5 序数分析

5.1 序数记号与函数

5.1.1 递归序数

\(\text{Veblen}\) 函数：设 \(\alpha, \beta\) 为序数，则定义映射 \(\varphi_\alpha: \mathbf{On} \to \mathbf{On}\) 如下
1. \(\varphi_0(\beta) = \omega^{\beta}\)
2. 对于后继序数 \(\alpha + 1\)，定义 \(\varphi_{\alpha+1}(\beta)\) 为 \(\xi \mapsto \varphi_{\alpha}(\xi)\) 的第 \(\beta\) 个不动点
3. 对于极限序数 \(\alpha\)，定义 \(\varphi_{\alpha}(\beta)\) 为所有序数 \(\gamma < \alpha\) 对应的 \(\text{Veblen}\) 函数 \(\xi \mapsto \varphi_{\gamma}(\xi)\) 的第 \(\beta\) 个公共不动点
所有 \(\text{Veblen}\) 函数的不动点均从 \(0\) 开始计数
1. 扩展 \(\text{Veblen}\) 函数：将 \(\varphi_{\alpha}(\gamma)\) 视作二元函数 \(\varphi(\alpha, \gamma)\)，令 \(z\) 为多于或等于 \(0\) 个 \(0\) 的简记，\(s\) 为多于或等于 \(0\) 个序数变量 \(\alpha_1, \alpha_2, \cdots, \alpha_n \ (\alpha_1 > 0)\) 的简记，则定义
  1. \(\varphi(\gamma)=\omega^\gamma\)
  2. \(\varphi(z,s,\gamma)=\varphi(s,\gamma)\)
  3. 若 \(\alpha_{n+1} > 0\)，则 \(\varphi(s, \alpha_{n+1}, z, \gamma)\) 表示「对所有序数 \(\beta < \alpha_{n+1}\) 对应的函数 \(\xi \mapsto \varphi(s, \beta, \xi, z)\)」的第 \(\gamma\) 个公共不动点
2. 超限 \(\text{Veblen}\) 函数：利用数阵 \(\varphi \begin{pmatrix} \alpha_1 & \alpha_2 & \cdots & \alpha_n \\ \beta_1 & \beta_2 & \cdots & \beta_n \\ \end{pmatrix}\) 表示 \(\text{Veblen}\) 函数，其中 \(\beta_i\) 表示 \(\alpha_i\) 在扩展 \(\text{Veblen}\) 函数的位置
\(\text{Kleene } \mathcal{O}-\)记号：枚举所有部分递归函数 \(\Phi_1, \Phi_2, \cdots\)，设 \(K\) 表示所有记号，\(<_{\mathcal O}\) 表示 \(K\) 的良基严格偏序．定义 \(\mathcal O: K \to \mathbf{On}\)
1. \(0 \in K\) 且 \(\mathcal O(0) = 0\)
2. 若 \(n \in K\) 且 \(\mathcal O(n) = \alpha\)，则 \(n <_{\mathcal O} 2^{n} \in K\) 且 \(\mathcal O(2^{n}) = \alpha + 1\)
3. 若对任意 \(n \in \mathbf N\) 有 \(\Phi_i(n) \in K\) 且 \(\Phi_i(n) <_{\mathcal O} \Phi_i(n + 1)\)，则 \(3 \cdot 5^i \in K, {\displaystyle \mathcal{O}(3 \cdot 5^i) = \sup_{k \in \omega} \mathcal{O}(\Phi_i(k))}\) 且对所有 \(k\) 有 \(\Phi_i(k) <_\mathcal{O} 3 \cdot 5^i\)
易知可由 \(\mathcal{O}\) 表示的序数集合等价于全体递归序数集合 \(\Omega\)

