The post is meant to extend https://googology.fandom.com/wiki/User_blog:Hyp_cos/Attempt_of_OCF_up_to_Stability.
Firstly, I’ll verify that it’s worth extending. I compared it with the other OCFs (Stegert, Duchhardt). It is virtually identical to Stegert, if we associate the second reflection instance element be the position in the hierarchy. The only worrying difference is that Stegert’s hierarchies require their elements to be “big” relative to \(\mathbb{X}\). This shouldn’t be a major problem, as the ability of Hyp cos’s system to intersect collapsing hierarchies should make up for it. Duchhardt uses ordinals instead, but as he needs to perform cut-elimination his hierarchies can’t differ too much from Stegert’s.
Up to \(\lambda x.\Gamma_{x+1}\)-stable
We use a \((+)\)-stable ordinal to diagonalize. \(K\) is the least such ordinal. The reason why we need this ordinal is that we have weakened the \(S\) function, without \(K\) it the OCF would only reach \(+\text{constant}\)-stable.
Reflection instances are defined by:
- \(()\) is a reflection instance
- If \(\mathbb{X}\) and \(\mathbb{Y}\) are reflection instances, then \((\mathbb{X},\alpha,\mathbb{Y},\beta,n)\) is a reflection instance.
- \(v(())=\varnothing\)
- \(v((\mathbb{X},\alpha,\mathbb{Y},\beta,n))=v(\mathbb{X})\cup v(\mathbb{Y})\cup\{\alpha,\beta\}\)
- \(iv(())=\varnothing\)
- \(iv((\mathbb{X},\alpha,\mathbb{Y},\beta,n))=iv(\mathbb{X})\cup iv(\mathbb{Y})\cup\{\alpha\}\)
\(C(\alpha,\beta)\) is defined as follows:
- \(C_0(\alpha,\beta)=\beta\cup\{0,K\}\)
- \(C_{n+1}(\alpha,\beta)=\{\gamma,S_\gamma(\delta,n),\gamma+\delta,\phi_\gamma(\delta),\psi_\gamma(\mathbb{X},\epsilon):\gamma,\delta,\epsilon\in C_n(\alpha,\beta)\land\epsilon<\alpha\land\mathbb{X}\in r_\alpha[C_n(\alpha,\beta)]\}\)
- \(C(\alpha,\beta)=\bigcup C_n(\alpha,\beta)\)
\(\Pi^\beta_n[S]\) represents the set of ordinals \(\beta\)-\(\Pi_n\)-reflecting on \(S\). Define the function hierarchies \(f^\beta_\pi\) as follows:
- \(f^0_\pi=0\)
- \(f^\beta_\pi\) is a partial function on ordinals \(\kappa\) with \(C(\beta,\kappa)\cap\pi=\kappa\) such that \(f^\beta_\pi(\kappa)\) is the least ordinal for which for all \(\gamma\in C(\beta,\kappa)\cap\beta\), \(f^\gamma_\pi(\kappa)<f^\beta_\pi(\kappa)\).
Define collapsing sets \(A_\pi\mathbb{X}\) as follows:
- \(A_\pi()=\pi\)
- \(A_\pi(\mathbb{X},0,\mathbb{Y},\beta,n)=A_\pi\mathbb{X}\cap A_\pi\mathbb{Y}\)
-
If \(\alpha>0\), \(A_\pi(\mathbb{X},\alpha,\mathbb{Y},\beta,n)\) consists of ordinals \(\kappa\) such that:
- \(\kappa\in A_\pi\mathbb{X}\)
- \(C(\alpha,\kappa)\cap\pi=\kappa\)
- For all \(\gamma\in C(\alpha,\kappa)\cap\alpha\), \(\kappa\in\Pi^{f^\beta_\pi(\kappa)}_n[A_\pi(\mathbb{X},\gamma,\mathbb{Y},\beta,n)]\)
Define the collapsing functions \(\psi_\pi(\mathbb{X},\alpha)\) as the smallest member \(\beta\) of \(A_\pi\mathbb{X}\) such that \(C(\alpha,\beta)\cap\pi=\beta\).
