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These OCFs are more complicated than previous OCFs. They are quite different from the OCFs in some literature, which I do not really understand, so it is probable that I do not follow the right way.

Those OCF are defined based on some guesses at reflecting/indescribable ranges. If there are more things, such as

  • \(\Pi_3\)-reflecting ordinals that are also \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings, but are not "\(\Pi_3\)-reflecting ordinals that are also \(\Pi_2\)-reflecting on \(\Pi_2\)-reflectings on \(\Pi_3\)-reflectings"
  • ordinals that are both \(\Pi_3\)-reflecting on \(\Pi_3\)-reflectings, and \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings on \(\Pi_3\)-reflectings, but are not "\(\Pi_3\)-reflecting on \(\Pi_3\)-reflectings, and \(\Pi_2\)-reflecting on such ordinals that are both \(\Pi_3\)-reflecting and \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings on \(\Pi_3\)-reflectings"
  • ordinals \(\Pi_2\)-reflecting on \(\Pi_2\)-reflectings on \(\Pi_3\)-reflectings, but are not "\(\Pi_2\)-reflecting on \(\Pi_3\)-reflecting ordinals that are also limit of \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings"
  • ordinals \(\Pi_3\)-reflecting on \(\Pi_3\)-reflectings on \(\Pi_4\)-reflectings, but are not "\(\Pi_3\)-reflecting on \(\Pi_4\)-reflecting ordinals that are also \(\Pi_2\)-reflecting on \(\Pi_3\)-reflecting on \(\Pi_4\)-reflectings"

then the guesses underestimate the strength of \(\text{KP}+\Pi_n\text{-Ref}\) and beyond.

\(\Pi_4\)-Reflection

Not only the definition, but also the expressions of this OCF are highly inductive. A reflection instance is defined as

  • \((\alpha)\) is a reflection instance.
  • If \(\mathbb X\) is a reflection instance, then \((\alpha,\beta,\mathbb X)\) is also a reflection instance.

Inductive varibles of a reflection instance (denoted as a set \(IV\mathbb X\) where \(\mathbb X\) is the reflection instance) are defined as

  • \(IV(\alpha)=\{\alpha\}\)
  • \(IV(\alpha,\beta,\mathbb X)=IV\mathbb X\cup\{\alpha,\beta\}\)

Let \(\mathcal K\) be the least \(\Pi_4\)-reflecting ordinal, \begin{eqnarray*} C_0(\alpha,\beta)&=&\beta\cup\{0,\mathcal K\}\\ C_{i+1}(\alpha,\beta)&=&\{\gamma+\delta|\gamma,\delta\in C_i(\alpha,\beta)\}\\ &\cup&\{\omega^\gamma|\gamma\in C_i(\alpha,\beta)\land\gamma>\mathcal K\}\\ &\cup&\{\psi_\pi(\mathbb X,\gamma)|\pi,\gamma\in C_i(\alpha,\beta)\land\gamma<\alpha\land IV\mathbb{X}\subseteq C_i(\alpha,\beta)\cap\alpha\}\\ C(\alpha,\beta)&=&\bigcup_{i<\omega}C_i(\alpha,\beta) \\ A_\pi(\alpha)&=&\{\gamma<\pi|C(\alpha,\gamma)\cap\pi\subseteq\gamma\land\\ & &\forall\xi\in C(\alpha,\gamma)\cap\alpha(\gamma\text{ is }\Pi_3\text{-reflecting on }A_\pi(\xi))\}\\ A_\pi(\alpha,0,\mathbb X)&=&A_\pi(\alpha)\cap A_\pi\mathbb X\\ \text{For }\beta>0:& & \\ A_\pi(\alpha,\beta,\mathbb X)&=&\{\gamma\in A_\pi(\alpha)|C(\beta,\gamma)\cap\pi\subseteq\gamma\land\\ & &\forall\xi\in C(\beta,\gamma)\cap\beta(\gamma\text{ is }\Pi_2\text{-reflecting on }A_\pi(\alpha,\xi,\mathbb X))\}\\ \psi_\pi(\mathbb X,\alpha)&=&\min(\{\gamma\in A_\pi\mathbb X|C(\alpha,\gamma)\cap\pi\subseteq\gamma\}\cup\{\pi\}) \end{eqnarray*} Superficially the definition of this OCF is simpler than the one in the literature, but it has more parameters, while the literature resulted a function with 6 parameters.