5.1.2 非递归序数

非递归序数：记 \(\{\Omega_{\alpha} \mid \alpha \in \mathbf{On}\}\) 是所有非递归序数的类
1. \(\Omega_0 = 0\)
2. \(\Omega_{\alpha + 1} = {\displaystyle \sup_{X} \{X \mid X = \{x \mid x = \varphi(x)^{L_{\Omega_{\alpha}}}，\varphi \textsf{ 为 } \Sigma_1\textsf{ 公式}\}\}}\)
3. 若 \(\alpha\) 为极限序数，则 \(\omega_{\alpha} = \sup \{\omega_{\beta} \mid \beta < \alpha\}\)
通常将 \(\Omega_1\) 记作 \(\Omega\)
反射序数：设公式层级 \(\Phi \in \{\Pi_n, \Sigma_n\}\)
1. 设 \(L_{\alpha} \in \mathbf{L}\)，若 \(L_{\alpha} \vDash \varphi\) 蕴含 \(\beta \in (X \cap \alpha)\) 使得 \(L_{\beta} \vDash \varphi\)，则称 \(L_{\alpha}\) 在 \(X\) 上反射了公式 \(\varphi\)
  1. 若 \(L_{\alpha}\) 在 \(X\) 上反射了所有的 \(\Phi\) 公式，则称 \(\alpha\) 是 \(X\) 上的 \(\Phi\) 反射序数
  2. 若 \(L_{\alpha}\) 在全体序数上反射了所有的 \(\Phi\) 公式，则称 \(\alpha\) 是 \(\Phi\) 反射序数
2. 反射序数的性质
  1. \(\alpha\) 是 \(X\) 上的 \(\Pi_{n}\) 反射序数等价于 \(\alpha\) 是 \(X\) 上的 \(\Sigma_{n+1}\) 反射序数
  2. \(\alpha\) 是 \(X\) 上的 \(\Pi_{0}\) 反射序数等价于 \(\alpha\) 是 \(X\) 上的 \(\Pi_{1}\) 反射序数，等价于 \(\alpha\) 是 \(X\) 上的上确界
  3. \(\alpha\) 是 \(X\) 上的 \(\Pi_{2}\) 反射序数等价于 \(\alpha\) 是 \(X\) 上的容许序数
3. 通常用反射序数的模式表示满足相应条件的最小反射序数，其集合 \(P\) 递归定义如下
  1. 符号 \(\varnothing \in P\)
  2. 若 \(s, t \in P\)，则 \(s \wedge t, \sigma_1(s), \pi_1(s) \in P\)
  将 \(\sigma_1(\varnothing), \pi_1(\varnothing)\) 简记作 \(\sigma_1\) 与 \(\pi_1\)
  1. 序数 \(\alpha\) 是 \(\sigma_1(\varnothing)\) 反射或 \(\pi_1(\varnothing)\) 反射的当且仅当 \(\alpha\) 反射是 \(\Sigma_1\) 或 \(\Pi_1\) 反射的
  2. 若 \(s \in P\)，则序数 \(\alpha\) 是 \(\sigma(s)\) 反射的当且仅当 \(\alpha\) 在 \(s\) 反射序数上是 \(\Sigma_1\) 反射的
  3. 若 \(s \in P\)，则序数 \(\alpha\) 是 \(\pi(s)\) 反射的当且仅当 \(\alpha\) 在 \(s\) 反射序数上是 \(\Pi_1\) 反射的
  4. 若 \(s, t \in P\)，则序数 \(\alpha\) 是 \(s \wedge t\) 反射的当且仅当 \(\alpha\) 既是 \(s\) 反射的，也是 \(t\) 反射的
\(\text{Gandy}\) 序数
1. 定义 \(\delta_{\alpha} = \sup \{\delta \mid \delta\) 是 \(\alpha\) 子集上 \(\alpha-\)递归良序的序型\(\}\)，即大于 \(\alpha\) 的最小非递归序数
2. 定义 \(\alpha^{+}\) 为大于 \(\alpha\) 的最小容许序数
则必有 \(\delta_{\alpha} \leqslant \alpha^{+}\)
1. 若 \(\delta_{\alpha} = \alpha^{+}\)，则称 \(\alpha\) 是 \(\text{Gandy}\) 的
2. \(\sigma_1\) 是最小的非 \(\text{Gandy}\) 序数
稳定序数：若序数 \(\alpha\) 是关于 \(f(\alpha)\) 是 \(\Sigma_{1}\) 稳定的，则称 \(\alpha\) 是 \(f-\)稳定的．通常用括号函数或 \(\lambda\) 表达式表示 \(f\)
1. 若序数 \(\alpha \leqslant \beta\) 且 \(L_{\alpha} \prec_{\Sigma_{n}} L_{\beta}\)，则称 \(\alpha\) 关于 \(\beta\) 是 \(\Sigma_{n}\) 稳定的
  1. 若 \(L_{\alpha} \prec_{\Sigma_{n}} L_{\beta}\) 且 \(L_{\beta} \prec_{\Sigma_{n}} L_{\gamma}\)，则 \(L_{\alpha} \prec_{\Sigma_{n}} L_{\gamma}\)
  2. 若 \(L_{\alpha} \prec_{\Sigma_{n}} L_{\gamma}\) 且 \(L_{\beta} \prec_{\Sigma_{n}} L_{\gamma}\)，则 \(L_{\alpha} \prec_{\Sigma_{n}} L_{\beta}\)
  3. 若 \(L_{\alpha} \prec_{\Sigma_{1}} L_{\gamma}\) 且 \(\beta<\gamma\)，则 \(L_{\alpha} \prec_{\Sigma_{1}} L_{\beta}\)
  4. 若对任意 \(\alpha \in A\) 都有 \(L_{\alpha} \prec_{\Sigma_{1}} L_{\beta}\)，则 \(L_{\sup A} \prec_{\Sigma_{1}} L_{\beta}\)
  5. 若对任意 \(\beta \in B\) 都有 \(L_{\alpha} \prec_{\Sigma_{1}} L_{\beta}\)，则 \(L_{\alpha} \prec_{\Sigma_{1}} L_{\mathrm{sup} B}\)
2. 若 \(\alpha\) 是 \((\cdot + 1)-\)稳定的，则 \(\alpha\) 是 \(\Pi_{n}\) 反射序数对任意 \(n \in \mathbf N\) 都成立

