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>

# <math>\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[0]=0</math>。
# <math>\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[n+1]=\varphi(\#,\alpha</math>@<math>(\beta+1),\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[n]</math>@<math>\beta)</math>。
# 对于极限序数<math>\alpha</math>，<math>\varphi(\#,\alpha</math>@<math>\beta)[n]=\varphi(\#,\alpha[n]</math>@<math>\beta)</math>。
# 对于极限序数<math>\beta</math>，<math>\varphi(\#,(\alpha+1)</math>@<math>\beta,0</math>@<math>0)[n]=\varphi(\#,\alpha</math>@<math>\beta,1</math>@<math>\beta[n])</math>。
# <math>\varphi(\#,(\gamma+1)</math>@<math>0)=\varphi(\#,\gamma</math>@<math>0)+1</math>。

<big>'''<math>\times\omega\ \phi</math>'''</big>
# <math>\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[0]=0</math>。
# <math>\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[n+1]=\varphi(\#,\alpha</math>@<math>(\beta+1),\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[n]</math>@<math>\beta)</math>。
# 对于极限序数<math>\alpha</math>，<math>\varphi(\#,\alpha</math>@<math>\beta)[n]=\varphi(\#,\alpha[n]</math>@<math>\beta)</math>。
# 对于极限序数<math>\beta</math>，<math>\varphi(\#,(\alpha+1)</math>@<math>\beta,0</math>@<math>0)[n]=\varphi(\#,\alpha</math>@<math>\beta,1</math>@<math>\beta[n])</math>。
# <math>\varphi(\#,(\gamma+1)</math>@<math>0)=\varphi(\#,\gamma</math>@<math>0)\cdot\varphi(1)</math>。

=== 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>。

<math>\varphi(0)=\varphi(0)=\varphi(0)</math>

<math>\varphi(0)\cdot2=\varphi(0)\cdot2=\varphi(\varphi(0))</math>

<math>\varphi(0)\cdot3=\varphi(0)\cdot3=\varphi(\varphi(\varphi(0)))</math>

<math>\varphi(1)=\varphi(1)=\varphi(1,0)=</math> [[FTO|ω]]

<math>\varphi(1)+1=\varphi(1)+1=\varphi(1,1)</math>

<math>\varphi(1)+2=\varphi(1)+2=\varphi(1,2)</math>

<math>\varphi(1)\cdot2=\varphi(1)\cdot2=\varphi(1,\varphi(1,0))</math>

<math>\varphi(1)\cdot2+1=\varphi(1)\cdot2+1=\varphi(1,\varphi(1,1))</math>

<math>\varphi(1)\cdot2+2=\varphi(1)\cdot2+2=\varphi(1,\varphi(1,2))</math>

<math>\varphi(1)\cdot3=\varphi(1)\cdot3=\varphi(1,\varphi(1,\varphi(1,0)))</math>

<math>\varphi(2)=\varphi(2)=\varphi(2,0)</math>

<math>\varphi(2)+1=\varphi(2)+1=\varphi(2,1)</math>

<math>\varphi(2)+2=\varphi(2)+2=\varphi(2,2)</math>

<math>\varphi(2)+\varphi(1)=\varphi(2)+\varphi(1)=\varphi(2,\varphi(1,0))</math>

<math>\varphi(2)\cdot2=\varphi(2)\cdot2=\varphi(2,\varphi(2,0))</math>

<math>\varphi(2)\cdot3=\varphi(2)\cdot3=\varphi(2,\varphi(2,\varphi(2,0)))</math>

<math>\varphi(3)=\varphi(3)=\varphi(3,0)</math>

<math>\varphi(3)+\varphi(2)=\varphi(3)+\varphi(2)=\varphi(3,\varphi(2,0))</math>

<math>\varphi(4)=\varphi(4)=\varphi(4,0)</math>

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

<math>\varphi(\varphi(1))+1=\varphi(\varphi(1))+1=\varphi(\varphi(1,0),1)</math>

<math>\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1,0),\varphi(1,0))</math>

<math>\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1,0),\varphi(2,0))</math>

<math>\varphi(\varphi(1))\cdot2=\varphi(\varphi(1))\cdot2=\varphi(\varphi(1,0),\varphi(\varphi(1,0),0))</math>

