查看“︁稳定序数”︁的源代码

=== 定义 ===
<math>L_{\alpha}</math> 是 <math>L_{\beta}</math> 的 <math>\Sigma_{n}</math> 初等子结构，如果任取 <math>\Sigma_{n}</math> 公式 <math>\varphi</math> 均有单射 <math>j</math> 满足<math>L_\alpha\models\varphi(x_1,x_2,\cdots)</math> 等价于 <math>L_\beta\models\varphi(j(x_1),j(x_2),\cdots)</math>，也称其为 <math>L_\alpha\ \Sigma_n\text{稳定到}\ L_\beta</math>。<br>

除此外，我们还有 <math>L_{\alpha}</math> 是 <math>L_\beta-\Pi_n\text{-反射}</math> 用于表达一些精细的层级，其中 <math>L_\alpha\ \Sigma_1\text{稳定到}\ L_\beta</math><br>（如未特别说明，下文的稳定到均为 <math>\Sigma_{1}</math> 稳定到）

函数式定义：   

<math>L_{\alpha}</math> 是 <math>L_{f(\alpha)}\text{-}\Pi_{n}</math> 反射 onto <math>X</math>，当且仅当对于任意 <math>\Pi_{n}</math> 公式 <math>\varphi</math> 及参数 <math>\gamma \in L_{\alpha}</math>、<math>\gamma' \in L_{\alpha'}</math>（其中 <math>\alpha' \in \alpha \cap X</math>），有  
<math>L_{f(\alpha)} \models \varphi(\alpha, \gamma) \rightarrow L_{f(\alpha')} \models \varphi(\alpha', \gamma')</math>。  

序数式定义：    

<math>L_{\alpha}</math> 是 <math>L_{\beta}\text{-}\Pi_{n}</math> 反射 onto <math>X</math>，当且仅当对于任意 <math>\Pi_{n}</math> 公式 <math>\varphi</math> 及参数 <math>\gamma \in \alpha</math>、<math>\gamma' \in \alpha'</math>（其中 <math>\beta' \in \alpha</math> 且 <math>\alpha' \in \alpha \cap X</math>），有  
<math>L_{\beta} \models \varphi(\alpha, \gamma) \rightarrow L_{\beta'} \models \varphi(\alpha', \gamma')</math>。  

关于函数式定义，由于 <math>\omega</math>-ply 的顶点下成员均为 <math>\omega</math>-ply，这会触发 <math>f</math> 与 <math>\alpha</math> 的某种不动点，导致无法继续推进。

=== 结构讲解 ===
参见词条 [[Σ1稳定序数|Σ1 稳定序数]]、[[方括号稳定]]。

=== 枚举 ===
稳定序数有如下路径：

<math>\beta=\min\ \alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}=\text{psd.}\Pi_\omega</math>

<math>\beta=\sup_{n\in\omega}\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}=\min\ \Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\omega^2}\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}=\min\ \Pi_1\ \text{onto}\ \Pi_1\ \text{onto}\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}=\min\ \Pi_1\ \text{onto}^{(1,0)}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\quad(\beta\in\Pi_2)=\min\ \Pi_2\cap\Pi_1\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\omega}\{\alpha:\Pi_2\cap\Pi_1\ \text{onto}\ \{L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}=\min\ \Pi_1\ \text{onto}\ \Pi_2\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\Pi_2\cap\Pi_1\ \text{onto}\ \{L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}\quad(\beta\in\Pi_2)=\min\ \Pi_2\cap\Pi_1\ \text{onto}\ \Pi_2\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\{n:\Pi_2\cap\Pi_1\ \text{onto}^n\ \{L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}=\min\ \Pi_2\cap\Pi_1\ \text{onto}^{(1,0)}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\quad(\beta\in\Pi_2\ \text{onto}\ \Pi_2)=\min\ (\Pi_2\ \text{onto}\ \Pi_2)\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\quad(\beta\in\Pi_3)=\min\ \Pi_3\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\quad(L_\beta\prec_{\Sigma_1}L_{\beta+1})=\min\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\quad(\beta\in\Pi_n)</math>

<math>\beta=\sup_{n\in\omega}\{\alpha:\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}=\min\ \Pi_1\ \text{onto}\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}\quad(\beta\in\Pi_2)=\min\ \Pi_2\cap\Pi_1\ \text{onto}\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}\quad(L_\beta\prec_{\Sigma_1}L_{\beta+1})=\min\ \{\gamma:L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\}\cap\Pi_1\ \text{onto}\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\{n:(\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto})^n\}\}=\min\ (\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto})^{(1,0)}</math>

<math>\beta=\min\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}=\min\ \Pi_1\ \text{onto}^{(1,0)}\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}\quad(\beta\in\Pi_2)=\min\ \Pi_2\cap\Pi_1\ \text{onto}\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}\quad(L_\beta\prec_{\Sigma_1}L_{\beta+1})=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\}\quad(\beta\in\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})=\min\ \Pi_2\ \text{onto}\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\sup_{n\in\beta}\{\alpha:\{x:(\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})\cap\Pi_1\ \text{onto})^x\}\}=\min\ (\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\cap\Pi_1\ \text{onto})^{(1,0)}</math>

<math>\beta=\min\ \Pi_2\ \text{onto}\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\quad(\beta\in\Pi_3)=\min\ \Pi_3\cap\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\quad(L_\beta\prec_{\Sigma_1}L_{\beta+1})=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\min\ \Pi_3\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\land L_\alpha=\min\ L_{\alpha+1}-\Pi_2\})</math>

<math>\beta=\min\ (\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto})^{(1,0)}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\land L_\alpha=\min\ L_{\alpha+1}-\Pi_2\})</math>