Up to (+constant)-stable

It seems to be a big step because this section exceeds \(\Pi_n\)-reflection and \(\Pi_\omega\)-reflection, but the notation up to this follows the same way.

A reflection instance is defined as

  • \(()\) is a reflection instance.
  • If \(\mathbb X\) and \(\mathbb Y\) are reflection instances, then \((\mathbb X,\alpha,\mathbb Y,\beta)\) is also a reflection instance.

Varibles of a reflection instance are defined as

  • \(v()=\varnothing\)
  • \(v(\mathbb X,\alpha,\mathbb Y,\beta)=v\mathbb X\cup v\mathbb Y\cup\{\alpha,\beta\}\)

Inductive varibles of a reflection instance are defined as

  • \(IV()=\varnothing\)
  • \(IV(\mathbb X,\alpha,\mathbb Y,\beta)=IV\mathbb X\cup IV\mathbb Y\cup\{\alpha\}\)

Let \(p_1(\alpha)=\sup\{\beta\le\alpha|\omega\cdot\beta\le2+\alpha\}\), \(p_2(\alpha)=\sup\{n<\omega|\omega\cdot p_1(\alpha)+n\le2+\alpha\}\), and \(S(\alpha)\) is the least \(p_1(\alpha)\)-\(\Pi_{p_2(\alpha)}\)-reflecting ordinal.

The notation of \(p_1,\ p_2,\ S\) come from transfinite levels of Reflection. An ordinal \(\alpha\) is \(\beta\)-\(\Pi_n\)-reflecting on set or class \(X\) if for every \(\Pi_n\) formula \(\phi(\vec p)\), \(\forall\vec p\in L_\alpha(L_{\alpha+\beta}\models\phi(\vec p)\rightarrow\exists\gamma\in\alpha\cap X(\vec p\in L_\gamma\land L_\gamma\models\phi(\vec p)))\), denoted \(\alpha\Pi^\beta_n[X]\). An ordinal is \(\beta\)-\(\Pi_n\)-reflecting if it is \(\beta\)-\(\Pi_n\)-reflecting on \(\text{Ord}\). Begin with \(\Pi_2\)-reflection, S(\alpha) then is the \(1+\alpha\)-th level of reflection, e.g. \(S(0)\) is the least \(\Pi_2\)-reflecting, \(S(1)\) is the least \(\Pi_3\)-reflecting, \(S(2)\) is the least \(\Pi_4\)-reflecting, \(S(\omega)\) is the least 1-\(\Pi_0\)-reflecting, \(S(\omega2)\) is the least 2-\(\Pi_0\)-reflecting, etc. \begin{eqnarray*} C_0(\alpha,\beta)&=&\beta\cup\{0\}\\ C_{i+1}(\alpha,\beta)&=&\{\gamma+\delta|\gamma,\delta\in C_i(\alpha,\beta)\}\\ &\cup&\{S(\gamma)|\gamma\in C_i(\alpha,\beta)\}\\ &\cup&\{\psi_\pi(\mathbb X,\gamma)|\pi,\gamma\in C_i(\alpha,\beta)\land\gamma<\alpha\land v\mathbb{X}\subseteq C_i(\alpha,\beta)\land IV\mathbb{X}\subseteq\alpha\}\\ C(\alpha,\beta)&=&\bigcup_{i<\omega}C_i(\alpha,\beta) \\ A_\pi()&=&\pi\\ A_\pi(\mathbb X,0,\mathbb Y,\beta)&=&A_\pi\mathbb X\cap A_\pi\mathbb Y\\ \text{For }\alpha>0:& & \\ A_\pi(\mathbb X,\alpha,\mathbb Y,\beta)&=&\{\gamma\in A_\pi\mathbb X|C(\alpha,\gamma)\cap\pi\subseteq\gamma\land\\ & &\forall\xi\in C(\alpha,\gamma)\cap\alpha(\gamma\Pi^{p_1(\beta)}_{p_2(\beta)}[A_\pi(\mathbb X,\xi,\mathbb Y,\beta)])\}\\ \psi_\pi(\mathbb X,\beta)&=&\min(\{\gamma\in A_\pi\mathbb X|C(\beta,\gamma)\cap\pi\subseteq\gamma\}\cup\{\pi\}) \end{eqnarray*} The notation for \(\Pi_n\)-reflection and \(\Pi_\omega\)-reflection are derived by limiting the \(S\) to \(S(n)\) or \(S(\omega)\).