5.1.3 序数折叠函数

\(\psi\) 型定义
1. \(\text{Bachmann} \ \psi-\)函数：令 \(\Omega\) 为不可数序数（例如 \(\omega_1\)），定义 \(C^{\Omega}(\alpha, \beta)\) 为 \(\beta \cup\{0, \Omega\}\) 在 \(+,\left(\xi \mapsto \omega^{\xi}\right),\left(\xi \mapsto \psi_{\Omega}(\xi)\right)_{\xi<\alpha}\) 下的闭包，则有 \(\psi_{\Omega}(\alpha)=\min \left\{\rho<\Omega: C^{\Omega}(\alpha, \rho) \cap \Omega=\rho\right\}\)
2. \(\text{Buchholz} \ \psi-\)函数
  1. \(C_{\nu}^{0}(\alpha)=\left\{\beta \mid \beta<\Omega_{\nu}\right\}\)
  2. \(C_{\nu}^{n+1}(\alpha)=\left\{\beta+\gamma, \psi_{\mu}(\eta) \mid \mu, \beta, \gamma, \eta \in C_{\nu}^{n}(\alpha) \wedge \eta<\alpha\right\}\)
  3. \(C_{\nu}(\alpha)={\displaystyle \bigcup_{n<\omega} C_{\nu}^{n}(\alpha)}\)
  4. \(\psi_{\nu}(\alpha)=\min \left\{\gamma \mid \gamma \notin C_{\nu}(\alpha)\right\}\)
  其中 \(\Omega_{\nu} = \left\{\begin{aligned} & 1, & \nu=0 \\ & \aleph_{\nu}, & \nu>0 \end{aligned}\right.\)
3. \(\text{Madore} \ \psi-\)函数
  1. \(C_{\nu}^{0}(\alpha)=\left\{\xi \mid \xi<\Omega_{\nu}\right\} \cup\left\{\Omega_{\mu} \mid \mu \in \mathbf{Ord}\right\}\)
  2. \(C_{\nu}^{n+1}(\alpha)=\left\{\gamma+\delta, \gamma \cdot \delta, \gamma^{\delta}, \psi_{\mu}(\eta) \mid \gamma, \delta, \mu, \eta \in C_{\nu}^{n}(\alpha), \mu<\nu, \eta<\alpha\right\}\)
  3. \(C_{\nu}(\alpha)={\displaystyle \bigcup_{n<\omega} C_{\nu}^{n}(\alpha)}\)
  4. \(\psi_{\nu}(\alpha)=\min \left\{\beta<\Omega_{\nu+1} \mid \beta \notin C_{\nu}(\alpha)\right\}\)
  通常记 \(\psi_0\) 为 \(\psi\)
4. \(\text{J}\ddot{\mathrm{a}}\text{ger} \ \psi-\)函数：设 \(M\) 是最小的 \(\text{Mahlo}\) 基数，\(R\) 为全体小于 \(M\) 的不可数正则序数集合，定义
  
  \[ I_{\alpha} = \operatorname{enum}\left(\operatorname{cl} \left(\left\{\beta \in R \mid \forall \gamma<\alpha\left(I_{\gamma}(\beta)=\beta\right)\right\}\right)\right) \]
  
