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weak Veblen 函数

2025-08-26T10:30:37Z

U3Vzc3liYWth：撤销U3Vzc3liYWth（讨论）的修订版本2511

'''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。

{{默认排序:序数记号}}
[[分类:记号]]
[[分类:分析]]

weak Veblen 函数

2025-08-26T09:21:16Z

U3Vzc3liYWth：修复公式错误

'''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。

{{默认排序:序数记号}}
[[分类:记号]]
[[分类:分析]]

Rayo函数

2025-08-25T16:10:26Z

U3Vzc3liYWth：修复公式错误和引用错误

Rayo函数，是目前被命名的增长速度最快速的函数之一。

== 定义 ==

=== 通俗定义 ===
在集合论中我们知道，每个自然数 n 都可以用一个集合来表示，而一个一阶逻辑语句的自由变元所能够取为的值同样能够构成一个集合。因此，如果这个一阶逻辑语句所描述的集合恰好对应于一个自然数，那么我们就用这个一阶逻辑语句成功地描述了一个自然数。

通俗地说，Rayo 函数 Rayo(n) 的定义为：不能够用包含字符数为 n 的[[一阶逻辑]]语句定义的最小的自然数。

Rayo数定义为<math>\text{Rayo}(10^{100})</math>

关于 Rayo 函数，有几点值得说明一下。首先，它不是用自然语言表述出的“不能用 n个字确定的最小自然数”。例如，我们可以考虑“不能用二十个字确定的最小自然数”，这句话确定了一个自然数。但是这句话本身又少于 20 个字，因此这个数也能够用小于 20 个字来确定。因此这句话产生了悖论，我们称之为 Perry 悖论。究其原因，这种悖论是由于自然语言的模糊性导致的。相比之下，Rayo 数的规则是明确的，因此不会产生这一问题。

另外，它也与“用 n 个字符所不能得到的最小自然数”不同。因为计算机程序总是递归的，而一阶逻辑不是递归的，它的描述能力要比计算机语言强得多。尽管初看起来，一阶逻辑的语言十分低效，但是一旦字符数量增加，我们就可以利用它定义一些非常高效的函数，它们的表示能力将大得惊人。

=== 正式定义 ===
令<math>[\phi]</math>和<math>[\psi]</math>为 Gödel 编码公式，s 和 t 为变量。定义<math>\text{Sat}([\phi], s)</math>如下:

<math>\forall R \{</math>
{
<math>\forall[\psi],t:R([\psi],t)\leftrightarrow([\psi]=''x_i\in x_j''\land t(x_i)\in t(x_j))</math>
<math>\lor([\psi]=''x_i=x_j''\land t(x_i)=t(x_j))</math>
<math>\lor([\psi]=''(\neg\theta)''\land\neg R([\theta],t))</math>
<math>\lor([\psi]=''(\theta\land\xi)''\land R([\theta],t)\land R([\xi],t))</math>
<math>\lor([\psi]='\exists x_i(\theta)''\land\exists t':R([\theta],t'))</math>
(where t′ is a copy of t with <math>x_i</math> changed)
<math>\} \Rightarrow R([\phi],s)</math>
}

我们称自然数 m 为 Rayo 可命名的，如果存在一个公式 <math>\phi(x_1)</math> 小于 n 个符号且<math>x_1</math>作为其唯一的自由变量满足以下性质：

# 存在一个变量赋值 s，赋值 x1 := m，使得<math>\text{Sat} ([\phi (x_1)] , s)</math>。
# 对于任何变量赋值 t，如果 <math>\text{Sat} ([\phi (x_1)] , t)</math>，则 t 必须有 <math>x_1 = m</math>。

Rayo(n) 是大于使用 n 个符号的 Rayo 可命名的所有数的最小数。

对 Rayo 函数定义的阐述如下：

# 原子公式“<math>x_a \in x_b</math>”表示第 a 变量是第 b 变量的元素。
# 原子公式<math>x_a=x_b</math>”表示第 a 变量等于第 b 变量。
# 公式 e 的公式“(<math>\neg e</math>) ”表示 e 的否定。
# 公式 e 和 f 的公式“(<math>e \land f</math>) ”表示 e 和 f 的合取（逻辑与）。
# 公式 e 的公式“<math>\exist x_a(e)</math>”意味着我们可以修改 a 变量的自由出现，即用 e 中所有集合的 V 类的另一个成员替换<math>x_a</math>使得公式 e 为真。

