User blog:Syst3ms/Stellar OCF Formalization

This is an attempt to formally define "Stellar OCF" (originally created by Ecl1psed). Some of the techniques used here to define the ordinal notation originate from Buchholz

Set of formal strings \(T\)
Definition of \(T\):
 * \(0 \in T\)
 * For \(a,b \in T\), \(a+b\in T\)
 * For \(a,b \in T\), \(\psi_a(b)\in T\)
 * For \(a \in T\), \(\psi^1(a)\in T\)

Base of the notation :
Definition of \(PT\), the set of principal terms :
 * For \(a,b \in T\), \(\psi_a(b)\in PT\)
 * For \(a \in T\), \(\psi^1(a)\in PT\)

Definition of \(ST\), the set of successor terms :
 * \(1\in ST\)
 * For \(a\in T\), \(a+1\in ST\)

Definition of \(\text{pre}(s)\), the predecessor function :
 * 1) If \(s\in \{0\}\cap PT\), \(\text{pre}(s)=0\)
 * 2) Else if \(s=s_1+s_2\) with \(s_1\in PT,s_2\in T\) :
 * 3) If \(\text{pre}(s_2)=0\) then  \(\text{pre}(s)=s_1\)
 * 4) Otherwise, \(\text{pre}(s)=s_1+\text{pre}(s_2)\)

Definition of \(\text{cof}(s)\), the cofinality function:
 * 1) If \(s=0\vee s=1\) then \(\text{cof}(s)=s\)
 * 2) Else if \(s=\psi^1(a)\):
 * 3) If \(\text{cof}(a)\leq\omega\) then \(\text{cof}(s)=s\)
 * 4) Otherwise, \(\text{cof}(s)=\text{cof}(a)\)
 * 5) Else if \(0<\text{lev}(s)\) then \(\text{cof}(s)=s\)
 * 6) Else if \(s=s_1+s_2\) with \(s_1\in T,s_2\in PT\) then \(\text{cof}(s)=\text{cof}(s_2)\)
 * 7) Else if \(s=\psi_a(b)\) :
 * 8) If \(\text{cof}(b)=1\vee a\leq\text{cof}(b)\) then \(\text{cof}(s)=\omega\)
 * 9) Otherwise, \(\text{cof}(s)=\text{cof}(b)\)

Definition of \(\text{lev}(s)\), the level function :
 * 1) If \(s=s_1+s_2\) with \(s_1\in T, s_2 \in PT\) then \(\text{lev}(s)=0\)
 * 2) Else if \(s=\psi^1(a)\) :
 * 3) If \(a=0\) then \(\text{lev}(s)=1\)
 * 4) Else if \(a=a_1+a_2\) with \(a_1\in T,a_2 \in PT\) then \(\text{lev}(s)=\text{lev}(\psi^1(a_2))\)
 * 5) Else if \(a=\psi_b(c)\) and \(\text{lev}(b)=1\wedge \text{cof}(\psi_b(c))\leq\omega\):
 * 6) If \(b\leq c\) then \(\text{lev}(s)=\nu(c)\), where \(\nu(s)\) is defined as such :
 * 7) \(\nu(0)=0\)
 * 8) Else if \(s=a+b\) with \(a\in PT,b\in T\) then \(\nu(s)=\nu(a)+\nu(b)\)
 * 9) Else if \(b\leq\text{cof}(s)\) then \(\nu(s)=\nu(S(\psi_b(s)))\)
 * 10) Otherwise, \(\nu(s)=s\)
 * 11) Else if \(\omega\leq c\) then \(\text{lev}(s)=c\)
 * 12) Else if \(c<\omega\) then \(\text{lev}(s)=c+1\)
 * 13) Else if \(a=\psi^1(b)\) then \(\text{lev}(s)=0\)
 * 14) Otherwise, \(\text{lev}(s)=0\)
 * 15) Else if \(s=\psi_a(b)\) :
 * 16) If \(\text{lev}(a)\in ST\) then \(\text{lev}(s)=\text{pre}(\text{lev}(a))\)
 * 17) Else if \(b=b_1+b_2\) with \(b_2<\omega\) and \(\text{cof}(\text{lev}(a))=\omega\) then \(\text{lev}(s)=\text{lev}(a)[b_n]\)
 * 18) Otherwise, \(\text{lev}(s)=0\)