  其中枚举函数 \(\operatorname{enum}(X): \operatorname{type}(X) \to X\) 是使得 \(X = \operatorname{ran}(\operatorname{enum}(X))\) 的唯一严格递增函数
  1. \(\left\{\kappa^{-}\right\} \cup \kappa^{-} \subseteq C_{\kappa}^{n}(\alpha)\)
  2. \(S(\gamma) \subseteq C_{\kappa}^{n}(\alpha) \to \gamma \in C_{\kappa}^{n+1}(\alpha)\)
  3. \(\beta, \gamma \in C_{\kappa}^{n}(\alpha) \to I_{\beta}(\gamma) \in C_{\kappa}^{n+1}(\alpha)\)
  4. \(\gamma<\pi<\kappa \wedge \pi \in C_{\kappa}^{n}(\alpha) \to \gamma \in C_{\kappa}^{n+1}(\alpha)\)
  5. \(\gamma<\alpha \wedge \gamma, \pi \in C_{\kappa}^{n}(\alpha) \wedge \gamma \in C_{\pi}(\gamma) \to \psi_{\pi}(\gamma) \in C_{\kappa}^{n+1}(\alpha)\)
  6. \(C_{\kappa}(\alpha)=\bigcup\left\{C_{\kappa}^{n}(\alpha) \mid n<\omega\right\}\)
  7. \(\psi_{\kappa}(\alpha)=\min \left\{\xi \mid \xi \notin C_{\kappa}(\alpha)\right\}\)
5. \(\text{J}\ddot{\mathrm{a}}\text{ger}-\text{Buchholz} \ \psi-\)函数：假设存在弱不可达基数，设 \(I\) 为最小的弱不可达基数，\(\mathrm{Reg}_{I}\) 表示不大于 \(I\) 的全体正则基数集合
  1. \(\mathrm{Cl}^{0}(\alpha, \beta)=\beta \cup\{0, I\}\)
  2. \(\mathrm{Cl}^{n+1}(\alpha, \beta)\) 是以下四个集合的并集
    1. \(\left\{\xi+\zeta \mid \xi, \zeta \in \mathrm{Cl}^{n}(\alpha, \beta)\right\}\)
    2. \(\left\{\varphi_{\xi}(\zeta) \mid \xi, \zeta \in \mathrm{Cl}^{n}(\alpha, \beta)\right\}\)
    3. \(\left\{\Omega_{\xi} \mid \xi \in \mathrm{Cl}^{n}(\alpha, \beta)\right\}\)
    4. \(\left\{\psi_{\pi}(\xi) \mid \xi \in \mathrm{Cl}^{n}(\alpha, \beta) \cap \alpha \wedge \pi \in \mathrm{Cl}^{n}(\alpha, \beta) \cap \operatorname{Reg}_{I}\right\}\)
  3. \(\mathrm{Cl}(\alpha, \beta)={\displaystyle \bigcup_{n<\omega} \mathrm{Cl}^{n}(\alpha, \beta)}\)
  4. \(\psi_{\kappa}(\alpha):=\min \{\xi \in \mathbf{On} \mid \kappa \in \mathrm{Cl}(\alpha, \xi) \wedge \mathrm{Cl}(\alpha, \xi) \cap \kappa \subseteq \xi\}\)
  不加说明时，默认用此函数作为默认 \(\psi\) 型函数
\(\theta\) 型定义
1. \(\text{Bird} \ \theta-\)函数： \(\theta\left(\Omega^{n-1} a_{n-1}+\cdots+\Omega^{2} a_{2}+\Omega a_{1}+a_{0}, b\right)=\varphi\left(a_{n-1}, \ldots, a_{2}, a_{1}, a_{0}, b\right)\)
2. \(\text{Feferman} \ \theta-\)函数
  1. \(C_{0}(\alpha, \beta)=\beta \cup\left\{0, \omega_{1}, \omega_{2}, \ldots, \omega_{\omega}\right\}\)
  2. \(C_{n+1}(\alpha, \beta)=\left\{\gamma+\delta, \theta_{\xi}(\eta) \mid \gamma, \delta, \xi, \eta \in C_{n}(\alpha, \beta) \wedge \xi<\alpha\right\}\)
  3. \(C(\alpha, \beta)={\displaystyle \bigcup_{n<\omega} C_{n}(\alpha, \beta)}\)
  4. \(\theta_{\alpha}(\beta)=\min \left\{\gamma \mid \gamma \notin C(\alpha, \gamma) \wedge \forall(\delta<\beta)\left(\theta_{\alpha}(\delta)<\gamma\right)\right\}\)