<math>\beta=\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\land L_\alpha=L_{\alpha+1}-\Pi_2\})\quad(\beta\in\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\land L_\alpha=L_{\alpha+1}-\Pi_2\})=\min\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\land L_\alpha=L_{\alpha+1}-\Pi_2\}\cap\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto}\ \{\gamma:L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\land L_\gamma=L_{\gamma+1}-\Pi_2\}</math>

<math>\beta=\min\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\land L_\alpha=L_{\alpha+1}-\Pi_2\}\ \text{onto}\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\land L_\beta=L_{\beta+1}-\Pi_2\}</math>

<math>\beta=\min\ (\{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\land L_\alpha=L_{\alpha+1}-\Pi_2\}\ \text{onto})^{(1,0)}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\quad(L_\beta=L_{\beta+1}-\Pi_3)</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_2}L_{\beta+1}\}\quad(L_\beta=L_{\beta+1}-\text{psd.}\Pi_\omega)</math><math>\beta=\min\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+2}\}\cap\Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\}</math>

<math>\beta=\min\ \Pi_3\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\}</math>

<math>\beta=\min\ \{\gamma:L_\gamma\prec_{\Sigma_1}L_{\gamma+2}\}\cap\{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\land L_\beta=L_{\beta+1}-\Pi_2\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+2}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+2}\land L_\beta=L_{\beta+2}-\Pi_2\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+3}\land L_\beta=L_{\beta+2}-\text{psd.}\Pi_\omega\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+\omega}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta\times2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+\alpha+1}\}\quad(\alpha=\min\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\})</math>

<math>\beta=\text{2nd}\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta\times2}\}</math>

<math>\beta=\min\ \Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}</math>

<math>\beta=\min\ \Pi_2\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+\gamma}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}\quad(\gamma=\min\ \{\gamma:L_\gamma\prec_{\Sigma_1}L_{\gamma\times2}\})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+\gamma}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}\quad(\gamma\text{ 是上一条中的 }\beta)</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta\times2}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}</math>

<math>\beta=\min\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+\gamma}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}\quad(\gamma=\min\ \{\gamma:L_\gamma\prec_{\Sigma_1}L_{\gamma\times2}\})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta\times2}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha\times2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta=L_{\beta\times2}-\Pi_2\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta\times2+1}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta^2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\Omega_{\beta+1}}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\land L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\}\quad(L_\gamma\prec_{\Sigma_1}L_{\gamma+1})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\land L_\beta=L_{\gamma+1}-\Pi_2\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\land L_\beta\prec_{\Sigma_1}L_{\gamma+2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\land L_\beta\prec_{\Sigma_1}L_{\gamma+\omega}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\land L_\beta\prec_{\Sigma_1}L_{\gamma\times2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\land L_\beta\prec_{\Sigma_1}L_{\Omega_{\gamma+1}}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+1}\land L_\beta\prec_{\Sigma_1}L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\}\quad(\gamma<\alpha)</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\gamma_n}\prec_{\Sigma_1}L_{\gamma_n+1}\}\quad(\gamma=\min\ \Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\}\quad(\gamma=\min\ \Pi_1\ \text{onto}\ \Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\}\quad(\gamma=\min\ \Pi_2\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\}\quad(\gamma=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\cap\Pi_1\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\}\quad(\gamma=\min\ \Pi_2\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\}\quad(\gamma=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_{\beta+1}\}\ \text{onto}\ \{\alpha:L_\alpha\prec_{\Sigma_1}L_{\alpha+1}\})</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\}\quad(L_\gamma=L_{\gamma+1}-\Pi_2)</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+\beta}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma\times2}\}</math>

<math>\beta=\min\ \{\beta:L_\beta\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_\zeta\prec_{\Sigma_1}L_{\zeta+1}\}</math>

<math>\beta=\min\ \{\beta:L_{\beta_x}\prec_{\Sigma_1}L_{\beta_{x+1}}\quad(\forall n\in\omega(\beta_n=\omega\text{-ply}))</math>

到达常规稳定链的终点，在此后需要涉及更高阶的反射。

=== 与 BMS 的关系 ===
Racheline 证明 BMS 良序的文章中，给出了 [[BMS]] 到 <math>\Sigma_n</math>-稳定的一个单射。

我们把 BMS 中第 n 行的父项关系记作 <n，每个列当成一个单独的序数。如此翻译，就得到了一个 <math>\Sigma_n</math> 稳定的表达式。

如 <math>(0,0)(1,1)</math>，(0,0) 记作 α，(1,1) 记作 β，注意到第一行上 <math>\alpha<1\beta</math>，第二行上 <math>\alpha<2\beta</math>，翻译过来可只写<math>\alpha<2\beta</math>。

又如 <math>(0,0)(1,1)(2,1)</math> 翻译成 <math>a<2(b,c),b<1c</math>。

又如 <math>(0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,0)(4,1)(5,1)(6,1)</math>，翻译成 <math>a<2(b,c,d,e,f),b<1c<1d,b,c,d\in e,e<1f<1g,g<2(j,k,l),j<1k<1l</math>。

又如 <math>a<2(b,d),b<2(c,e),c<1e,(b,c,e)\in d</math>，翻译为 <math>(0,0)(1,1)(2,2)(3,2)(1,1)</math>。其中属于关系对应的是 BMS 对应项的位置，然后 a 稳定到 b 暗含 a 属于 b。

注意并非满射。如 <math>a<1b<2c</math> 在稳定中标准而在 BMS 中是 <math>(0,0)(1,0)(2,1)</math> 不标准。

{{默认排序:非递归记号}}
[[分类:记号]]