weak Veblen 函数

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。