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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*}