Recursive comparison algorithm
Comparison algorithm for \(s<t\):
 * We always have \(s_1,t_1\in PT, s_2,t_2 \in T\)
 * 1) If \(s=0\), \(t\neq0\iff s<t\)
 * 2) Else if \(t=0\), \(s<t\) does not hold
 * 3) Else if \(s\in PT \wedge t=t_1+t_2\) then \(s\leq t_1 \iff s<t\)
 * 4) Else if \(s=\psi_\kappa(\alpha)\):
 * 5) If \(t=\psi_\pi(\beta)\):
 * 6) If \(\kappa=\pi\) then \(\alpha<\beta \iff s<t\)
 * 7) Else if \(\text{lev}(\kappa)=\text{lev}(\pi)\) then \(\kappa<\pi \iff s<t\)
 * 8) Else \(\text{lev}(\kappa)<\text{lev}(\pi)\iff s<t\)
 * 9) Else if \(t=\psi^1(\beta)\) then \(\kappa\leq t\iff s<t\)
 * 10) Else if \(s=\psi^1(\alpha)\):
 * 11) If \(t=\psi_\kappa(\beta)\) then \(s<\kappa \iff s<t\)
 * 12) Else if \(t=\psi^1(\beta)\) then \(\alpha<\beta \iff s<t\)
 * 13) Else if \(s=s_1+s_2\) :
 * 14) If \(t\in PT\) then \(s_1<t \iff s<t\)
 * 15) Else if \(t=t_1+t_2\):
 * 16) If \(s_1\neq t_1\) then \(s_1<t_1 \iff s<t\)
 * 17) Else if \(s_1=t_1\) then \(s_2<t_2 \iff s<t\)

Standard form
Check for constructibility \(G_\kappa(s)\ :
 * \(G_\kappa(0)=0\)
 * \(G_\kappa(a_1+\ldots+a_n)=\max(G_\kappa(a_1),\ldots,G_\kappa(a_n))\)
 * \(G_\kappa(\psi_\pi(a))=\begin{cases}\max(a,G_\kappa(a)) & \kappa\leq\pi \\ 0 & \pi<\kappa\end{cases}\)
 * \(G_\kappa(\psi^1(a))=\begin{cases}\max(a,G_\kappa(a)) & \kappa\leq\psi^1(a) \\ 0 & \psi^1(a)<\kappa\end{cases}\)

Standard form \(OT\):
 * 1) \(0\in OT\)
 * 2) For \(\alpha \in OT\), \(\psi^1(\alpha)\in OT\)
 * 3) For \(m\geq 2\), \(\alpha_1,\ldots,\alpha_m\in OT\cap PT\) and \(\alpha_m\leq\ldots\leq\alpha_1\) then \(\alpha_1+\ldots+\alpha_m\in OT\)
 * 4) For \(\kappa,\alpha\in OT\), \(\psi_\kappa(\alpha)\) iff:
 * 5) \(0<\text{lev}(\kappa)\)
 * 6) \(G_\kappa(\alpha)\leq\alpha\)
 * 7) \(\text{cof}(a)\leq \kappa\)
 * 8) \(\kappa\neq\psi^1(0)\iff\alpha\neq 0\)
 * 9) \(\kappa=\psi^1(\beta)\wedge\text{cof}(\beta)=\omega\implies \omega\leq\alpha\)