Superficially this definition is much simpler than the one in the literature.

(EDIT: there is a simpler OCF for \(\Pi_n\)-reflection)

Up to \(\lambda x.x^\omega\)-stable

Next we can simply extend the previous section by finitely many parameters for reflection. A reflection instance is defined as

  • \(()\) is a reflection instance.
  • If \(\mathbb X\) and \(\mathbb Y\) are reflection instances, then \((\mathbb X,\alpha,\mathbb Y,\beta_1,\cdots,\beta_n)\) is also a reflection instance.

Varibles of a reflection instance are defined as

  • \(v()=\varnothing\)
  • \(v(\mathbb X,\alpha,\mathbb Y,\beta_1,\cdots,\beta_n)=v\mathbb X\cup v\mathbb Y\cup\{\alpha,\beta_1,\cdots,\beta_n\}\)

Inductive varibles of a reflection instance are defined as

  • \(IV()=\varnothing\)
  • \(IV(\mathbb X,\alpha,\mathbb Y,\beta_1,\cdots,\beta_n)=IV\mathbb X\cup IV\mathbb Y\cup\{\alpha\}\)

Let \(p_1(\alpha)=\sup\{\beta\le\alpha|\omega\cdot\beta\le2+\alpha\}\), \(p_2(\alpha)=\sup\{n<\omega|\omega\cdot p_1(\alpha)+n\le2+\alpha\}\), and \(S(\alpha_n,\cdots,\alpha_1,\alpha_0)\) is the least ordinal \(\beta\) that is \(p_1(\beta^n\alpha_n+\cdots+\beta\alpha_1+\alpha_0)\)-\(\Pi_{p_2(\beta^n\alpha_n+\cdots+\beta\alpha_1+\alpha_0)}\)-reflecting ordinal. The notation \(\alpha\Pi^\beta_n[X]\) still has the same meaning as previous section. \begin{eqnarray*} C_0(\alpha,\beta)&=&\beta\cup\{0\}\\ C_{i+1}(\alpha,\beta)&=&\{\gamma+\delta|\gamma,\delta\in C_i(\alpha,\beta)\}\\ &\cup&\{S(\gamma_1,\cdots,\gamma_m)|\gamma_1,\cdots,\gamma_m\in C_i(\alpha,\beta)\}\\ &\cup&\{\psi_\pi(\mathbb X,\gamma)|\pi,\gamma\in C_i(\alpha,\beta)\land\gamma<\alpha\land v\mathbb{X}\subseteq C_i(\alpha,\beta)\land IV\mathbb{X}\subseteq\alpha\}\\ C(\alpha,\beta)&=&\bigcup_{i<\omega}C_i(\alpha,\beta) \\ A_\pi()&=&\pi\\ A_\pi(\mathbb X,0,\mathbb Y,\beta_n,\cdots,\beta_1,\beta_0)&=&A_\pi\mathbb X\cap A_\pi\mathbb Y\\ \text{For }\alpha>0:& & \\ A_\pi(\mathbb X,\alpha,\mathbb Y,\beta_n,\cdots,\beta_1,\beta_0)&=&\{\gamma\in A_\pi\mathbb X|C(\alpha,\gamma)\cap\pi\subseteq\gamma\land\\ & &\forall\xi\in C(\alpha,\gamma)\cap\alpha(\gamma\Pi^{p_1(\gamma^n\beta_n+\cdots+\gamma\beta_1+\beta_0)}_{p_2(\gamma^n\beta_n+\cdots+\gamma\beta_1+\beta_0)}[A_\pi(\mathbb X,\xi,\mathbb Y,\beta_n,\cdots,\beta_1,\beta_0)])\}\\ \psi_\pi(\mathbb X,\beta)&=&\min(\{\gamma\in A_\pi\mathbb X|C(\beta,\gamma)\cap\pi\subseteq\gamma\}\cup\{\pi\}) \end{eqnarray*}

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