User blog:Syst3ms/STON 2 : BMS and the Star

This is an attempt at extending STON.

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 \(a \in T\), \(\mu(a)\in T\)

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

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: We let \(H(s) :\!\iff 0<\text{lev}(\text{lev}(s))\)
 * 1) If \(s=0\vee s=1\) then \(\text{cof}(s)=s\)
 * 2) Else if \(s=\mu(a)\):
 * 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 \(\text{lev}(s)\) : Comparison algorithm for \(s<t\), total order on \(T\) :
 * 1) \(\text{lev}(0)=0\)
 * 2) If \(s=s_1+s_2\) with \(s_1\in T,s_2\in PT\) then \(\text{lev}(s)=0\)
 * 3) Else if \(s=\mu(a)\) :
 * 4) If \(\text{cof}(a)\leq\omega\) then \(\text{lev}(s)=\psi_{\mu(0)}(a)\)
 * 5) Otherwise, \(\text{lev}(s)=0\)
 * 6) Else if \(s=\psi_a(b))\) :
 * 7) If \(H(a)\) :
 * 8) If \(\omega<\text{cof}(b)\wedge\text{cof}(n)\leq a\) then \(\text{lev}(s)=0\)
 * 9) Else if \(a=V\wedge b=0\) then \(\text{lev}(s)=1\)
 * 10) Else if \(a=\mu(c)\wedge\text{cof}(c)=\omega\) :
 * 11) If \(b=b_1+b_2\) with \(b_1 \notin ST\wedge b_2<\omega\) then \(\text{lev}(s)=\text{lev}(a)[\text{lev}(\mu(\text{pred}(b_2)))]\)
 * 12) Otherwise, \(\text{lev}(s)=0\)
 * 13) Else if \(b=b_1+b_2\) with \(b_1\in T,b_2\in PT\) then \(\text{lev}(s)=\text{lev}(\psi_a(b_2))\)
 * 14) Else if \(b=\psi_c(d)\wedge\text{lev}(c)=1\wedge\text{cof}(\psi_c(d))\leq\omega\) :
 * 15) If \(a=\mu(e)\wedge e \in ST\) :
 * 16) If \(c\leq d\) then \(\text{lev}(s)=\mu(\text{pre}(e))+\nu_c(d)\), where \(\nu_p(s)\) is defined as such :
 * 17) \(\nu_p(0)=0\)
 * 18) Else if \(s=d+e\) with \(d\in PT,e\in T\) then \(\nu(s)=\nu_p(d)+\nu_p(e)\)
 * 19) Else if \(p\leq\text{cof}(s)\) then \(\nu_p(s)=\nu_p(S(\psi_p(s)))\)
 * 20) Otherwise, \(\nu_p(s)=s\)
 * 21) Else if \(d=0\) then \(\mu(\text{pre}(e))\)
 * 22) Otherwise, \(\text{lev}(s)=\mu(\text{pre}(e))+d\)
 * 23) Else if \(a=\mu(e)\wedge\text{cof}(e)=\omega\wedge\omega<\text{cof}(d)\) then \(\text{lev}(s)=0\)
 * 24) Else if \(d=0\) then \(\text{lev}(s)=0\)
 * 25) Else if \(c\leq d\) then \(\text{lev}(s)=\nu_c(d)\)
 * 26) Else if \(\omega\leq d\) then \(\text{lev}(s)=d\)
 * 27) Otherwise, \(\text{lev}(s)=d+1\)
 * 28) Else if \(\text{lev}(a) \in ST\) then \(\text{lev}(s)=\text{pre}(\text{lev}(a))\)
 * 29) Else if \(b=b_1+b_2\) with \(b_1 \notin ST\wedge b_2<\omega\wedge\text{cof}(\text{lev}(a))=\omega\) then \(\text{lev}(s)=\text{lev}(a)[b_2]\)
 * 30) Otherwise, \(\text{lev}(s)=0\)
 * We always have \(s_1,t_1\in T,s_2,t_2\in PT\)
 * 1) If \(s=0\) then \(t\neq0 \iff s<t\)
 * 2) If \(t=0\) then \(s<t\) does not hold
 * 3) If \(s \in PT\) and \(t=t_1+t_2\) then \(s\leq t_1 \iff s<t\)
 * 4) If \(s=\psi_\kappa(\alpha)\) :
 * 5) If \(t=\psi_\pi(\beta)\):
 * 6) If \(\kappa=\pi\) then \(\alpha<\beta \iff s<t\)
 * 7) If \(\text{lev}(\kappa)=\text{lev}(\pi)\) then \(\kappa<\pi \iff s<t\)
 * 8) If \(\kappa<\pi\wedge t<\alpha\) then \(s<t\) does not hold
 * 9) If \(H(\kappa)\) :
 * 10) If \(H(\pi)\) then \(\kappa<\pi \iff s<t\)
 * 11) Otherwise \(s<\pi \iff s<t\)
 * 12) If \(H(\pi)\) then \(\kappa<t \iff s<t\)
 * 13) Otherwise \(\text{lev}(\kappa)<\text{lev}(\pi) \iff s<t\)
 * 14) If \(t=\mu(\beta)\) :
 * 15) If \(\kappa<t\wedge t<\alpha\) then \(s<t\) does not hold
 * 16) Otherwise \(\kappa\leq t \iff s<t\)
 * 17) If \(s=\mu(\alpha)\):
 * 18) If \(t=\psi_\kappa(\beta)\):
 * 19) If \(\kappa<s\wedge s<\beta\) then \(s<t\)
 * 20) Otherwise \(s<\kappa \iff s<t\)
 * 21) If \(t=\mu(\beta)\) then \(\alpha<\beta \iff s<t\)
 * 22) If \(s=s_1+s_2\) :
 * 23) If \(t \in PT\) then \(s_1<t \iff s<t\)
 * 24) If \(t=t_1+t_2\) :
 * 25) If \(s_1\neq t_1\) then \(s_1<t_1 \iff s<t\)
 * 26) If \(s_1=t_1\) then \(s_2<t_2 \iff s<t\)