以上五条定义只是更加精确地给出了 ∈, =, ¬, ∧, ∃ 这五个逻辑的含义。这样我们就给出了用公式定义自然数的方法。若一个自然数是不能够以上述方式从长度为 n 的语句中定义的最小数，那么它就是 Rayo(n)。

== 取值 ==
我们有：

<math>\begin{align}&0 = &\{\} = (\neg\exists x_2(x_2 \in x_1)) \ \textrm{(10 symbols)} \\&1 = &\{\{\}\} = (\exists x_2(x_2 \in x_1) \land (\neg\exists x_3((x_3 \in x_1 \land \exists x_4(x_4 \in x_3))))) \ \textrm{(30 symbols)} \\&2 = &\{\{\},\{\{\}\}\} = (\exists x_2(\exists x_3((x_3 \in x_2 \land (x_2 \in x_1 \land x_3 \in x_1)))) \land (\neg \exists x_4(\exists x_2(\exists x_3((x_4 \in x_3 \land (x_3 \in x_2 \land x_2 \in x_1))))))) \ \textrm{(56 symbols)}\end{align}</math>

因此我们有：

<math>\begin{align}\text{Rayo}(0) &=& 0 \\\text{Rayo}(10) &\ge& 1 \\\text{Rayo}(30) &\ge& 2 \\\text{Rayo}(56) &\ge& 3 \\\end{align}</math>

这只是给出了下限，精确值是由Plain'N'Simple、Emk 和 Ytosk给出的<ref>Plain'N'Simple, [https://googology.fandom.com/wiki/User_blog:Plain%27N%27Simple/Proof_that_Rayo(n)_is_0_for_n_less_than_10 Proof that Rayo(n) is 0 for n less than 10], Googology Wiki user blog.</ref><ref>Emk, [https://googology.fandom.com/wiki/User_blog:Emk/Rayo(10)_is_exactly_1_(not_a_lower_bound) Rayo(10) is exactly 1 (not a lower bound)], Googology Wiki user blog.</ref><ref>Plain'N'Simple, [https://googology.fandom.com/wiki/User_blog:Plain%27N%27Simple/Proof_that_Rayo(n)_is_1_for_n_between_10_to_19 Proof that Rayo(n) is 1 for n between 10 and 19], Googology Wiki user blog.</ref><ref name="Yto">Ytosk, [https://googology.fandom.com/wiki/User_blog:Ytosk/Small_improvements_to_bounds_on_values_of_Rayo%27s_function Small improvements to bounds on values of Rayo's function], Googology Wiki user blog.</ref>

<math>\begin{align}\text{Rayo}(0) &=& 0 \\&\vdots& \\\text{Rayo}(9) &=& 0 \\\text{Rayo}(10) &=& 1 \\&\vdots& \\\text{Rayo}(29) &=& 1 \\\text{Rayo}(30) &\ge& 2 \\\end{align}</math>

除此之外，Ytosk还展示了<math>Rayo(88)\geq4</math>以及<math>Rayo(34+20n)>n</math>.<ref name="Yto"/>

从 2020 年 4 月开始，[[googology]]爱好者关于如何使用Rayo字符串表示数字<math>65536=2\uparrow\uparrow 4</math>以及类似的，2的指数塔，做了大量的研究。Plain'N'Simple 是第一个完成这项任务的人，表明<math>\text{Rayo}(835+96n) > 2 \uparrow\uparrow n</math>.当然，这是早期成果，还有很多改进要做。最后，通过 Plain'N'Simple、P进大好きbot 和 Ytosk 的各种努力，表明<math>\text{Rayo}(260+20n) > 2\uparrow\uparrow n</math><ref name="Yto"/>，因此我们有：

<math>\begin{align}\text{Rayo}(320) &>& 16 \\\text{Rayo}(340) &>& 65536 \\\text{Rayo}(360) &>& 2^{65536} \\\text{Rayo}(380) &>& 2^{2^{65536}} \\ \end{align}</math>

Emk 已经证明<math>\text{Rayo}(7901)>\text{S}(2^{65536}-1)</math>,其中S(n)是[[忙碌海狸函数#疯狂青蛙函数 S(n)|疯狂青蛙函数]]<ref>Emk, [https://googology.fandom.com/wiki/User_blog:Emk/A_Rayo_name_larger_than_BB(10%5E100) A Rayo name larger than BB(10^100)], Googology Wiki user blog.</ref>。然而，由于他的博客文章使用过时的 Rayo 字符串，这个下限没有应有的那么强。目前我们有<math>\text{Rayo}(7339)>\text{S}(2^{65536}-1)</math>。