  对任意 \(\alpha, \beta<\Gamma_{0}\)，有 \(\theta_{\alpha}(\beta)=\varphi_{\alpha}(\beta)\)；不加说明时，默认用此函数作为默认 \(\theta\) 型函数
\(\vartheta\) 型定义
1. \(\text{Wilken} \ \vartheta-\)函数：令 \(\Omega_{0}\) 为 \(1\) 或形如 \(\varepsilon_{\alpha}\) 形式的序数，\(\Omega_{1}>\Omega_{0}\) 为不可数正则基数；对 \(0<i<\omega\)，令 \(\Omega_{i+1}\) 为 \(\Omega_{i}\) 的后继基数．对 \(0<n<\omega, 0 \leqslant m<n\) 与 \(\beta<\Omega_{m+1}\)，定义
  1. \(\Omega_{m} \cup \beta \subseteq C_{m}^{n}(\alpha, \beta)\)
  2. \(\xi, \eta \in C_{m}^{n}(\alpha, \beta) \rightarrow \xi+\eta \in C_{m}^{n}(\alpha, \beta)\)
  3. \(\xi \in C_{m}^{n}(\alpha, \beta) \cap \Omega_{k+2} \rightarrow \vartheta_{k}^{n}(\xi) \in C_{m}^{n}(\alpha, \beta), m<k<n\)
  4. \(\xi \in C_{m}^{n}(\alpha, \beta) \cap \alpha \rightarrow \vartheta_{m}^{n}(\xi) \in C_{m}^{n}(\alpha, \beta)\)
  5. \(\vartheta_{m}^{n}(\alpha)=\min \left(\left\{\xi<\Omega_{m+1} \mid C_{m}^{n}(\alpha, \xi) \cap \Omega_{m+1} \subseteq \xi \wedge \alpha \in C_{m}^{n}(\alpha, \xi)\right\} \cup\left\{\Omega_{m+1}\right\}\right)\)
2. \(\text{Wilken}-\text{Weiermann} \ \vartheta-\)函数：令 \(\Omega_{0}\) 为 \(1\) 或形如 \(\varepsilon_{\alpha}\) 的序数，设 \(\Omega_{1}>\Omega_{0}\) 为不可数正则基数；对于 \(0<i<\omega\)，设 \(\Omega_{i+1}\) 为 \(\Omega_{i}\) 的后继基数；设 \(\Omega_{\omega}=\sup _{i<\omega} \Omega_{i}\)．对于所有 \(\beta \leqslant \Omega_{i+1}\)，定义
  1. \(\Omega_{i} \cup \beta \subseteq \overline{C}_{i}(\alpha, \beta)\)
  2. \(\xi, \eta \in \overline{C}_{i}(\alpha, \beta) \to \xi+\eta \in \overline{C}_{i}(\alpha, \beta)\)
  3. \(j \leqslant i<\omega \wedge \xi \in \overline{C}_{j}\left(\xi, \Omega_{j+1}\right) \cap \overline{C}_{i}(\alpha, \beta) \cap \alpha \to \overline{\vartheta}_{j}(\xi) \in \overline{C}_{i}(\alpha, \beta)\)
  4. \(\overline{\vartheta}_{i}(\alpha) =\min (\{\xi<\Omega_{i+1} \mid \alpha \in \overline{C}_{i}(\alpha, \xi) \wedge \overline{C}_{i}(\alpha, \xi) \cap \Omega_{i+1} \subseteq \xi\} \cup\left\{\Omega_{i+1}\right\})\)
3. \(\text{Weiermann} \ \vartheta-\)函数
  1. \(C_{0}(\alpha, \beta)=\beta \cup\{0, \Omega\}\)
  2. \(C_{n+1}(\alpha, \beta)=\left\{\gamma+\delta, \omega^{\gamma}, \vartheta(\eta) \mid \gamma, \delta, \eta \in C_{n}(\alpha, \beta) ; \eta<\alpha\right\}\)
  3. \(C(\alpha, \beta)={\displaystyle \bigcup_{n<\omega} C_{n}(\alpha, \beta)}\)
  4. \(\vartheta(\alpha)=\min \{\beta<\Omega \mid C(\alpha, \beta) \cap \Omega \subseteq \beta \wedge \alpha \in C(\alpha, \beta)\}\)
  不加说明时，默认用此函数作为默认 \(\vartheta\) 型函数

