User blog:Syst3ms/A formalization of Pi Notation as an ordinal notation

Today I am going to propose my formal definition of Username5243's Pi Notation. This notation is infamous for being extremely ill-defined, and even the more-than-informal explanation is confusing. However, I made a little attempt, and while it wasn't any more acceptable than Username's page in terms of an ordinal notation, it was exploitable, and after seeing this blog post by P進大好きbot, I got the idea of trying to finish what i started, and he helped me tremendously with making this post possible.

We are going to define an ordinal notation \((OT,[])\) based on a system of fundamental sequences. We are first going to define the base of the ordinal notation with sets of ordinal terms \(T, PT, N\) as well as a relation \(<\). From there, we will define a function \(\text{rank}\), the sets \(\text{Lis},\text{Li},\text{Su},L_\omega\) and the subset \(OT\subset T\) which contains standard forms. We are then going to define the fundamental sequence system using the functions \(F,S\). The expansion rules will rely on the function symbol L as well as two processes. Finally, we will define the fundamental sequence system \(\alpha[n]\) using the function \(U\).

Ordinal terms :
Definition of \(T\), the set of all ordinal terms: Definition of \(PT\), the set of "non-addition" ordinal terms : Definition of \(N\), the set of "natural numbers": We let \(1\) be a shorthand for \(\pi_0(0,0)\) and \(\omega\) a shorthand for \(\pi_0(0,1)\)
 * 1) \(0 \in T\)
 * \(a,b \in T : a+b \in T\)
 * \(a,b,c \in T : \pi_a(b,c) \in T\)
 * \(a,b \in T : L(a,b) \in T\)
 * \(a,b,c \in T : \pi_a(b,c) \in PT\)
 * \(a,b \in T : L(a,b) \in PT\)
 * \(0,\pi_0(0,0) \in N\)
 * 1) \(a \in N, a\neq 0 : a+\pi_0(0,0) \in N\)

Total order \(<\) :
Definition of the relation \(<\), which is a total order on \(T\):
 * Let \(s\leq t \iff (s<t \vee s=t)\)
 * 1) If \(t=0\), \(s<t\) does not hold
 * 2) Else if \(s=0\), \(s<t\)
 * 3) Else if \(s=L(a,b)\):
 * 4) If \(t=L(c,d)\):
 * 5) If \(a<c\) then \(s<t\)
 * 6) Else if \(a=c\) then \(b<d \iff s<t\)
 * 7) Otherwise, \(s<t\) does not hold
 * 8) Else if \(b<\omega\), \(a\leq t \iff s<t\)
 * 9) Otherwise, \(a<t \iff s<t\)
 * 10) Else if \(t=L(a,b)\) :
 * 11) If \(s<a\) then \(s<t\)
 * 12) Else if \(s=a\) then \(\omega<b \iff s<t\)
 * 13) Otherwise, \(s<t\) does not hold
 * 14) Else if \(s=\pi_a(b,c)\):
 * 15) If \(t=\pi_d(e,f)\):
 * 16) If \(a\neq d \wedge e\leq\pi_a(b,c) \wedge f\leq\pi_a(b,c)\) then \(s<t \iff a<d\)
 * 17) Else if \(b\neq e \wedge f\leq\pi_a(b,c)\) then \(s<t \iff b<e\)
 * 18) Else if \(c\neq f\) then \(s<t \iff c<f\)
 * 19) Otherwise, \(s<t\) does not hold
 * 20) Else, \(t=t_1+t_2\) with \(t_1\in PT,t_2 \in T\) then \(s<t \iff s\leq t_1\)
 * 21) Else, \(s=s_1+s_2\) with \(s_1 \in PT,s_2 \in T\):
 * 22) If \(t \in PT\) then \(s<t \iff s_1<t\)
 * 23) Else, \(t=t_1+t_2\) with \(t_1 \in PT, t_2 \in T\) :
 * 24) If \(s_1<t_1\) then \(s<t\)
 * 25) Else if \(s_1=t_1\) then \(s_2<t_2 \iff s<t\)
 * 26) Otherwise, \(s<t\) does not hold