== 历史 ==
2007 年，Elga 与 Rayo 在麻省理工大学进行了一场别开生面的比赛[[Big Number Duel]]，其内容是尽可能地写出一个有限的自然数。比赛的规则是两人轮流写出一个更大的数，要求不能够使用原始语义词汇，例如不能说“比迄今为止人类命名的任何数字都大的最小数字”等；且后一个被提出的数字不能够容易地由前面的数字的构造方法容易地抵达，例如不能够在前一个数字上简单地 +1 来得到更大的数字。这场比赛的获胜者是 Rayo，他在这场比赛中提出的最大数字就是著名的 Rayo 数

据信Rayo和Elga的大数对决的灵感来自斯科特·亚伦森（Scott Aaronson）的文章[https://www.scottaaronson.com/writings/bignumbers.html 谁能说出更大的数字？]

== 讨论 ==
为了在集合论中定义一个自然数，我们需要明确基于哪些公理进行定义。Rayo数定义中的一个问题在于，Rayo并未阐明所使用的公理。在数学中，只要我们在ZFC集合论的框架下工作，通常会省略对公理的声明。按照这一传统，许多大数研究者认为Rayo数是在ZFC集合论中定义的，或者与公理选择无关，但这种观点是错误的。

至少，由于ZFC集合论无法在[[冯诺依曼宇宙|冯·诺伊曼宇宙]]中形式化真值谓词，除非我们将Rayo数的定义解释为可证性，否则它在ZFC集合论中是不良定义的。即使我们以这种方式解释其定义，所得到的大数也不会显著大于例如<math>\Sigma(10^{100})</math>（其中<math>\Sigma</math> 为[[忙碌海狸函数]]），因为递归可枚举理论中的可证性可以通过图灵机的停机信息判定。要显著超越忙碌海狸函数，我们必须放弃可证性，转而讨论特定模型中的真值——而该模型的存在性在ZFC集合论的一致性前提下是无法被证明的。

另一方面，FOST（一阶集合论语言）本质上只是一种形式语言，根据其定义与公理系统无关——但这并不意味着Rayo数的定义也与公理无关。FOST与公理的无关联性，或者忙碌海狸函数与Rayo数基于可证性解释之间的关系，可能是导致人们误认为"Rayo数与公理无关"的主要原因。

正如Rayo在其原始描述中所述，他使用二阶集合论来形式化原始语义词汇。因此，Rayo数实际上是基于某种未明确说明的二阶集合论公理系统定义的。在不可计算的大数研究中，明确公理系统至关重要，因为只有当两个不可计算大数采用相同的定义公理时，它们之间才能进行有意义的比较。幸运的是，存在多种二阶集合论公理系统可供选择来定义Rayo数。由此可以得出结论：'''对于那些关注公理明确性的研究者来说，Rayo函数是不良定义的'''。

2020年，Rayo新增了以下关于如何处理Rayo数的说明<ref>[https://web.mit.edu/arayo/www/bignums.html A. Rayo, "Big Number Duel", 2007]</ref><blockquote>"注：哲学家有时会采用集合论的实在论解释。在这种解释下，集合论表达式具有'标准'含义，能为语言中的每个句子确定明确的真值，而无需考虑这些真值是否可能被认知（例如参见Vann McGee的这篇文章）。在竞赛期间，Adam和我默认（二阶）集合论语言采用标准解释，这保证了最终条目对应确定的数。如果该语言基于某个公理系统进行解释，最终条目就会失效。这是因为每个（一致的）公理化系统都存在非同构模型，无法保证最终条目在不同模型中对应相同的数。"</blockquote>这意味着Rayo采用了一种哲学上的"解释"方式，将集合论公式与现实中"真值"相对应——这种做法在数学上是无法形式化的，且并未涉及具体公理系统的选择。这是大数研究在数学框架之外的一种合理探索方向。然而，最后一句中的辩解（关于为何要接受这种无法形式化的"真值"）更像是一种托词：虽然数值对模型的依赖性确实存在，但这与"无效性"并无逻辑关联。