<math>\varphi(\varphi(1)+1)=\varphi(\varphi(1)+1)=\varphi(\varphi(1,1),0)</math>

<math>\varphi(\varphi(1)+2)=\varphi(\varphi(1)+2)=\varphi(\varphi(1,2),0)</math>

<math>\varphi(\varphi(1)+3)=\varphi(\varphi(1)+3)=\varphi(\varphi(1,3),0)</math>

<math>\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1,\varphi(1,0)),0)</math>

<math>\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1,\varphi(1,1)),0)</math>

<math>\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1,\varphi(1,\varphi(1,0))),0)</math>

<math>\varphi(\varphi(2))=\varphi(\varphi(2))=\varphi(\varphi(2,0),0)</math>

<math>\varphi(\varphi(2)+1)=\varphi(\varphi(2)+1)=\varphi(\varphi(2,1),0)</math>

<math>\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2,\varphi(1,0)),0)</math>

<math>\varphi(\varphi(3))=\varphi(\varphi(3))=\varphi(\varphi(3,0),0)</math>

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

<math>\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1,0),1),0)</math>

<math>\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1,0),\varphi(1,0)),0)</math>

<math>\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1,0),\varphi(\varphi(1,0),0)),0)</math>

<math>\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1,1),0),0)</math>

<math>\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2,0),0),0)</math>

<math>\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3,0),0),0)</math>

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

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

<math>\varphi(1,0)=\varphi(1,0)=\varphi(1,0,0)=</math> [[SCO|ε0]]

<math>\varphi(1,0)+1=\varphi(1,0)+1=\varphi(1,0,1)</math>

<math>\varphi(1,0)+\varphi(1)=\varphi(1,0)+\varphi(1)=\varphi(1,0,\varphi(1,0))</math>

<math>\varphi(1,0)\cdot2=\varphi(1,0)\cdot2=\varphi(1,0,\varphi(1,0,0))</math>

<math>\varphi(1,0)\cdot3=\varphi(1,0)\cdot3=\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))</math>

<math>\varphi(1,0)\cdot\varphi(1)=\varphi(1,1)=\varphi(1,1,0)</math>

<math>\varphi(1,0)\cdot\varphi(1)\cdot2=\varphi(1,1)\cdot2=\varphi(1,1,\varphi(1,1,0))</math>

<math>\varphi(1,0)\cdot\varphi(2)=\varphi(1,2)=\varphi(1,2,0)</math>

<math>\varphi(1,0)\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1))=\varphi(1,\varphi(1,0),0)</math>

<math>\varphi(1,0)\cdot\varphi(\varphi(1)+1)=\varphi(1,\varphi(1)+1)=\varphi(1,\varphi(1,1),0)</math>

<math>\varphi(1,0)\cdot\varphi(\varphi(2))=\varphi(1,\varphi(2))=\varphi(1,\varphi(2,0),0)</math>

<math>\varphi(1,0)\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1,0),0),0)</math>

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

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

<math>\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)</math>

<math>\varphi(1,0)^{3}=\varphi(1,\varphi(1,0)\cdot2)=\varphi(1,\varphi(1,0,\varphi(1,0,0),0),0)</math>

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

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

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

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

<math>\varphi(1,0)^{(\varphi(1)\cdot2)}=\varphi(1,\varphi(1,1)\cdot2)=\varphi(1,\varphi(1,1,\varphi(1,1,0)),0)</math>

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

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

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

<math>\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)</math>

<math>\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)</math>

<nowiki><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></nowiki>

<nowiki><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></nowiki>

<nowiki><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></nowiki>

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

<math>\varphi(1,1)\cdot\varphi(1,0)=\varphi(2,\varphi(1,0))=\varphi(2,\varphi(1,0,0),0)</math>

<math>\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)</math>

<math>\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)</math>

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

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

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

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

<math>\varphi(1,2)=\varphi(3,0)=\varphi(3,0,0)</math>

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

<math>\varphi(1,\varphi(1))\cdot\varphi(1,0)=\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0),\varphi(1,0,0),0)</math>

<math>\varphi(1,\varphi(1))\cdot\varphi(1,1)=\varphi(\varphi(1),\varphi(1,1))=\varphi(\varphi(1,0),\varphi(1,1,0),0)</math>

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

<math>\varphi(1,\varphi(1)+1)=\varphi(\varphi(1)+1,0)=\varphi(\varphi(1,1),0,0)</math>

<math>\varphi(1,\varphi(1)+2)=\varphi(\varphi(1)+2,0)=\varphi(\varphi(1,2),0,0)</math>

