查看“︁weak Veblen 函数”︁的源代码

'''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。

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 + 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 + 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：

<big>'''<math>+1\ \phi</math>'''</big>

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

<big>'''<math>\times\omega\ \phi</math>'''</big>
# \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[0]=0\)。
# \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)。[n+1]=\varphi(\#,\alpha\text{@}(\beta+1),\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[n]\text{@}\beta)\)。
# 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 <math>\mathrm{FP}\ \phi</math>，中为 <math>\times\omega\ \phi</math>，右为 <math>+1\ \phi</math>。

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

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

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

\(\varphi(1)=\varphi(1)=\varphi(1,0)=\) [[FTO|ω]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(\varphi(1,0)=\varphi(1,0)=\varphi(1,0,0)=\) [[SCO|ε0]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<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>

<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>

<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>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(\varphi(2,0)=\varphi(1,0,0)=\varphi(1,0,0,0)=\) [[CO|ζ0]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\(\varphi(3,0)=\varphi(1,0,0,0)=\varphi(1,0,0,0,0)=\) [[LCO|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 <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。

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