Set of standard forms \(OT\) :
Definition of \(\text{rank}(s)\), with \(s \in T\) : Definition of \(\text{Lis}\), the set of all "strictly limit" ordinal terms: Definition of \(\text{Li}\), the set of all "limit" ordinal terms : Definition of \(\text{Su}\), the set of all "successor" ordinal terms: We let \(L_\omega=\{L(a,\omega) |a\in \text{Lis}\cap OT\}\)
 * 1) \(\text{rank}(0)=0\)
 * 2) If \(s=\pi_a(b,c)\):
 * 3) If \(b=0 \wedge c=0\) then \(\text{rank}(s)=a\)
 * 4) If \(\text{rank}(b)\leq a\) then \(\text{rank}(s)=\text{rank}(b)\)
 * 5) Otherwise, \(\text{rank}(s) = 0\)
 * 6) If \(s=L(a,b)\) then \(\text{rank}(s)=\text{rank}(b)\)
 * 7) Else, \(s=s_1+s_2\) with \(s_1\in PT, s_2 \in T\) then \(\text{rank}(s)=\text{rank}(s_2)\)
 * 1) For \(s \in T\) :
 * 2) If \(s=0\) then \(s \notin \text{Li}\)
 * 3) If \(s=\pi_n(a,b) \wedge\neg(a= 0\wedge b=0)\) then \(s \in \text{Li}\)
 * 4) If \(s=s_1+s_2\) with \(s_1\in PT,s_2\in T\)then \(s_2 \in \text{Li} \iff s \in \text{Li}\)
 * 1) For \(s \in \text{Lis}\), \(s \in \text{Li}\)
 * 2) For \(s \in T\), if \(s=L(a,b)\) then \(b \in \text{Li} \iff s \in \text{Li}\)
 * 1) For \(s,s_1 \in T\), if \(s=s_1+1\) then \(s \in \text{Su}\)

Definition of \(OT \subset T\), the set of standard ordinal terms: For any set \(S\), we let \(S_O=S\cap OT\)
 * 1) \(0 \in OT\)
 * \(a,b \in OT, n \in N : \pi_n(a,b) \in OT\) iff :
 * 1) \(a,b\notin L_\omega\)
 * 2) \(\text{rank}(a)\leq n+1\)
 * 3) \(\text{rank}(b)=0\)
 * \(a,b \in \text{Lis}\cap OT \wedge b\neq\omega : L(a,b) \in OT\)
 * 1) For any integer \(m\geq 2\) and any \(s_1,\ldots,s_m \in PT\cap OT \cup L_\omega\), if \(s_{i+1}\leq s_i\) for any \(i<m\), then \(s_1+...+s_m \in OT\)