Up to a nonprojectible
We can massively increase the strength by adding an argument to \(\psi\). We will also need to modify the function hierarchies to prevent them from diverging from their intended meaning. \(K\) is the least nonprojectible.
Reflection instances are defined by:
- \(()\) is a reflection instance
- If \(\mathbb{X}\) and \(\mathbb{Y}\) are reflection instances, then \((\mathbb{X},\alpha,\mathbb{Y},\beta,n)\) is a reflection instance.
- \(v(())=\varnothing\)
- \(v((\mathbb{X},\alpha,\mathbb{Y},\beta,n))=v(\mathbb{X})\cup v(\mathbb{Y})\cup\{\alpha,\beta\}\)
- \(iv(())=\varnothing\)
- \(iv((\mathbb{X},\alpha,\mathbb{Y},\beta,n))=iv(\mathbb{X})\cup iv(\mathbb{Y})\cup\{\alpha\}\)
- \(d(())=\varnothing\)
- \(d((\mathbb{X},\alpha,\mathbb{Y},\beta,n))=d(\mathbb{X})\cup d(\mathbb{Y})\cup\{\beta\}\)
\(C(\alpha,\beta)\) is defined as follows:
- \(C_0(\alpha,\beta)=\beta\cup\{0,K\}\)
- \(C_{n+1}(\alpha,\beta)=\{\gamma,S(\gamma,n),\gamma+\delta,\psi_\gamma^{\mathbb{X},\epsilon}(\delta):\gamma,\delta,\epsilon\in C_n(\alpha,\beta)\land\epsilon<\alpha\land\mathbb{X}\in r_\alpha[C_n(\alpha,\beta)]\}\)
- \(C(\alpha,\beta)=\bigcup C_n(\alpha,\beta)\)
\(\Pi^\beta_n[S]\) represents the set of ordinals \(\beta\)-\(\Pi_n\)-reflecting on \(S\). Define the function hierarchies \(f^\beta_\pi\) as follows:
- \(f^0_\pi=0\)
- \(f^\beta_\pi\) is a partial function on ordinals \(\kappa\) with \(C(\beta,\kappa)\cap\pi=\kappa\) such that \(f^\beta_\pi(\kappa)\) is the least ordinal for which for all \(\gamma\in C(\beta,\kappa)\cap\beta\), \(f^\gamma_\pi(\kappa)<f^\beta_\pi(\kappa)\). In addition, if \(\beta\neq\pi\) is a value of \(\psi_{\pi^\prime}^{\mathbb{X},\delta}\) with \(d(\mathbb{X})\subseteq\beta\) then so is \(f^\beta_\pi(\kappa)\)
Define collapsing sets \(A_\pi\mathbb{X}\) as follows:
- \(A_\pi()=\pi\)
- \(A_\pi(\mathbb{X},0,\mathbb{Y},\beta,n)=A_\pi\mathbb{X}\cap A_\pi\mathbb{Y}\)
-
If \(\alpha>0\), \(A_\pi(\mathbb{X},\alpha,\mathbb{Y},\beta,n)\) consists of ordinals \(\kappa\) such that:
- \(\kappa\in A_\pi\mathbb{X}\)
- \(C(\alpha,\kappa)\cap\pi=\kappa\)
- For all \(\gamma\in C(\alpha,\kappa)\cap\alpha\), \(\kappa\in\Pi^{f^\beta_\pi(\kappa)}_n[A_\pi(\mathbb{X},\gamma,\mathbb{Y},\beta,n)]\)
Define the collapsing functions \(\psi_\pi^{\mathbb{X},\alpha}(\beta)\) as the smallest member \(\gamma\) of \(A_\pi\mathbb{X}\) above \(\beta\) such that \(C(\alpha,\gamma)\cap\pi=\gamma\).
Up to \(\alpha\)-stable (and beyond)
I turns out that another minor modification vastly increases the strength. \(K\) is the least 2-1-stable.