Expansion
We define the expansion function \(E(\alpha,n)\) with \(\alpha,n\in OT\) :
 * 1) If \(\alpha=\alpha_1+\alpha_2\) with \(\alpha_1\in T,\alpha_2\in PT\) then \(E(\alpha,n)=\alpha_1+E(\alpha_2,n)\)
 * 2) Else if \(\text{lev}(n)<\text{lev}(\alpha)\)  then \(E(\alpha,n)=n\)
 * 3) Else if \(\alpha=\psi_\kappa(\beta)\) :
 * 4) If \(\text{lev}(\kappa)=1\wedge \beta\in ST\):
 * 5) If \(n=0\), \(E(\alpha,0)=0\)
 * 6) Otherwise, \(E(\alpha,n)=\psi_\kappa(\text{pre}(\beta))+E(\psi_\kappa(\beta),\text{pre}(n))\)
 * 7) Else if \(\beta=\omega\wedge\kappa=\psi^1(\delta)\wedge\text{cof}(\delta)=\omega\) then \(E(\alpha,n)=\psi^1(E(\delta,n))\)
 * 8) Else if \(\text{cof}(\beta)<\kappa\) then \(E(\alpha,n)=\psi_\kappa(E(\beta,n))\)
 * 9) Else if \(n=0\), \(E(\alpha,0)=0\)
 * 10) Else if \(\text{cof}(\beta)=\kappa\) then \(E(\alpha,n)=\psi_\kappa(E(\beta,E(\alpha,\text{pre}(n))))\)

We then define a standardization function \(S(a)\) with \(a\in T\)
 * 1) \(S(0)=0\)
 * 2) \(S(\psi^1(b))=\psi^1(S(b))\)
 * 3) If \(a=a_1+a_2\) with \(a_1\in T,a_2\in PT\) :
 * 4) If \(a_2=0\) then \(S(a)=S(a_1)\)
 * 5) Otherwise, \(S(a)=S(a_1)+S(a_2)\)
 * 6) Else if \(a=\psi_b(c)\)
 * 7) If \(b=\psi^1(d)\wedge d\in ST\wedge c=0\) then \(S(a)=\psi^1(S(\text{pre}(d)))\)
 * 8) Else if \(b=\psi^1(d)\wedge\text{cof}(d)=\omega\wedge c<\omega\) then \(S(a)=\psi^1(S(d[c]))\)
 * 9) Else if \(b=\psi_d(e)\wedge e\in ST\wedge c=0\) then \(S(a)=S(\psi_d(\text{pre}(e)))\)
 * 10) Else if \(b<\text{cof}(c)\) then \(S(a)=S(\psi_b(\psi_{\text{cof}(c)}(c)))\)
 * 11) Otherwise, \(S(a)=\psi_{S(b)}(S(c))\)

Fundamental sequence system
We finally define the fundamental sequence system \(\alpha[n]\) as \(S(E(\alpha,n))\) under the restriction that \(\text{cof}(\alpha)=\omega\wedge n<\omega\)

Large computable number
We create a mapping from \(\mathbb{N}\) to \(OT\) :
 * \(N(0)=0\)
 * \(N(1)=1\)
 * For \(m\geq1\), \(N(m+1)=N(m)+1\)

We also create a mapping \(\tau\) from \(\mathbb{N}\) to \(\text{OT}\) :
 * \(\tau(0)=0\)
 * \(\tau(n+1)=\psi^1(\tau(n))\)

We define a version of the FGH for this ordinal notation, \(F_\alpha(n)\) with \(\alpha\in OT, n\in\mathbb{N}\) and \(\text{cof}(\alpha)\leq\omega\) :
 * \(F_0(n)=n+1\)
 * If \(\alpha\in ST\) then \(F_\alpha(n)=F^n_{\text{pre}(\alpha)}(n)\), where \(F^n\) denotes iteration
 * Otherwise, \(F_\alpha(n)=F_{\alpha[N(n)]}(n)\)

Finally, we define our large computable number as \(F_{\psi(\tau(10^{100}))}(10^{100})\). Assuming the well-foundedness of the ordinal notation system immediately implies the termination of the computation.

Explanation of the notation
[WIP]