在数学中，存在许多良定义但非绝对的概念（即依赖于模型），例如满足条件<math>(CH\rightarrow n=0)\land(\neg CH\rightarrow n=1)</math>的唯一自然数n。同样在大数领域，许多数值本身就具有模型依赖性，比如忙碌海狸函数的值。值得注意的是，在"大数对决"竞赛中，从未禁止过依赖模型的数值定义，甚至允许参与者不指定公理系统。

== 引证 ==
{{默认排序:相关问题}}
[[分类:记号]]

用户讨论:Tabelog

2025-08-25T15:37:49Z

U3Vzc3liYWth：创建页面，内容为“为什么以“意义不明”为由删除了一个用户页😰 ~~~~”

为什么以“意义不明”为由删除了一个用户页😰 [[用户:U3Vzc3liYWth|U3Vzc3liYWth]]（[[用户讨论:U3Vzc3liYWth|留言]]） 2025年8月25日 (一) 23:37 (CST)

用户:U3Vzc3liYWth

2025-08-25T15:16:48Z

U3Vzc3liYWth：清空全部内容

用户:U3Vzc3liYWth

2025-08-25T15:12:01Z

U3Vzc3liYWth：

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a</math> 是 <math>\Pi^1_0</math>-反射。

确实很果糕。

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用户:U3Vzc3liYWth

2025-08-25T15:09:13Z

U3Vzc3liYWth：页面内容被替换为“果糕的页面。 <math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a</math> 是 <math>\Pi^1_0</math>-反射。确实很过高”

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a</math> 是 <math>\Pi^1_0</math>-反射。

确实很过高

用户:U3Vzc3liYWth

2025-08-25T15:08:37Z

U3Vzc3liYWth：

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a</math> 是 <math>\Pi^1_0</math>-反射。

大数花园数（英语：The Large Number Garden Number，缩写：LNGN，日语：巨大数庭園数）是 <math>f^{10}(10 \uparrow^{10} 10)</math> 大数的缩写名称。这里 <math>f(\cdot)</math> 是超越高阶集合论的一阶理论中定义的函数。

'''理论'''

首先，通过在具有可数多个变量项符号和集合隶属关系符号 <math>\in</math> 的一阶集合论语言中加入一个一元函数符号 <math>U</math> 来定义语言 L。将 ZF<sub>L</sub> 定义为属于 ZF 集合论公理的 L 公式集合。在这里，ZF<sub>L</sub> 中理解和替换的公理模式由所有 L-公式（即可以包含 <math>U</math> 的公式）参数化。通过使用显式 Gödel 对应关系将可数多个常数项符号、可数多个函数符号、可数多个关系符号和一个新的一元函数符号 \Theta 添加到 L 的显式形式化中，定义一阶逻辑的形式语言 L。然后，我们用 ZFC<sub>L</sub> 表示属于 ZFC 集合论公理的 L-公式集合，在这里，ZFC<sub>L</sub> 中理解和替换的公理模式由所有 L-公式参数化，即公式可以包括 <math>U</math> 的形式化，附加常数项符号，附加函数符号，附加关系符号和 <math>\Theta</math>。我们将 <math>\varepsilon_0</math> 和 L-公式下面的序数显式编码为 ZF<sub>L</sub> 中的自然数，并将 Henkin 公理“如果存在满足 <math>P</math> 的 x，则 <math>\Theta(n)</math> 满足 <math>P</math>”的形式化，对于每个变量项符号 x，每个带有代码 n 的 L-公式 <math>P</math> 通过重复后继运算形式化为 ZFC<sub>L</sub>，