<math>\varphi(1,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0)=\varphi(\varphi(1,\varphi(1,0)),0,0)</math>

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

<math>\varphi(1,\varphi(1)^{3})=\varphi(\varphi(3),0)=\varphi(\varphi(3,0),0,0)</math>

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

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

<math>\varphi(1,\varphi(1,0)+1)=\varphi(\varphi(1,0)+1,0)=\varphi(\varphi(1,0,1),0,0)</math>

<math>\varphi(1,\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,1),0)=\varphi(\varphi(1,1,0),0,0)</math>

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

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

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

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

<math>\varphi(2,0)=\varphi(1,0,0)=\varphi(1,0,0,0)=</math> [[CO|ζ0]]

<math>\varphi(1,\varphi(2,0)+1)=\varphi(\varphi(1,0,0),0)=\varphi(\varphi(1,0,0,0),0,0)</math>

<math>\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)</math>

<math>\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)</math>

<math>\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)</math>

<math>\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)</math>

<math>\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))</math>

<math>\varphi(1,\varphi(2,0)\cdot2)=\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(\varphi(1,0,1,0),0,0)</math>

<math>\varphi(1,\varphi(2,0)\cdot\varphi(1))=\varphi(\varphi(1,1,0),0)=\varphi(\varphi(1,0,\varphi(1,0),0),0,0)</math>

<math>\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)</math>

<math>\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)</math>

<math>\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)</math>

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

<math>\varphi(1,\varphi(2,1)+1)=\varphi(\varphi(2,0,0),0)=\varphi(\varphi(2,0,0,0),0,0)</math>

<math>\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)</math>

<math>\varphi(2,2)=\varphi(3,0,0)=\varphi(3,0,0,0)</math>

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

<math>\varphi(2,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0,0)=\varphi(\varphi(1,\varphi(1,0)),0,0,0)</math>

<math>\varphi(2,\varphi(2))=\varphi(\varphi(2),0,0)=\varphi(\varphi(2,0),0,0,0)</math>

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

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

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

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

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

<math>\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)</math>

<math>\varphi(3,0)=\varphi(1,0,0,0)=\varphi(1,0,0,0,0)=</math> [[LCO|η0]]

<math>\varphi(1,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0)=\varphi(\varphi(1,0,0,0,0),0,0)</math>

<math>\varphi(2,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0)</math>

<math>\varphi(3,1)=\varphi(2,0,0,0)=\varphi(2,0,0,0,0)</math>

<math>\varphi(3,\varphi(1))=\varphi(\varphi(1),0,0,0)=\varphi(\varphi(1,0),0,0,0,0)</math>

<math>\varphi(3,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0,0)</math>

<math>\varphi(3,\varphi(1,0))=\varphi(\varphi(1,0),0,0,0)=\varphi(\varphi(1,0,0),0,0,0,0)</math>

<math>\varphi(3,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0,0)=\varphi(\varphi(1,0,0,0),0,0,0,0)</math>

<math>\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)</math>

<math>\varphi(4,0)=\varphi(1,0,0,0,0)=\varphi(1,0,0,0,0,0)</math>

<math>\varphi(\varphi(1),0)=\varphi(1</math>@<math>\varphi(1))=\varphi(1</math>@<math>\varphi(1,0))=</math> [[HCO]]

<math>\varphi(1,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1</math>@<math>\varphi(1)),0)=\varphi(\varphi(1</math>@<math>\varphi(1,0)),0,0)</math>

<math>\varphi(2,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1</math>@<math>\varphi(1)),0,0)=\varphi(\varphi(1</math>@<math>\varphi(1,0)),0,0,0)</math>

<math>\varphi(3,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1</math>@<math>\varphi(1)),0,0,0)=\varphi(\varphi(1</math>@<math>\varphi(1,0)),0,0,0,0)</math>

<math>\varphi(\varphi(1),1)=\varphi(2</math>@<math>\varphi(1))=\varphi(2</math>@<math>\varphi(1,0))</math>

<math>\varphi(1,\varphi(\varphi(1),1)+1)=\varphi(\varphi(2</math>@<math>\varphi(1)),0)=\varphi(\varphi(2</math>@<math>\varphi(1,0)),0,0)</math>