5.2 可数序数

5.2.1 递归序数

\(\text{Cantor}\) 序数系列
1. 小 \(\text{Cantor}\) 序数：\(\text{SCO} = \varepsilon_{0}\)，其中 \(\varepsilon_{\gamma} = \varphi_{1}(\gamma)\) 是满足 \(\omega^\xi = \xi\) 的解
2. \(\text{Cantor}\) 序数：\(\text{CO} = \zeta_{0}\)，其中 \(\zeta_{\gamma} = \varphi_2(\gamma)\) 是满足 \(\varepsilon_{\xi} = \xi\) 的解
3. 大 \(\text{Cantor}\) 序数：\(\text{LCO} = \eta_{0}\)，其中 \(\eta_{\gamma} = \varphi_3(\gamma)\) 是满足 \(\zeta_{\xi} = \xi\) 的解
4. 超 \(\text{Cantor}\) 序数：\(\text{HCO} = \varphi_{\omega}(0)\)
\(\text{Feferman}-\text{Sch}\ddot{\mathrm u}\text{tte}\) 序数与 \(\text{Ackermann}\) 序数
1. \(\text{Feferman}-\text{Sch}\ddot{\mathrm u}\text{tte}\) 序数：\(\text{FSO} = \Gamma_0\)，其中 \(\Gamma_{\gamma} = \varphi(1, 0, \gamma)\)
  1. \(\Gamma_0=\varphi(1,0,0)=\varphi \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix}=\theta_{\Omega}(0)=\psi\left(\Omega^{\Omega}\right)\)
  2. \(\Gamma_0\) 是满足 \(\varphi_{\alpha}(0) = \alpha\) 的最小序数，且有 \(\Gamma_0 = \varphi_{\varphi_{\varphi_{._{._..}.}(0)}(0)}(0) = \varphi(\varphi(\varphi(\cdots),0),0),0)\)
2. \(\text{Ackermann}\) 序数：\(\text{AO} = \varphi(1, 0, 0, 0)\)
  1. \(\varphi(1,0,0,0)=\theta_{\Omega^3}(0)=\psi\left(\Omega^{\Omega^2}\right)=\vartheta\left(\Omega^3\right)\)
  2. \(\text{Ackermann}\) 序数是 \(\xi \mapsto \varphi(\xi, 0, 0)\) 的最小不动点
\(\text{Veblen}\) 序数系列
1. 小 \(\text{Veblen}\) 序数：\(\text{SVO} = \sup \left\{\varphi(1, 0), \varphi(1, 0, 0), \varphi(1, 0, 0, 0), \cdots\right\}=\varphi \begin{pmatrix} 1 \\ \omega \\ \end{pmatrix}=\theta_{\Omega^\omega}(0)=\psi\left(\Omega^{\Omega^\omega}\right)=\vartheta\left(\Omega^\omega\right)\)
2. 大 \(\text{Veblen}\) 序数：\(\text{LVO}= \sup \left\{\varphi \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix}, \varphi \begin{pmatrix} 1 \\ \varphi \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix} \\ \end{pmatrix}, \cdots\right\}= \varphi \begin{pmatrix} 1 \\ \varphi \begin{pmatrix} 1 \\ \varphi \begin{pmatrix} 1 \\ \vdots \\ \end{pmatrix} \\ \end{pmatrix} \\ \end{pmatrix}=\theta_{\Omega^{\Omega}}(0)=\psi\left(\Omega^{\Omega^{\Omega}}\right)=\vartheta\left(\Omega^{\Omega}\right)\)，是映射 \(\xi \mapsto \varphi \begin{pmatrix} 1 \\ \xi \\ \end{pmatrix}\) 的最小不动点
\(\text{OCF}\) 定义的序数：设递归不可达序数 \(I = \pi_2 \wedge \pi_1(\pi_2)\)
1. \(\text{Bachmann}-\text{Howard}\) 序数：\(\text{BHO} = \psi\left(\psi_{1}(0)\right)\)
2. \(\text{Buchholz}\) 序数：\(\text{BO} = \psi\left(\Omega_{\omega}\right)\)
3. \(\text{Takeuti}-\text{Feferman}-\text{Buchholz}\) 序数：\(\text{TFBO} = \psi\left(\varepsilon_{\Omega_{\omega}}+1\right)\)
4. 扩展 \(\text{Buchholz}\) 序数：\(\text{EBO} = \psi\left(\psi_{I}(0)\right)\)
5. \(\text{J}\ddot{\mathrm{a}}\text{ger}\) 序数：\(\text{JO} = \psi\left(\psi_{\Omega_{I+1}}(0)\right)\)

5.2.2 大可数序数

\(\text{Church}-\text{Kleene}\) 序数：\(\omega_{1}^{\mathrm{CK}} = \Omega_1\)，即最小的容许序数
反射序数
1. 递归不可达序数：\(\pi_2 \wedge \pi_1(\pi_2)\)
2. 递归 \(\text{Mahlo}\) 序数：\(\pi_2(\pi_2)\)
3. 不可转换序数：\(\pi_2(\pi_2(\pi_2))\)
4. 递归弱紧致序数：\(\pi_3\)
稳定序数：设 \(\alpha\) 为任意序数
1. 不可达稳定序数：定义 \(f(\alpha)\) 是大于 \(\alpha\) 的最小递归不可达序数，则定义不可达稳定序数为 \(f-\)稳定序数
2. \(\text{Mahlo}\) 稳定序数：定义 \(f(\alpha)\) 是大于 \(\alpha\) 的最小递归 \(\text{Mahlo}\) 序数，则定义 \(\text{Mahlo}\) 稳定序数为 \(f-\)稳定序数
3. 不可投影序数：使得 \(\alpha = \sup \{\beta < \alpha \mid L_{\beta} \prec_{\Sigma_1} L_{\alpha}\}\) 的序数，即使得 \(\alpha\) 是 \(\alpha-\)稳定序数的序数 \(\alpha\)

