稳定序数：修订间差异

2025年8月30日 (六) 21:58的最新版本

定义

$L_{α}$ 是 $L_{β}$ 的 $Σ_{n}$ 初等子结构，如果任取 $Σ_{n}$ 公式 $φ$ 均有单射 $j$ 满足 $L_{α} ⊨ φ (x_{1}, x_{2}, \dots)$ 等价于 $L_{β} ⊨ φ (j (x_{1}), j (x_{2}), \dots)$ ，也称其为 $L_{α} Σ_{n} 稳定到 L_{β}$ 。

除此外，我们还有 $L_{α}$ 是 $L_{β} - Π_{n} -反射$ 用于表达一些精细的层级，其中 $L_{α} Σ_{1} 稳定到 L_{β}$
（如未特别说明，下文的稳定到均为 $Σ_{1}$ 稳定到）

函数式定义：

$L_{α}$ 是 $L_{f (α)} - Π_{n}$ 反射 onto $X$ ，当且仅当对于任意 $Π_{n}$ 公式 $φ$ 及参数 $γ \in L_{α}$ 、 $γ^{'} \in L_{α^{'}}$ （其中 $α^{'} \in α \cap X$ ），有 $L_{f (α)} ⊨ φ (α, γ) \to L_{f (α^{'})} ⊨ φ (α^{'}, γ^{'})$ 。

序数式定义：

$L_{α}$ 是 $L_{β} - Π_{n}$ 反射 onto $X$ ，当且仅当对于任意 $Π_{n}$ 公式 $φ$ 及参数 $γ \in α$ 、 $γ^{'} \in α^{'}$ （其中 $β^{'} \in α$ 且 $α^{'} \in α \cap X$ ），有 $L_{β} ⊨ φ (α, γ) \to L_{β^{'}} ⊨ φ (α^{'}, γ^{'})$ 。

关于函数式定义，由于 $ω$ -ply 的顶点下成员均为 $ω$ -ply，这会触发 $f$ 与 $α$ 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、方括号稳定。

枚举

稳定序数有如下路径：

$β = \min α : L_{α} ≺_{Σ_{1}} L_{α + 1} = psd. Π_{ω}$

$β = \sup_{n \in ω} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} = \min Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in ω^{2}} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} = \min Π_{1} onto Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} = \min Π_{1} {onto}^{(1, 0)} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} (β \in Π_{2}) = \min Π_{2} \cap Π_{1} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in ω} {α : Π_{2} \cap Π_{1} onto {L_{α} ≺_{Σ_{1}} L_{α + 1}}} = \min Π_{1} onto Π_{2} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : Π_{2} \cap Π_{1} onto {L_{α} ≺_{Σ_{1}} L_{α + 1}}} (β \in Π_{2}) = \min Π_{2} \cap Π_{1} onto Π_{2} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : {n : Π_{2} \cap Π_{1} {onto}^{n} {L_{α} ≺_{Σ_{1}} L_{α + 1}}} = \min Π_{2} \cap Π_{1} {onto}^{(1, 0)} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} (β \in Π_{2} onto Π_{2}) = \min (Π_{2} onto Π_{2}) \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} (β \in Π_{3}) = \min Π_{3} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} (L_{β} ≺_{Σ_{1}} L_{β + 1}) = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} (β \in Π_{n})$

$β = \sup_{n \in ω} {α : {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}} = \min Π_{1} onto {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}} (β \in Π_{2}) = \min Π_{2} \cap Π_{1} onto {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}} (L_{β} ≺_{Σ_{1}} L_{β + 1}) = \min {γ : L_{γ} ≺_{Σ_{1}} L_{γ + 1}} \cap Π_{1} onto {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : {n : ({β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto)^{n}}} = \min ({β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto)^{(1, 0)}$

$β = \min Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}})$

$β = \sup_{n \in β} {α : Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}} = \min Π_{1} {onto}^{(1, 0)} Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}} (β \in Π_{2}) = \min Π_{2} \cap Π_{1} onto Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}} (L_{β} ≺_{Σ_{1}} L_{β + 1}) = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}} (β \in Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}) = \min Π_{2} onto {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \sup_{n \in β} {α : {x : (Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}) \cap Π_{1} onto)^{x}}} = \min (Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} \cap Π_{1} onto)^{(1, 0)}$

$β = \min Π_{2} onto Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} (β \in Π_{3}) = \min Π_{3} \cap Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}} (L_{β} ≺_{Σ_{1}} L_{β + 1}) = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \min Π_{3} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1} \land L_{α} = \min L_{α + 1} - Π_{2}})$

$β = \min ({β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto)^{(1, 0)} {α : L_{α} ≺_{Σ_{1}} L_{α + 1} \land L_{α} = \min L_{α + 1} - Π_{2}})$

$β = {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1} \land L_{α} = L_{α + 1} - Π_{2}}) (β \in {α : L_{α} ≺_{Σ_{1}} L_{α + 1} \land L_{α} = L_{α + 1} - Π_{2}}) = \min {α : L_{α} ≺_{Σ_{1}} L_{α + 1} \land L_{α} = L_{α + 1} - Π_{2}} \cap {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto {γ : L_{γ} ≺_{Σ_{1}} L_{γ + 1} \land L_{γ} = L_{γ + 1} - Π_{2}}$