<math>\varphi(\varphi(1),2)=\varphi(3</math>@<math>\varphi(1))=\varphi(3</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1),\varphi(1))=\varphi(\varphi(1)</math>@<math>\varphi(1))=\varphi(\varphi(1,0)</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1),\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2</math>@<math>\varphi(1))=\varphi(\varphi(1,0)\cdot2</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1),\varphi(2))=\varphi(\varphi(2)</math>@<math>\varphi(1))=\varphi(\varphi(2,0)</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1),\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1))</math>@<math>\varphi(1))=\varphi(\varphi(\varphi(1,0),0)</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0)</math>@<math>\varphi(1))=\varphi(\varphi(1,0,0)</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1),\varphi(2,0))=\varphi(\varphi(1,0,0)</math>@<math>\varphi(1))=\varphi(\varphi(1,0,0,0)</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1),\varphi(3,0))=\varphi(\varphi(1,0,0,0)</math>@<math>\varphi(1))=\varphi(\varphi(1,0,0,0,0)</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1),\varphi(\varphi(1),0))=\varphi(\varphi(1</math>@<math>\varphi(1))</math>@<math>\varphi(1))=\varphi(\varphi(1</math>@<math>\varphi(1,0))</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1)+1,0)=\varphi(1</math>@<math>\varphi(1)+1)=\varphi(1</math>@<math>\varphi(1,1))</math>

<math>\varphi(\varphi(1)+1,\varphi(\varphi(1)+1,0))=\varphi(\varphi(1</math>@<math>\varphi(1)+1)</math>@<math>\varphi(1)+1)=\varphi(\varphi(1</math>@<math>\varphi(1,1))</math>@<math>\varphi(1,1))</math>

<math>\varphi(\varphi(1)+2,0)=\varphi(1</math>@<math>\varphi(1)+2)=\varphi(1</math>@<math>\varphi(1,2))</math>

<math>\varphi(\varphi(1)\cdot2,0)=\varphi(1</math>@<math>\varphi(1)\cdot2)=\varphi(1</math>@<math>\varphi(1,\varphi(1)))</math>

<math>\varphi(\varphi(1)\cdot3,0)=\varphi(1</math>@<math>\varphi(1)\cdot3)=\varphi(1</math>@<math>\varphi(1,\varphi(1,\varphi(1,0))))</math>

<math>\varphi(\varphi(2),0)=\varphi(1</math>@<math>\varphi(2))=\varphi(1</math>@<math>\varphi(2,0))</math>

<math>\varphi(\varphi(\varphi(1)),0)=\varphi(1</math>@<math>\varphi(\varphi(1)))=\varphi(1</math>@<math>\varphi(\varphi(1,0),0))</math>

<math>\varphi(\varphi(1,0),0)=\varphi(1</math>@<math>\varphi(1,0))=\varphi(1</math>@<math>\varphi(1,0,0))</math>

<math>\varphi(\varphi(2,0),0)=\varphi(1</math>@<math>\varphi(1,0,0))=\varphi(1</math>@<math>\varphi(1,0,0,0))</math>

<math>\varphi(\varphi(3,0),0)=\varphi(1</math>@<math>\varphi(1,0,0,0))=\varphi(1</math>@<math>\varphi(1,0,0,0,0))</math>

<math>\varphi(\varphi(\varphi(1),0),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1)))=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0)))</math>

<math>\varphi(\varphi(\varphi(1,0),0),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0)))=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0,0)))</math>

<math>\varphi(\varphi(\varphi(2,0),0),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0,0)))=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0,0,0)))</math>

<math>\varphi(1,0,0)=\varphi(1</math>@<math>(1,0))=\varphi(1</math>@<math>(1,0))=</math> [[FSO]]

<math>\varphi(1,\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0)),0)=\varphi(\varphi(1</math>@<math>(1,0)),0,0)</math>

<math>\varphi(2,\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0)),0,0)=\varphi(\varphi(1</math>@<math>(1,0)),0,0,0)</math>

<math>\varphi(\varphi(1),\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0))</math>

<math>\varphi(\varphi(1,0),\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0,0))</math>

<math>\varphi(\varphi(2,0),\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0,0))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0,0,0))</math>

<math>\varphi(\varphi(1,0,0),1)=\varphi(2</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(2</math>@<math>\varphi(1</math>@<math>(1,0)))</math>

<math>\varphi(\varphi(1,0,0),2)=\varphi(3</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(3</math>@<math>\varphi(1</math>@<math>(1,0)))</math>

