2024 年下旬及以前的 Weak Veblen 都是指的 <math>\times\omega\ \varphi</math>，但在这之后 <math>\times\omega\ \varphi</math> 的生态位快速被 <math>+1\ \varphi</math> 所代替了，所以出现了一些较为混乱的局面。
+{{默认排序:序数记号}}
 [[分类:记号]]
 [[分类:分析]]