Reflection instances are defined by:
- \(()\) is a reflection instance
- \((\alpha)\) is a reflection instance
- If \(\mathbb{X}\) and \(\mathbb{Y}\) are reflection instances, then \((\mathbb{X},\alpha,\mathbb{Y},\beta,n)\) is a reflection instance.
- \(v(())=\varnothing\)
- \(v((\alpha))=\{\alpha\}\)
- \(v((\mathbb{X},\alpha,\mathbb{Y},\beta,n))=v(\mathbb{X})\cup v(\mathbb{Y})\cup\{\alpha,\beta\}\)
- \(iv(())=\varnothing\)
- \(iv((\alpha))=\{\alpha\}\)
- \(iv((\mathbb{X},\alpha,\mathbb{Y},\beta,n))=iv(\mathbb{X})\cup iv(\mathbb{Y})\cup\{\alpha\}\)
- \(d(())=\varnothing\)
- \(d((\alpha))=\{\alpha\}\)
- \(d((\mathbb{X},\alpha,\mathbb{Y},\beta,n))=d(\mathbb{X})\cup d(\mathbb{Y})\cup\{\beta\}\)
\(C(\alpha,\beta)\) is defined as follows:
- \(C_0(\alpha,\beta)=\beta\cup\{0,K\}\)
- \(C_{n+1}(\alpha,\beta)=\{\gamma,S(\gamma,n),\gamma+\delta,\psi_\gamma^{\mathbb{X},\epsilon}(\delta):\gamma,\delta,\epsilon\in C_n(\alpha,\beta)\land\epsilon<\alpha\land\mathbb{X}\in r_\alpha[C_n(\alpha,\beta)]\}\)
- \(C(\alpha,\beta)=\bigcup C_n(\alpha,\beta)\)
\(\Pi^\beta_n[S]\) represents the set of ordinals \(\beta\)-\(\Pi_n\)-reflecting on \(S\). Define the function hierarchies \(f^\beta_\pi\) as follows:
- \(f^0_\pi=0\)
- \(f^\beta_\pi\) is a partial function on ordinals \(\kappa\) with \(C(\beta,\kappa)\cap\pi=\kappa\) such that \(f^\beta_\pi(\kappa)\) is the least ordinal for which for all \(\gamma\in C(\beta,\kappa)\cap\beta\), \(f^\gamma_\pi(\kappa)<f^\beta_\pi(\kappa)\). In addition, if \(\beta\neq\pi\) is a value of \(\psi_{\pi^\prime}^{\mathbb{X},\delta}\) with \(d(\mathbb{X})\subseteq\beta\) then so is \(f^\beta_\pi(\kappa)\)
Define collapsing sets \(A_\pi\mathbb{X}\) as follows:
- \(A_\pi()=\pi\)
- \(A_\pi(\alpha)\) consists of the \(\kappa\) that are \(f^\alpha_\pi(\kappa)\)-ply-stable.
- \(A_\pi(\mathbb{X},0,\mathbb{Y},\beta,n)=A_\pi\mathbb{X}\cap A_\pi\mathbb{Y}\)
-
If \(\alpha>0\), \(A_\pi(\mathbb{X},\alpha,\mathbb{Y},\beta,n)\) consists of ordinals \(\kappa\) such that:
- \(\kappa\in A_\pi\mathbb{X}\)
- \(C(\alpha,\kappa)\cap\pi=\kappa\)
- For all \(\gamma\in C(\alpha,\kappa)\cap\alpha\), \(\kappa\in\Pi^{f^\beta_\pi(\kappa)}_n[A_\pi(\mathbb{X},\gamma,\mathbb{Y},\beta,n)]\)
Define the collapsing functions \(\psi_\pi^{\mathbb{X},\alpha}(\beta)\) as the smallest member \(\gamma\) of \(A_\pi\mathbb{X}\) above \(\beta\) such that \(C(\alpha,\gamma)\cap\pi=\gamma\).