用 ZFCH<sub>L</sub> 表示由 Henkin 公理模式增强的理论 ZFC<sub>L</sub>。新的函数符号 <math>\Theta</math> 起着“Henkin 常数族”的作用。请不要混淆基本理论 ZF<sub>L</sub> 和形式化理论 ZFCH<sub>L</sub>。用 <math>U1</math> 表示 L-公式“对于任何序数 <math>\alpha</math>，<math>U(\alpha) \vDash \text{ZFCH}_{\text{L}}</math>”。在 <math>U1</math> 增强的 ZFC<sub>L</sub> 下，<math>U(\alpha)</math> 形成了 ZFC<sub>L</sub> 的模型，从而形成了任何序数 <math>\alpha</math> 的 L-结构。我们用 <math>U^{U(\alpha)}</math> 表示 <math>U(\alpha)</math> 中 <math>U</math> 的解释。我们用 <math>U2</math> 表示 L-公式“对于任何序数 <math>\alpha</math> 和任何 <math>\beta \in \alpha</math>，<math>U^{U(\alpha)} = U(\beta)</math>”，用 <math>U3</math> 表示 L-公式“对于任何序数 <math>\alpha</math>，存在一个序数 <math>\beta</math> 使得 <math>\vert U(\alpha)\vert = \text{V}_\beta</math>，对于任何 <math>x \in \text{V}_\beta</math> 和任何 <math>y \in \text{V}_\beta</math>，<math>x \in^{U(\alpha)} y</math> 等价于 <math>x \in y</math>”，其中 <math>\text{V}_\beta</math> 表示 von Neumann 层次。将 T 定义为 L-公式的集合 <math>\text{ZF}_{\text{L}} \cup \{U1, U2, U3\}</math>。

'''嵌入'''

通过给 ZFC 集合论中每个原子式 <math>x_i \in x_j</math> 赋值 L-公式 <math>\{x_i \in x_j\} \wedge \{x_j \in U(0)\}</math>，理论 T 可以看作是 ZFC 集合论的扩展。特别是，ZFC 集合论中定义的集合 <math>N</math> 在 <math>U(0)</math> 处被解释为 T 的项，这与 ZF<sub>L</sub> 下定义的项 <math>N</math> 一致，因为 <math>U(0)</math> 是 ZFC<sub>L</sub> 的传递模型。因此，在 ZFC 集合论中可定义的大数也可以在 T 中定义，并形成一个大数项。此外，由于 L 承认可数无限多的常数项符号、函数符号和关系符号，因此，即使是在 ZFC 集合论中加入可数多个常数项符号、函数符号和关系符号的理论中的闭式，也可以在 <math>U(0)</math> 处解释为 T 中的闭式。此外，通过给未排序的 MK 集合论中的每个原子公式 <math>x_i \in x_j</math> 赋值 L-公式 <math>\{x_i \in x_j\} \wedge \{x_j \in U(0)\}</math>，理论 T 可以看作是 MK 集合论的扩展。粗略地说，<math>U(0)</math> 在形式上起着一阶集合论宇宙的作用，<math>U(0)</math> 的幂集在形式上起着二阶集合论和一阶类理论的宇宙的作用，它的幂集在形式上起着三阶集合论宇宙的作用。由于它们都包含在 <math>U(1)</math> 中，因此 <math>U</math> 正式扮演着高阶集合论宇宙的严格递增序列的角色。请注意，这种严格递增序列的存在可以在 ZFC 集合论中构造，并由 Grothendieck 宇宙公理增强，这在通常的数学中出现。

'''大数'''

显式定义一个满射映射：<math>\text{CNF}: N \rightarrow \varepsilon_0</math> 使用 Cantor 范式。对于 L-公式 <math>P</math>，用 <math>\text{IsDefinition}(P)</math> 表示 L-公式“存在一个 x，使得 <math>P</math> 并且对于任何 <math>i</math>，<math>(P)[i/x]</math> 意味着 <math>i = x</math>”。用 <math>\text{Definable}(m, i, P)</math> 表示 L-公式“<math>i \in N</math>，<math>P</math> 是 L-公式，<math>U(\text{CNF}(i)) \vDash \text{IsDefinition}(P)</math>，<math>U(\text{CNF}(i)) \vDash (P)[m/x]</math>”，其中 <math>P[/x]</math> 中的 m<math>m</math> 以显式方式被视为参数。对于 <math>n \in N</math>，将 <math>f(n)</math> 定义为满足 <math>i \in N, P \in N</math> 和 <math>\text{Definable}(m, i, P)</math> 的 <math>m \in N</math> 之和，这样，最终就有一个不可计算的大函数 <math>f: N \rightarrow N, n \mapsto N</math>。从这里开始，大数花园数是 <math>f^{10}(10 \uparrow^{10} 10)</math>。

{{默认排序:相关问题}}
[[分类:经典大数]]
[[分类:记号]]

用户:U3Vzc3liYWth

2025-08-25T15:07:49Z

U3Vzc3liYWth：测试

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a</math> 是 <math>\Pi^1_0</math>-反射。

'''理论'''

