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 $\mathrm{F}\mathrm{P}\phi$，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 $\times \omega$，则称其为 $\times \omega \varphi$。

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

以下是两种 weak Veblen 直到序元级别的定义：

$+1\varphi$

1. $\phi (\mathrm{\#},(\alpha +1)$@$(\beta +1),0$@$0)[0]=0$。
2. $\phi (\mathrm{\#},(\alpha +1)$@$(\beta +1),0$@$0)[n+1]=\phi (\mathrm{\#},\alpha$@$(\beta +1),\phi (\mathrm{\#},(\alpha +1)$@$(\beta +1),0$@$0)[n]$@$\beta )$。
3. 对于极限序数$\alpha$，$\phi (\mathrm{\#},\alpha$@$\beta )[n]=\phi (\mathrm{\#},\alpha [n]$@$\beta )$。
4. 对于极限序数$\beta$，$\phi (\mathrm{\#},(\alpha +1)$@$\beta ,0$@$0)[n]=\phi (\mathrm{\#},\alpha$@$\beta ,1$@$\beta [n])$。
5. $\phi (\mathrm{\#},(\gamma +1)$@$0)=\phi (\mathrm{\#},\gamma$@$0)+1$。

$\times \omega \varphi$

1. $\phi (\mathrm{\#},(\alpha +1)$@$(\beta +1),0$@$0)[0]=0$。
2. $\phi (\mathrm{\#},(\alpha +1)$@$(\beta +1),0$@$0)[n+1]=\phi (\mathrm{\#},\alpha$@$(\beta +1),\phi (\mathrm{\#},(\alpha +1)$@$(\beta +1),0$@$0)[n]$@$\beta )$。
3. 对于极限序数$\alpha$，$\phi (\mathrm{\#},\alpha$@$\beta )[n]=\phi (\mathrm{\#},\alpha [n]$@$\beta )$。
4. 对于极限序数$\beta$，$\phi (\mathrm{\#},(\alpha +1)$@$\beta ,0$@$0)[n]=\phi (\mathrm{\#},\alpha$@$\beta ,1$@$\beta [n])$。
5. $\phi (\mathrm{\#},(\gamma +1)$@$0)=\phi (\mathrm{\#},\gamma$@$0)\cdot \phi (1)$。

分析

以下分析中，左为 $\mathrm{F}\mathrm{P}\varphi$，中为 $\times \omega \varphi$，右为 $+1\varphi$。

$\phi (0)=\phi (0)=\phi (0)$

$\phi (0)\cdot 2=\phi (0)\cdot 2=\phi (\phi (0))$

$\phi (0)\cdot 3=\phi (0)\cdot 3=\phi (\phi (\phi (0)))$

$\phi (1)=\phi (1)=\phi (1,0)=$ ω

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

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

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

$\phi (1)\cdot 2+1=\phi (1)\cdot 2+1=\phi (1,\phi (1,1))$

$\phi (1)\cdot 2+2=\phi (1)\cdot 2+2=\phi (1,\phi (1,2))$

$\phi (1)\cdot 3=\phi (1)\cdot 3=\phi (1,\phi (1,\phi (1,0)))$

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

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

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

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

$\phi (2)\cdot 2=\phi (2)\cdot 2=\phi (2,\phi (2,0))$

$\phi (2)\cdot 3=\phi (2)\cdot 3=\phi (2,\phi (2,\phi (2,0)))$

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

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

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

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

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

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

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

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

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

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

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

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

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

$\phi (\phi (1)\cdot 3)=\phi (\phi (1)\cdot 3)=\phi (\phi (1,\phi (1,\phi (1,0))),0)$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\phi (1,0)\cdot 3=\phi (1,0)\cdot 3=\phi (1,0,\phi (1,0,\phi (1,0,0)))$

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

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

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

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

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

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

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

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

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

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

$\phi (1,0{)}^3=\phi (1,\phi (1,0)\cdot 2)=\phi (1,\phi (1,0,\phi (1,0,0),0),0)$

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

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

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

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

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

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

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

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

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

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

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

<math>\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)</math>

<math>\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)</math>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\phi (\phi (1),\phi (1)\cdot 2)=\phi (\phi (1)\cdot 2$@$\phi (1))=\phi (\phi (1,0)\cdot 2$@$\phi (1,0))$

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

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

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

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

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

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

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

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

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

$\phi (\phi (1)\cdot 2,0)=\phi (1$@$\phi (1)\cdot 2)=\phi (1$@$\phi (1,\phi (1)))$

$\phi (\phi (1)\cdot 3,0)=\phi (1$@$\phi (1)\cdot 3)=\phi (1$@$\phi (1,\phi (1,\phi (1,0))))$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\phi (\phi (1,0,0)\cdot 2,0)=\phi (1$@$\phi (1$@$(1,0))\cdot 2)=\phi (1$@$\phi (1$@$(1,0),\phi (1$@$(1,0))$@$0))$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\phi (1$@$\phi (1)\cdot 2)=\phi (1$@$(1$@$\phi (1)\cdot 2))=\phi (1$@$(1$@$\phi (1,\phi (1,0))))$

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

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

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

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

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

历史

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