Standard form
Check for constructibility \(G_\kappa(s)\) : Standard form \(OT\):
 * \(G_\kappa(0)=0\)
 * \(G_\kappa(a_1+\ldots+a_n)=\max(G_\kappa(a_1),\ldots,G_\kappa(a_n))\)
 * \(G_\kappa(\psi_\pi(a))=\begin{cases}\max(a,G_\kappa(a)) & \kappa=\pi \\ 0 & \text{otherwise}\end{cases}\)
 * \(G_\kappa(\mu(a))=\begin{cases}\max(a,G_\kappa(a)) & \kappa=\mu(a) \\ 0 & \text{otherwise}\end{cases}\)
 * 1) \(0\in OT\)
 * 2) For \(\alpha \in OT\), \(\mu(\alpha)\in OT\)
 * 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 \(\kappa,\alpha\in OT\), \(\psi_\kappa(\alpha)\in OT\) iff:
 * 5) \(0<\text{lev}(\kappa)\)
 * 6) \(G_\kappa(\alpha)\leq\alpha\)
 * 7) \(\text{cof}(\alpha)\leq \kappa\)
 * 8) \((\kappa\neq\Omega\vee\kappa\neq V)\implies\alpha\neq 0\)
 * 9) \(\kappa=\mu(\beta)\wedge\text{cof}(\beta)=\omega\implies \omega\leq\alpha\)