5.3 证明论序数

设 \(I\) 为最小的弱不可达基数，设 \(M\) 为最小的 \(\text{Mahlo}\) 基数

5.3.1 一阶算术

\(\text{Robinson}\) 算术：\(\operatorname{PTO}(\mathbf{Q}) = \omega\)
1. \(\operatorname{PTO}(\mathbf{I \Delta}_{0})=\omega^{2}, \operatorname{PTO}(\mathbf{I \Delta}_{0}^{+})=\operatorname{PTO}(\mathbf{I \Delta}_{0}+\exp )=\omega^{3}\)
2. \(\operatorname{PTO}(\mathbf{I \Sigma}_{n+1})=\omega \uparrow \uparrow (n+2)\)
\(\text{Peano}\) 算术：\(\operatorname{PTO}(\mathbf{PA}) = \varepsilon_0\)
1. \(\operatorname{PTO}(\mathbf{ID}_{1}\#) = \varphi(\omega, 0)\)
2. \(\operatorname{PTO}(\widehat{\mathbf{ID}}_{1}) = \varphi(\varepsilon_0, 0), \operatorname{PTO}(\widehat{\mathbf{ID}}_{<\omega}) = \Gamma_{0}, \operatorname{PTO}(\widehat{\mathbf{ID}}_{\omega}) = \Gamma_{\varepsilon_0}, \operatorname{PTO}(\widehat{\mathbf{ID}}_{<\varepsilon_0}) = \varphi(1, \varepsilon_0, 0), \operatorname{PTO}(\widehat{\mathbf{ID}}_{<\Gamma_0}) = \varphi(1, \Gamma_0, 0)\)
3. \(\operatorname{PTO}(\mathbf{ID}_{1}) = \psi(\varepsilon_{\Omega+1}), \operatorname{PTO}(\mathbf{ID}_{2}) = \psi(\varepsilon_{\Omega_{2}+1}), \operatorname{PTO}(\mathbf{ID}_{<\omega}) = \psi(\Omega_{\omega}), \operatorname{PTO}(\mathbf{ID}_{\omega}) = \psi(\varepsilon_{\Omega_{\omega}+1}), \operatorname{PTO}(\mathbf{ID}_{\varepsilon_0}) = \psi(\Omega_{\varepsilon_{0}})\)

5.3.2 二阶算术

\(\mathbf{RCA}_0\) 子系统：\(\operatorname{PTO}(\mathbf{RCA}_0) = \omega^{\omega}\)
\(\mathbf{WKL}_0\) 子系统：\(\operatorname{PTO}(\mathbf{WKL}_0) = \omega^{\omega}\)
\(\mathbf{ACA}_0\) 子系统：\(\operatorname{PTO}(\mathbf{ACA}_0) = \varepsilon_0\)
\(\mathbf{ATR}_0\) 子系统：\(\operatorname{PTO}(\mathbf{ATR}_0) = \Gamma_0\)
\(\mathbf{\Phi}-\mathbf{CA}_0\) 子系统
1. \(\operatorname{PTO}(\mathbf{\Delta}_{1}^{1}-\mathbf{CA}_{0})=\varepsilon_{0}, \operatorname{PTO}(\mathbf{\Delta}_{1}^{1}-\mathbf{CA})=\psi(\Omega^{\varepsilon_{0}})\)
2. \(\operatorname{PTO}(\mathbf{\Pi}_{1}^{1}-\mathbf{CA}_{0})=\psi(\Omega_{\omega}), \operatorname{PTO}(\mathbf{\Pi}_{1}^{1}-\mathbf{CA})=\psi(\Omega_{\omega} \cdot \varepsilon_{0})\)
3. \(\operatorname{PTO}(\mathbf{\Delta}_{2}^{1}-\mathbf{CA}_{0})=\psi(\Omega_{\omega}), \operatorname{PTO}(\mathbf{\Delta}_{2}^{1}-\mathbf{CA})=\psi(\Omega_{\varepsilon_{0}})\)
\(\mathbf{\Phi}-\mathbf{TR}_0\) 子系统
1. \(\operatorname{PTO}(\mathbf{\Pi}_{1}^{1}-\mathbf{TR}_{0})=\psi(\psi_{I}(0)), \operatorname{PTO}(\mathbf{\Pi}_{1}^{1}-\mathbf{TR}_{0})=\psi(\psi_{I}(0) \cdot \varepsilon_{0})\)
2. \(\operatorname{PTO}(\mathbf{\Delta}_{2}^{1}-\mathbf{TR}_{0})=\psi(\psi_{I}(I^{\omega})), \operatorname{PTO}(\mathbf{\Delta}_{2}^{1}-\mathbf{TR})=\psi(\psi_{I}(I^{\varepsilon_{0}}))\)
\(\mathbf{\Phi}-\mathbf{AC}_0\) 子系统
1. \(\operatorname{PTO}(\mathbf{\Sigma}_{1}^{1}-\mathbf{AC}_{0})=\varepsilon_{0}, \operatorname{PTO}(\mathbf{\Sigma}_{1}^{1}-\mathbf{AC})=\psi(\Omega^{\varepsilon_{0}})\)
2. \(\operatorname{PTO}(\mathbf{\Sigma}_{2}^{1}-\mathbf{AC})=\psi(\Omega_{\varepsilon_{0}})\)
\(\Phi-\mathrm{BI}\) 公理相关子系统
1. \(\operatorname{PTO}(\mathbf{ACA}+\mathbf{BI})=\psi(\varepsilon_{\Omega+1})\)
2. \(\operatorname{PTO}(\mathbf{\Pi}_{1}^{1}-\mathbf{CA}_{0}+\mathbf{\Pi}_{2}^{1}-\mathbf{BI})=\psi(\Omega_{\omega}{ }^{\omega}), \operatorname{PTO}(\mathbf{\Pi}_{1}^{1}-\mathbf{CA}+\mathbf{BI})=\psi(\varepsilon_{\Omega_{\omega}+1}), \operatorname{PTO}(\mathbf{\Pi}_{1}^{1}-\mathbf{TR}+\mathbf{BI})=\psi\left(\varepsilon_{\psi_{I}(0)+1}\right)\)
3. \(\operatorname{PTO}(\mathbf{\Delta}_{2}^{1}-\mathbf{CA}_{0}+\mathbf{\Sigma}_{2}^{1}-\mathbf{BI})=\psi(\psi_{I}(I^{\omega})), \operatorname{PTO}(\mathbf{\Delta}_{2}^{1}-\mathbf{CA}+\mathbf{\Sigma}_{2}^{1}-\mathbf{BI})=\psi(\psi_{I}(I^{\varepsilon_{0}}))\)
4. \(\operatorname{PTO}(\mathbf{\Delta}_{2}^{1}-\mathbf{CA}+\mathbf{BI})=\psi(\varepsilon_{I+1}), \operatorname{PTO}(\mathbf{\Sigma}_{2}^{1}-\mathbf{AC}+\mathbf{BI})=\psi(\varepsilon_{I+1})\)