$β = \min {α : L_{α} ≺_{Σ_{1}} L_{α + 1} \land L_{α} = L_{α + 1} - Π_{2}} onto {β : L_{β} ≺_{Σ_{1}} L_{β + 1} \land L_{β} = L_{β + 1} - Π_{2}}$

$β = \min ({α : L_{α} ≺_{Σ_{1}} L_{α + 1} \land L_{α} = L_{α + 1} - Π_{2}} onto)^{(1, 0)}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} (L_{β} = L_{β + 1} - Π_{3})$

$β = \min {β : L_{β} ≺_{Σ_{2}} L_{β + 1}} (L_{β} = L_{β + 1} - psd. Π_{ω})$ $β = \min Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 2}} \cap Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 2}}$

$β = \min Π_{3} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 2}}$

$β = \min {γ : L_{γ} ≺_{Σ_{1}} L_{γ + 2}} \cap {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1} \land L_{β} = L_{β + 1} - Π_{2}} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 2}} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 2} \land L_{β} = L_{β + 2} - Π_{2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 3} \land L_{β} = L_{β + 2} - psd. Π_{ω}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + ω}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β \times 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + α + 1}} (α = \min {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}})$

$β = 2nd {β : L_{β} ≺_{Σ_{1}} L_{β \times 2}}$

$β = \min Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}}$

$β = \min Π_{2} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + γ}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}} (γ = \min {γ : L_{γ} ≺_{Σ_{1}} L_{γ \times 2}})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + γ}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}} (γ 是上一条中的 β)$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β \times 2}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}}$

$β = \min Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β + γ}} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}} (γ = \min {γ : L_{γ} ≺_{Σ_{1}} L_{γ \times 2}})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β \times 2}} onto {α : L_{α} ≺_{Σ_{1}} L_{α \times 2}}$

$β = \min {β : L_{β} = L_{β \times 2} - Π_{2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β \times 2 + 1}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{β^{2}}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{Ω_{β + 1}}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} \land L_{γ} ≺_{Σ_{1}} L_{γ + 1}} (L_{γ} ≺_{Σ_{1}} L_{γ + 1})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ + 1} \land L_{β} = L_{γ + 1} - Π_{2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ + 1} \land L_{β} ≺_{Σ_{1}} L_{γ + 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ + 1} \land L_{β} ≺_{Σ_{1}} L_{γ + ω}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ + 1} \land L_{β} ≺_{Σ_{1}} L_{γ \times 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ + 1} \land L_{β} ≺_{Σ_{1}} L_{Ω_{γ + 1}}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ + 1} \land L_{β} ≺_{Σ_{1}} L_{α} ≺_{Σ_{1}} L_{α + 1}} (γ < α)$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ_{n}} ≺_{Σ_{1}} L_{γ_{n} + 1}} (γ = \min Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ}} (γ = \min Π_{1} onto Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ}} (γ = \min Π_{2} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ}} (γ = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} \cap Π_{1} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ}} (γ = \min Π_{2} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ}} (γ = \min {β : L_{β} ≺_{Σ_{1}} L_{β + 1}} onto {α : L_{α} ≺_{Σ_{1}} L_{α + 1}})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ}} (L_{γ} = L_{γ + 1} - Π_{2})$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ + 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ + β}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{γ \times 2}}$

$β = \min {β : L_{β} ≺_{Σ_{1}} L_{γ} ≺_{Σ_{1}} L_{ζ} ≺_{Σ_{1}} L_{ζ + 1}}$

$β = \min {β : L_{β_{x}} ≺_{Σ_{1}} L_{β_{x + 1}} (\forall n \in ω (β_{n} = ω -ply))$

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 $Σ_{n}$ -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 $Σ_{n}$ 稳定的表达式。

如 $(0, 0) (1, 1)$ ，(0,0) 记作 α，(1,1) 记作 β，注意到第一行上 $α < 1 β$ ，第二行上 $α < 2 β$ ，翻译过来可只写 $α < 2 β$ 。

又如 $(0, 0) (1, 1) (2, 1)$ 翻译成 $a < 2 (b, c), b < 1 c$ 。

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

又如 $a < 2 (b, d), b < 2 (c, e), c < 1 e, (b, c, e) \in d$ ，翻译为 $(0, 0) (1, 1) (2, 2) (3, 2) (1, 1)$ 。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 $a < 1 b < 2 c$ 在稳定中标准而在 BMS 中是 $(0, 0) (1, 0) (2, 1)$ 不标准。

@@ 第1行： / 第1行： @@
-L_{\alpha}是L<sub>β</sub>的∑<sub>n</sub>初等子结构，如果任取∑<sub>n</sub>公式φ均有单射j满足L<sub>α</sub>|=φ(x1,x2,…)等价于L<sub>β</sub>|=φ(j(x1),j(x2),…)，也称其为L<sub>α</sub>稳定到L<sub>β</sub>
+=== 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]]