Expansion
We define the expansion function \(E(\alpha,n)\) with \(\alpha,n\in T\) :
 * 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 \(\alpha=\mu(\beta)\) then \(E(\alpha,n)=\mu(E(\beta,n))\)
 * 4) Else if \(\alpha=\psi_\kappa(\beta)\) :
 * 5) If \(\text{lev}(\kappa)=1\wedge \beta\in ST\):
 * 6) If \(n=0\), \(E(\alpha,0)=0\)
 * 7) Otherwise, \(E(\alpha,n)=\psi_\kappa(\text{pre}(\beta))+E(\psi_\kappa(\beta),\text{pre}(n))\)
 * 8) Else if \(H(\kappa)\) then \(E(\alpha,n)=\psi_\kappa(E(\beta,n))\)
 * 9) Else if \(\beta=\omega\wedge\kappa=\mu(\delta)\wedge\text{cof}(\delta)=\omega\) then \(E(\alpha,n)=\mu(E(\delta,n))\)
 * 10) Else if \(\beta=\omega\wedge\kappa=\psi_\pi(\delta)\wedge H(\pi)\wedge\text{cof}(\delta)=\omega\) then \(E(\alpha,n)=\psi_\pi(E(\delta,n))\)
 * 11) Else if \(\text{cof}(\beta)<\kappa\) then \(E(\alpha,n)=\psi_\kappa(E(\beta,n))\)
 * 12) Else if \(n=0\), \(E(\alpha,0)=0\)
 * 13) Else if \(\text{cof}(\beta)=\kappa\) then \(E(\alpha,n)=\psi_\kappa(E(\beta,S(E(\alpha,\text{pre}(n)))))\)

We then define a standardization function \(S(a)\) with \(a\in T\)
 * 1) \(S(0)=0\)
 * 2) \(S(\mu(b))=\mu(S(b))\)
 * 3) If \(a=a_1+a_2\) with \(a_1\in T,a_2\in \{0\}\cup PT\) :
 * 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 \(S(b)=\mu(d)\wedge d\in ST\wedge c=0\) then \(S(a)=\mu(S(\text{pre}(d)))\)
 * 8) Else if \(S(b)=\mu(d)\wedge\text{cof}(d)=\omega\wedge S(c)<\omega\) then \(S(a)=\mu(S(d[c]))\)
 * 9) Else if \(S(b)=\psi_d(e)\wedge H(d)\wedge\text{cof}(e)=\omega\wedge S(c)<\omega\) then \(S(a)=\psi_d(S(e[c]))\)
 * 10) Else if \(S(b)=\psi_d(e)\wedge e\in ST\wedge S(c)=0\) then \(S(a)=S(\psi_d(\text{pre}(e)))\)
 * 11) Else if \(S(b)<\text{cof}(S(c))\) then \(S(a)=S(\psi_b(\psi_{\text{cof}(S(c))}(S(c))))\)
 * 12) 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 \(\alpha,n\in OT\), \(\text{cof}(\alpha)=\omega\) and \(n<\omega\)

Large computable number : the Star
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 \(\text{OT}\) :
 * \(\tau(0)=0\)
 * \(\tau(n+1)=\mu(\tau(n))\)

We define a version of the FGH for this ordinal notation, \(F_\alpha(n)\) with \(\alpha\in OT, n\in\mathbb{N}\), \(\alpha<\Omega\) 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 the Star as \(F_{S(\psi(\tau(10^{100})))}(10^{100})\). Assuming the well-foundedness of the ordinal notation system immediately implies the termination of the computation.

Strength
(Needless to say, assuming well-foundedness here)

The strength of the system is currently unknown. Ecl1psed has put it beyond the limit of BMS as a whole. I personally doubt that claim, so analysis will have to be performed. I can however say with some confidence that it is stronger than Rathjen's OCF based on a weakly compact cardinal and maybe stronger than a \(\Pi_\omega\)-reflection OCF, such as Stegert's or Arai's. However, I would still place it below TON (assuming well-foundedness) and Loader's function.

Long standard forms
This is just a little area to showcase how long and crazy the standard forms get in this extension. I will be going over a few well-known ordinals and their believed equivalent in STON.