Expansion processes :
We define the process \(F(E)\), for \(E \in T\): L Expansion rules : Definition of \(S(E)\) for \(E \in T'\), the "standardize" function:
 * 1)  Find the rightmost L in E, and let it be \(L(x,y)\) for some \(x,y \in T\)
 * 2)  Expand \(L(x,y)\) according to the L Expansion rules in order, and let the resulting expression be \(E'\)
 * 3)  Set \(E\) to be \(S(E')\)
 * 4)  If none of the L Expansion rules applied, find the first L to the left of the current one in \(E\), let it be \(L(x,y)\) again, and go back to step 2
 * 1) If \(x = s_1+...+s_m\) for \(s_1,\ldots,s_m \in OT \cap PT\) and \(s_m \in \text{Li}\) then \(L(x,y) = s_1+\ldots+s_{m-1}+L(s_m,y)\)
 * 2) If \(x=L(a,b)\) then \(L(x,y)=L(a,L(b,y))\)
 * 3) If \(x=\pi_n(a,b)\):
 * 4) If \(\text{rank}(y)<\text{rank}(x)\) :
 * 5) \(a=0\wedge b=0 \implies L(x,y)=y\)
 * 6) \(a\neq 0\vee b\neq 0 \implies L(x,y)=\pi_n(L(a,y),b)\)
 * 7) \(b \in \text{Li} \implies L(x,y) = \pi_n(a,L(b,y))\)
 * 8) \(a=0\):
 * 9) \(y=0;b \in \text{Su} \implies L(x,y) = 0\)
 * 10) \(b,y \in \text{Su},b=b_1+1;y=y_1+1 \implies L(x,y) = L(x,y_1)+\pi_n(0,b_1)\)
 * 11) \(a \in \text{Su},a=a_1+1,a_1 \notin \text{Lis}\):
 * 12) \(b,y=0 \implies L(x,y) = 0\)
 * 13) \(b\in\text{Su},b=b_1+1;y=0 \implies L(x,y) = \pi_n(a,b_1)+1\)
 * 14) \(y\in\text{Su},y=y_1+1\) :
 * 15) \(y_1 \notin \text{Lis}\implies L(x,y) = \pi_n(a_1,L(\pi_n(a,b),y_1))\)
 * 16) \(y_1 \in \text{Lis}\implies L(x,y) = \pi_n(a_1,L(\pi_n(a,b),y_1)+1)\)
 * 17) \(\text{rank}(a)=n+1\):
 * 18) \(b,y=0 \implies L(x,y) = \pi_n(0,0)\)
 * 19) \(b \in \text{Su},b=b_1+1;y=0 \implies L(x,y) = \pi_n(a,b_1)+1\)
 * 20) \(y \in \text{Su},y=y_1+1\):
 * 21) \(y_1\notin\text{Lis}\implies L(x,y) = \pi_n(L(a,L(\pi_n(a,b),y_1)),0)\)
 * 22) \(y_1\in\text{Lis}\implies L(x,y) = \pi_n(L(a,L(\pi_n(a,b),y_1)+1),0)\)
 * 23) \(a \in \text{Li}\):
 * 24) \(b=0 \implies L(x,y) = \pi_n(L(a,y),0)\)
 * 25) \(b \in \text{Su},b=b_1+1 \implies L(x,y) = \pi_n(L(a,y),\pi_n(a,b_1)+1)\)
 * 26) \(a \in \text{Su},a=a_1+1,a_1 \in \text{Lis}\):
 * 27) \(b,y=0 \implies L(x,y) = 0\)
 * 28) \(b\in\text{Su},b=b_1+1;y=0 \implies L(x,y) = \pi_n(a,b_1)+1\)
 * 29) \(y\in\text{Su},y=y_1+1 \implies L(x,y) = \pi_n(L(a_1,L(\pi_n(a,b),y_1)),0)\)
 * 1) \(S(0) = 0\)
 * 2) If \(E = L(a,b)\), for \(a,b\in T\):
 * 3) \(S(E) = 0 \iff S(a)=0\)
 * 4) \(S(E) = L(S(a),S(b))\) otherwise
 * 5) If \(E= \pi_n(a,b)\), for \(a,b\in T\):
 * 6) \(S(E) = S(b) \iff \text{rank}(S(a))<\text{rank}(S(b0))\)
 * 7) \(S(E)=\pi_n(S(\pi_m(a,0)),S(b)) \iff a=\pi_{m+1}(d,e)\wedge n<m\)
 * 8) \(S(E) = \pi_n(S(a),S(b))\) otherwise
 * 9) if \(E=s_1+\ldots+s_m\), for \(m\in\mathbb{N}\) and \(s_1,\ldots,s_m\in T\):
 * 10) \(S(E) = S(s_{n+1})+\ldots+S(s_m)\) where \(n=\max\{x\in\mathbb{N} : s_x<s_{x+1}\}\)
 * 11) \(S(E) = S(s_1)+\ldots+S(s_n)\) where \(n=\min\{x\in\mathbb{N} : s_{x+1}=0\)
 * 12) \(S(E) = S(s_1)+\ldots+S(s_m)\) otherwise

Fundamental sequence :
We define the function Finally, we define the fundamental sequence \(\alpha[n]\) for \(\alpha \in OT, n \in N\) : \(\alpha[n] = U^m(\alpha,n)\) where \(m = \min\{o:U^o(\alpha,n) \in OT\}\) and \(U:T \times N \to T\) is defined as : \(U(\alpha,n) =\begin{cases} \beta & \alpha\in\text{Su}\wedge\alpha=\beta+1\wedge n=0 \\ F(L(\alpha,n)) & \alpha\in\text{Li}\wedge F(\alpha)=\alpha \\ F(\alpha) & \alpha\in\text{Li} \end{cases}\)

To give this ordinal notation at least some credibility, proving the following would be nice : The first two are quite easy but also tedious to prove given the expansion rules and the total order \(<\), but I have some questions regarding the third one. I might prove those statements one day, but as of now this is not my priority.
 * 1) For any \(s \in \text{Li}_O\), \(s[n]<s\)
 * 2) For any \(s \in \text{Li}_O\) and \(n,m \in N\), \(n<m \iff s[n]<s[m]\)
 * 3) For any \(s \in \text{Li}_O\), \(\sup\limits_n(s[n]) = s\)