<math>\varphi(\varphi(1,0,0),\varphi(1))=\varphi(\varphi(1)</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(\varphi(1,0)</math>@<math>\varphi(1</math>@<math>(1,0)))</math>

<math>\varphi(\varphi(1,0,0),\varphi(1,0))=\varphi(\varphi(1,0)</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(\varphi(1,0,0)</math>@<math>\varphi(1</math>@<math>(1,0)))</math>

<math>\varphi(\varphi(1,0,0),\varphi(1,0,0))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1</math>@<math>(1,0)))</math>

<math>\varphi(\varphi(1,0,0)+1,0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0))+1)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),1</math>@<math>0))</math>

<math>\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0))+\varphi(1))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1,0)</math>@<math>0))</math>

<math>\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0))+\varphi(1,0))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1,0,0)</math>@<math>0))</math>

<math>\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0))\cdot2)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0))</math>@<math>0))</math>

<math>\varphi(\varphi(1,0,0)\cdot\varphi(1),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),1</math>@<math>0))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),1</math>@<math>1))</math>

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

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

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

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

<math>\varphi(\varphi(1,\varphi(1,0,0)+1),0)=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0)),0))=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0)),0,0))</math>

<math>\varphi(\varphi(1,\varphi(1,0,0)+\varphi(1)),0)=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0))+\varphi(1),0))=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0))+\varphi(1,0),0))</math>

<math>\varphi(\varphi(\varphi(1),\varphi(1,0,0)+1),0)=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1)))=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0)))</math>

<math>\varphi(1,0,1)=\varphi(1</math>@<math>(1,1))=\varphi(1</math>@<math>(1,1))</math>

<math>\varphi(1,0,2)=\varphi(2</math>@<math>(1,1))=\varphi(2</math>@<math>(1,1))</math>

<math>\varphi(1,0,\varphi(1))=\varphi(\varphi(1)</math>@<math>(1,1))=\varphi(\varphi(1,0)</math>@<math>(1,1))</math>

<math>\varphi(1,0,\varphi(1,0))=\varphi(\varphi(1,0)</math>@<math>(1,1))=\varphi(\varphi(1,0,0)</math>@<math>(1,1))</math>

<math>\varphi(1,0,\varphi(1,0,0))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>(1,1))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>(1,1))</math>

<math>\varphi(1,1,0)=\varphi(1</math>@<math>(1,1))=\varphi(1</math>@<math>(1,1))</math>

<math>\varphi(1,1,\varphi(1,1,0))=\varphi(\varphi(1</math>@<math>(1,1))</math>@<math>(1,1))=\varphi(\varphi(1</math>@<math>(1,1))</math>@<math>(1,1))</math>

<math>\varphi(1,2,0)=\varphi(1</math>@<math>(1,2))=\varphi(1</math>@<math>(1,2))</math>

<math>\varphi(1,\varphi(1),0)=\varphi(1</math>@<math>(1,\varphi(1)))=\varphi(1</math>@<math>(1,\varphi(1,0)))</math>

<math>\varphi(1,\varphi(1,0),0)=\varphi(1</math>@<math>(1,\varphi(1,0)))=\varphi(1</math>@<math>(1,\varphi(1,0,0)))</math>

<math>\varphi(1,\varphi(1,0,0),0)=\varphi(1</math>@<math>(1,\varphi(1</math>@<math>(1,0))))=\varphi(1</math>@<math>(1,\varphi(1</math>@<math>(1,0))))</math>

<math>\varphi(2,0,0)=\varphi(1</math>@<math>(2,0))=\varphi(1</math>@<math>(2,0))</math>

<math>\varphi(2,0,1)=\varphi(2</math>@<math>(2,0))=\varphi(2</math>@<math>(2,0))</math>

<math>\varphi(2,1,0)=\varphi(1</math>@<math>(2,1))=\varphi(1</math>@<math>(2,1))</math>

<math>\varphi(2,\varphi(2,0,0),0)=\varphi(1</math>@<math>(2,\varphi(1</math>@<math>(2,0))))=\varphi(1</math>@<math>(2,\varphi(1</math>@<math>(2,0))))</math>

<math>\varphi(3,0,0)=\varphi(1</math>@<math>(3,0))=\varphi(1</math>@<math>(3,0))</math>

<math>\varphi(\varphi(1),0,0)=\varphi(1</math>@<math>(\varphi(1),0))=\varphi(1</math>@<math>(\varphi(1,0),0))</math>