5.3.2 KP 集合论

\(\operatorname{PTO}(\mathbf{KP}^{-})=\omega, \operatorname{PTO}(\mathbf{KP}^{-\infty})=\varepsilon_0, \operatorname{PTO}(\mathbf{KP})=\psi(\varepsilon_{\Omega+1})\)
1. \(\mathrm{PTO}(\mathbf{KP}^{-}+\mathbf{\Pi}_{1}-\mathbf{Fnd}+\mathbf{\Sigma}_{1}-\mathbf{Ind})=\omega^{\omega}, \mathrm{PTO}(\mathbf{KP}^{-}+\mathbf{\Pi}_{1}-\mathbf{Fnd}+ \mathbf{Ind})=\varepsilon_{0}\)
2. \(\mathrm{PTO}(\mathbf{KP}^{-}+\mathbf{\Sigma}_{1}-\mathbf{Fnd})=\varphi(\varphi(\omega, 0), 0), \mathrm{PTO}(\mathbf{KP}^{-}+\mathbf{\Sigma}_{1}-\mathbf{Fnd}+ \mathbf{Ind})=\varphi(\varphi(\varepsilon_{0}, 0), 0)\)
3. \(\mathrm{PTO}(\mathbf{KP}^{-}+\mathbf{\Pi}_{2}-\mathbf{Fnd})=\psi\left(\Omega^{\Omega^{\omega}}\right), \mathrm{PTO}(\mathbf{KP}^{-}+\mathbf{\Pi}_{2}-\mathbf{Fnd}+ \mathbf{Ind})=\psi\left(\Omega^{\Omega_{0}^{\varepsilon}}\right), \mathrm{PTO}(\mathbf{KP}^{-}+\mathbf{\Pi}_{3}-\mathbf{Fnd})=\psi\left(\Omega^{\Omega^{\Omega^{\omega}}}\right)\)
\(\mathrm{PTO}(\mathbf{KPi})=\psi(\varepsilon_{I+1})\)
1. \(\mathrm{PTO}(\mathbf{KPi}^{-})=\Gamma_{0}, \mathrm{PTO}(\mathbf{KPi}^{-}+\mathbf{\Sigma}_{1}-\mathbf{Ind})=\varphi(1, \omega, 0), \mathrm{PTO}(\mathbf{KPi}^{-}+\mathbf{Ind})=\varphi(1, \varepsilon_{0}, 0)\)
2. \(\mathrm{PTO}(\mathbf{KPi}^{R})=\psi(\Omega_{\omega})\)
\(\mathrm{PTO}(\mathbf{KPl})=\psi(\varepsilon_{\Omega_{\omega}+1})\)
1. \(\mathrm{PTO}(\mathbf{KPl}^{-})=\Gamma_{0}, \mathrm{PTO}(\mathbf{KPl}^{-}+\mathbf{Ind})=\varphi(1,0, \varepsilon_{0})\)
2. \(\mathrm{PTO}(\mathbf{KPl}^{R})=\psi(\Omega_{\omega})\)
\(\mathrm{PTO}(\mathbf{KPM}^{-})=\varphi(\omega, 0,0), \mathrm{PTO}(\mathbf{KPM})=\psi(\varepsilon_{M+1})\)