首先，通过在具有可数多个变量项符号和集合隶属关系符号 <math>\in</math> 的一阶集合论语言中加入一个一元函数符号 <math>U</math> 来定义语言 L。将 ZF<sub>L</sub> 定义为属于 ZF 集合论公理的 L 公式集合。在这里，ZF<sub>L</sub> 中理解和替换的公理模式由所有 L-公式（即可以包含 <math>U</math> 的公式）参数化。通过使用显式 Gödel 对应关系将可数多个常数项符号、可数多个函数符号、可数多个关系符号和一个新的一元函数符号 \Theta 添加到 L 的显式形式化中，定义一阶逻辑的形式语言 L。然后，我们用 ZFC<sub>L</sub> 表示属于 ZFC 集合论公理的 L-公式集合，在这里，ZFC<sub>L</sub> 中理解和替换的公理模式由所有 L-公式参数化，即公式可以包括 <math>U</math> 的形式化，附加常数项符号，附加函数符号，附加关系符号和 <math>\Theta</math>。我们将 <math>\varepsilon_0</math> 和 L-公式下面的序数显式编码为 ZF<sub>L</sub> 中的自然数，并将 Henkin 公理“如果存在满足 <math>P</math> 的 x，则 <math>\Theta(n)</math> 满足 <math>P</math>”的形式化，对于每个变量项符号 x，每个带有代码 n 的 L-公式 <math>P</math> 通过重复后继运算形式化为 ZFC<sub>L</sub>，

用 ZFCH<sub>L</sub> 表示由 Henkin 公理模式增强的理论 ZFC<sub>L</sub>。新的函数符号 <math>\Theta</math> 起着“Henkin 常数族”的作用。请不要混淆基本理论 ZF<sub>L</sub> 和形式化理论 ZFCH<sub>L</sub>。用 <math>U1</math> 表示 L-公式“对于任何序数 <math>\alpha</math>，<math>U(\alpha) \vDash \text{ZFCH}_{\text{L}}</math>”。在 <math>U1</math> 增强的 ZFC<sub>L</sub> 下，<math>U(\alpha)</math> 形成了 ZFC<sub>L</sub> 的模型，从而形成了任何序数 <math>\alpha</math> 的 L-结构。我们用 <math>U^{U(\alpha)}</math> 表示 <math>U(\alpha)</math> 中 <math>U</math> 的解释。我们用 <math>U2</math> 表示 L-公式“对于任何序数 <math>\alpha</math> 和任何 <math>\beta \in \alpha</math>，<math>U^{U(\alpha)} = U(\beta)</math>”，用 <math>U3</math> 表示 L-公式“对于任何序数 <math>\alpha</math>，存在一个序数 <math>\beta</math> 使得 <math>\vert U(\alpha)\vert = \text{V}_\beta</math>，对于任何 <math>x \in \text{V}_\beta</math> 和任何 <math>y \in \text{V}_\beta</math>，<math>x \in^{U(\alpha)} y</math> 等价于 <math>x \in y</math>”，其中 <math>\text{V}_\beta</math> 表示 von Neumann 层次。将 T 定义为 L-公式的集合 <math>\text{ZF}_{\text{L}} \cup \{U1, U2, U3\}</math>。

用户:U3Vzc3liYWth

2025-08-25T15:06:22Z

U3Vzc3liYWth：

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a</math> 是 <math>\Pi^1_0</math>-反射。

用户:U3Vzc3liYWth

2025-08-25T15:04:45Z

U3Vzc3liYWth：

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a\textbf{is}\Pi^1_0\textbf{-Reflecting}</math>

用户:U3Vzc3liYWth

2025-08-25T15:03:47Z

U3Vzc3liYWth：

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a\, \textbf{is}\, \Pi^1_0\textbf{-Reflecting}</math>

用户:U3Vzc3liYWth

2025-08-25T15:02:40Z

U3Vzc3liYWth：

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a\ \textbf{is}\ \Pi^1_0\textbf{-Reflecting}</math>

用户:U3Vzc3liYWth

2025-08-25T15:00:54Z

U3Vzc3liYWth：创建页面，内容为“果糕的页面。 <math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a\textbf{\ is\ }\Pi^1_0\textbf{-Reflecting}</math>”

果糕的页面。

<math>\mathbf{L}_a \prec \mathbf{L}_{a+1} \Longrightarrow a\textbf{\ is\ }\Pi^1_0\textbf{-Reflecting}</math>