<math>\varphi(\varphi(1,0),0,0)=\varphi(1</math>@<math>(\varphi(1,0),0))=\varphi(1</math>@<math>(\varphi(1,0,0),0))</math>

<math>\varphi(\varphi(1,0,0),0,0)=\varphi(1</math>@<math>(\varphi(1</math>@<math>(1,0)),0))=\varphi(1</math>@<math>(\varphi(1</math>@<math>(1,0)),0))</math>

<math>\varphi(1,0,0,0)=\varphi(1</math>@<math>(1,0,0))=\varphi(1</math>@<math>(1,0,0))=</math> [[ACO]]

<math>\varphi(1,0,0,1)=\varphi(2</math>@<math>(1,0,0))=\varphi(2</math>@<math>(1,0,0))</math>

<math>\varphi(1,0,1,0)=\varphi(1</math>@<math>(1,0,1))=\varphi(1</math>@<math>(1,0,1))</math>

<math>\varphi(1,1,0,0)=\varphi(1</math>@<math>(1,1,0))=\varphi(1</math>@<math>(1,1,0))</math>

<math>\varphi(2,0,0,0)=\varphi(1</math>@<math>(2,0,0))=\varphi(1</math>@<math>(2,0,0))</math>

<math>\varphi(\varphi(1),0,0,0)=\varphi(1</math>@<math>(\varphi(1),0,0))=\varphi(1</math>@<math>(\varphi(1,0),0,0))</math>

<math>\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(1</math>@<math>(\varphi(1</math>@<math>(1,0,0)),0,0))=\varphi(1</math>@<math>(\varphi(1</math>@<math>(1,0,0)),0,0))</math>

<math>\varphi(1,0,0,0,0)=\varphi(1</math>@<math>(1,0,0,0))=\varphi(1</math>@<math>(1,0,0,0))</math>

<math>\varphi(1,0,0,0,0,0)=\varphi(1</math>@<math>(1,0,0,0,0))=\varphi(1</math>@<math>(1,0,0,0,0))</math>

<math>\varphi(1,1,4,5,1,4)=\varphi(5</math>@<math>(1,1,4,5,1))=\varphi(5</math>@<math>(1,1,4,5,1))</math>

<math>\varphi(1</math>@<math>\varphi(1))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0)))=</math> [[SVO]]

<math>\varphi(1,\varphi(1</math>@<math>\varphi(1))+1)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1)),1</math>@<math>1)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0)),1</math>@<math>2)</math>

<math>\varphi(1</math>@<math>\varphi(1),1</math>@<math>0)=\varphi(2</math>@<math>(1</math>@<math>\varphi(1)))=\varphi(2</math>@<math>(1</math>@<math>\varphi(1,0)))</math>

<math>\varphi(1</math>@<math>\varphi(1),1</math>@<math>1)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1),1</math>@<math>0))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0),1</math>@<math>0))</math>

<math>\varphi(1</math>@<math>\varphi(1),1</math>@<math>2)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1),1</math>@<math>1))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0),1</math>@<math>1))</math>

<math>\varphi(2</math>@<math>\varphi(1))=\varphi(1</math>@<math>(2</math>@<math>\varphi(1)))=\varphi(1</math>@<math>(2</math>@<math>\varphi(1,0)))</math>

<math>\varphi(\varphi(1)</math>@<math>\varphi(1))=\varphi(1</math>@<math>(\varphi(1)</math>@<math>\varphi(1)))=\varphi(1</math>@<math>(\varphi(1,0)</math>@<math>\varphi(1,0)))</math>

<math>\varphi(1</math>@<math>\varphi(1)+1)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1)+1))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,1)))</math>

<math>\varphi(1</math>@<math>\varphi(1)\cdot2)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1)\cdot2))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,\varphi(1,0))))</math>

<math>\varphi(1</math>@<math>\varphi(2))=\varphi(1</math>@<math>(1</math>@<math>\varphi(2)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(2,0)))</math>

<math>\varphi(1</math>@<math>\varphi(1,0))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0,0)))</math>

<math>\varphi(1</math>@<math>\varphi(2,0))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0,0)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0,0,0)))</math>

<math>\varphi(1</math>@<math>\varphi(1,0,0))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1</math>@<math>(1,0))))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1</math>@<math>(1,0))))</math>

<math>\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1</math>@<math>(1</math>@<math>\varphi(1)))))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0)))))=</math> [[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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[[分类:记号]]
[[分类:分析]]