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 which we will define a function \(\text{rank}\) and the recursive subset \(OT\subset T\) which contains standard forms. We are then going to define the fundamental sequence system by introducing the sets \(T',\text{Li},\text{Su},L\) and the functions \(F,S\). The expansion rules will rely on a new function symbol L as well as two processes. Finally, we will define the fundamental sequence system \(\alpha[n]\).

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 \(\pi_0(0,0)\)
 * 1) \(0 \in T\)
 * \(a,b \in T : a+b \in T\)
 * \(a,b,c \in T : \pi_a(b,c) \in T\)
 * \(a,b,c \in T : \pi_a(b,c) \in PT\)
 * 1) \(0 \in N\)
 * 2) \(a \in N : a+\pi_0(0,0) \in N\)

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) If \(s=0\), \(s<t\)
 * 3) Suppose \(s \in PT\), \(s=\pi_a(b,c)\):
 * 4) Suppose \(t \in PT\), \(t=\pi_d(e,f)\):
 * 5) If \(a\neq d \wedge\pi_a(b,c)\leq e \wedge \pi_a(b,c)\leq f\) then \(s<t \iff a<d\)
 * 6) If \(b\neq e \wedge\pi_a(b,c)\leq f\) then \(s<t \iff b<e\)
 * 7) If \(c\neq f\) then \(s<t \iff c<f\)
 * 8) Otherwise, \(s=t\)
 * 9) Else, \(t=t_1+t_2\) then \(s<t \iff s<t_1\)
 * 10) Else, \(s=s_1+s_2\):
 * 11) If \(t \in PT\) then \(s<t \iff s_1<t\)
 * 12) Else, \(t=t_1+t_2\) then \(s<t \iff s_1<t_1\)

Recursive subset \(OT\) :
Definition of \(\text{rank}(s)\), with \(s \in T\) : Definition of \(OT \subset T\), the set of standard ordinal terms:
 * 1) \(\text{rank}(0)=0\)
 * 2) If \(s \in PT\), \(s=\pi_a(b,c)\) then \(\text{rank}(s)=a\)
 * 3) Else, \(s=s_1+s_2\) then \(\text{rank}(s)=\text{rank}(s_1)\)
 * 1) \(0 \in OT\)
 * \(a,b \in T, n \in N : \pi_n(a,b) \in OT\) where :
 * 1) \(\text{rank}(a)\leq n+1\)
 * 2) if \(a=0 \iff \text{rank}(b)=0\)
 * 3) For any integer \(m\geq 2\) and any \(s_1,\ldots,s_m \in (OT \cap PT)\backslash\{0\}\), if \(s_{i+1}\leq s_i\) for any \(i<m\), then \(s_1+...+s_m \in OT\)

Special sets of ordinal terms :
Definition of \(\text{Li}\), the set of all "limit" ordinal terms: Definition of \(\text{Su}\), the set of all "successor" ordinal terms:
 * 1) For \(s \in OT\) :
 * 2) If \(s=0\) then \(s \notin \text{Li}\)
 * 3) If \(s=\pi_n(a,b) \wedge a\neq 0\wedge b\neq 0)\) then \(s \in \text{Li}\)
 * 4) If \(s=s_1+s_2\) then if \(s_2 \in \text{Li}\) then \(s \in \text{Li}\)
 * 1) If \(s=s_1+1\) then \(s \in \text{Su}\)

Ordinal terms during expansion :
Definition of \(T'\), the set of all ordinal terms during expansion: We can see \(OT \subset T'\)
 * 1) \(0 \in T'\)
 * \(a,b \in T' : a+b \in T'\)
 * \(a,b,c \in T' : \pi_a(b,c) \in T'\)
 * \(a,b \in T' : L(a,b) \in T'\)

Definition of \(L\):
 * \(s,t \in T' : L(s,t) \in L\)

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) \(\text{rank}(y)<\text{rank}(x) : L(x,y)=y\)
 * 2) \(x \in PT : x=\pi_n(a,b)\):
 * 3) \(b \in \text{Li} \implies L(x,y) = \pi_n(a,L(b,y))\)
 * 4) \(a \in L,a=L(a_1,a_2) \implies L(x,y) = \pi_n(L(a_1,L(a_2,y)),b)\)
 * 5) \(a,y=0;b \in \text{Su} \implies L(x,y) = 0\)
 * 6) \(a=0;b,y \in \text{Su},b=b_1+1;y=y_1+1 \implies L(x,y) = L(x,y_1)+\pi_n(0,b_1)\)
 * 7) \(a \in \text{Su},a=a_1+1,a_1 \notin \text{Li};b,y=0 \implies L(x,y) = 0\)
 * 8) \(a,b \in \text{Su},a=a_1+1,a_1 \notin \text{Li};b=b_1+1;y=0 \implies L(x,y) = \pi_n(a,b_1)+1\)
 * 9) \(a,y \in \text{Su},a=a_1+1,a_1 \notin \text{Li};y=y_1+1 \implies L(x,y) = \pi_n(a_1,L(\pi_n(a,b),y_1))\)
 * 10) \(\text{rank}(a)=n+1\):
 * 11) \(b,y=0 \implies L(x,y) = 0\)
 * 12) \(b \in \text{Su},b=b_1+1;y=0 \implies L(x,y) = \pi_n(a,b_1)+1\)
 * 13) \(y \in \text{Su},y=y_1+1 \implies L(x,y) = \pi_n(L(a,L(\pi_n(a,b),y_1)),0)\)
 * 14) \(a \in \text{Li};b=0 \implies L(x,y) = \pi_n(L(a,y),0)\)
 * 15) \(a \in \text{Li};b \in \text{Su},b=b_1+1 \implies L(x,y) = \pi_n(L(a,y),\pi_n(a,b_1)+1)\)
 * 16) \(a \in \text{Su},a=a_1+1,a_1 \in \text{Li};b,y=0 \implies L(x,y) = 0\)
 * 17) \(a,b \in \text{Su},a=a_1+1,a_1 \in \text{Li};b=b_1+1;y=0 \implies L(x,y) = \pi_n(a,b_1)+1\)
 * 18) \(a,y \in \text{Su},a=a_1+1,a_1 \in \text{Li};y=y_1+1 \implies L(x,y) = \pi_n(L(a_1,L(\pi_n(a,b),y_1)),0)\)
 * 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) \(S(E) = S(b) \iff \text{rank}(S(b))>\text{rank}(S(a))\)
 * 7) \(S(E) = \pi_n(S(a),S(b))\) otherwise
 * 8) if \(E=s_1+\ldots+s_m\), for \(m\in\mathbb{N}\) and \(s_1,\ldots,s_m\in T'\):
 * 9) \(S(E) = S(s_{n+1})+\ldots+S(s_m)\) where \(n=\max\{x\in\mathbb{N} : s_x<s_{x+1}\}\)
 * 10) \(S(E) = S(s_1)+\ldots+S(s_m)\) otherwise

Fundamental sequence :
Finally, we define the fundamental sequence \(\alpha[n]\) for \(\alpha \in OT, n \in \mathbb{N}\) : \(\alpha[n] = F^m(L(a,n))\) where \(m = \min\{o:{F'}^m(L(a,n)) \in OT\}\)

In order to prove this ordinal notation system is well-founded, we need to prove the following : The first two are quite easy yet quite 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 \(s \in OT\), \(s[n]<s\)
 * 2) For any \(s \in OT\) and \(n,m \in \mathbb{N}\), \(n<m \iff s[n]<s[m]\)
 * 3) For any \(s \in \text{Li}\), \(\sup\limits_n(s[n]) = s\)