User blog:Rpakr/Double Primitive Sequence System

日本語

In this blog post, I will explain double primitive sequence system (DPrSS), which is an extension of the primitive sequence system. The notation is intended to be a formalization of TY sequence, which is another extension of primitive sequence system by Yukito that only has some correspondences with ordinals and no definition.

Definition
For \(p\in\mathbb{N}\cup\{0\}\), let \(\mathbb{N}_p=\{i|i\in\mathbb{N} \land i\leq p\}\).

An expression is a string of the form \((a_1,a_2,...,a_k)[n]\) where \(\forall m\in\mathbb{N}_k(a_m\in\mathbb{N})\land n\in\mathbb{N}\).

The value of \((a_1,a_2,...,a_k)[n]\) is defined as follows:


 * If \(k=0\), \((a_1,a_2,...,a_k)[n]=n\) (Rule 1)
 * Else, if \(k>0\land\nexists i\in\mathbb{N}_{k-1}(a_i0\land\exists i\in\mathbb{N}_{k-1}(a_i1\),
 * \(r=\max\{i|i\in\mathbb{N}_{k-1}\land\exists d\in\mathbb{N}_{k-i}(p^d(k)=i)\land n(i)<n(k)\}\)
 * \(r_2=\max\{i|i\in\mathbb{N}_{r-1}\land\exists d\in\mathbb{N}_{r-i}(p^d(r)=i)\land n(i)<n(k)\land\)
 * \(\exists l\in\mathbb{N}_{k-r}(\forall j\in\mathbb{N}_{l-1}(a_{i+j}-a_i=a_{r+j}-a_r)\land a_{i+l}-a_i<a_{r+l}-a_r)\}\)
 * \(t=\min\{i|i\in\mathbb{N}_{k-r}\land a_{r_2+i}-a_{r_2}\neq a_{r+i}-a_r\}\)
 * If \(r+t=k\land a_{r_2+t}-a_{r_2}=a_{r+t}-a_r-1 \land\forall i\in\mathbb{N}_{k-r_2-t}(a_{p(r_2+t)}<a_{r_2+t+i})\),
 * \(\Delta=a_r-a_{r_2}\)
 * \(G=(a_1,a_2,...,a_{r_2-1})\)
 * \(B_i=(a_{r_2}+i\Delta,a_{r_2+1}+i\Delta,...,a_{r-1}+i\Delta)\)
 * \((a_1,a_2,...,a_k)[n]=GB_0B_1...B_{n-1}[n]\) (Rule 4)
 * Else, if \(r+t\neq k\lor a_{r_2+t}-a_{r_2}\neq a_{r+t}-a_r-1\lor\exists i\in\mathbb{N}_{k-r_2-t}(a_{p(r_2+t)}\geq a_{r_2+t+i})\),
 * \(r_3=\min\{i|i\in\mathbb{N}_k\land r_2< i\land n(i)<n(k)\land\exists d\in\mathbb{N}_{k-p(i)}(p^d(k)=p(i))\}\)
 * \(\Delta=a_k-a_{r_3}-1\)
 * \(G=(a_1,a_2,...,a_{r_3-1})\)
 * \(B_i=(a_{r_3}+i\Delta,a_{r_3+1}+i\Delta,...,a_{k-1}+i\Delta)\)
 * \(L=(a_{r_3}+(n-1)\Delta,a_{r_3+1}+(n-1)\Delta,...,a_{r-1}+(n-1)\Delta)\)
 * \((a_1,a_2,...,a_k)[n]=GB_0B_1...B_{n-2}L[n]\) (Rule 5)

Googologism
Let \(f(n)=(1,2,n)[n]\).

I define the double primitive sequence number as \(f^{10}(10)\).

Correspondence with UNOCF
DPrSS is conjectured to correspond to UNOCF as follows.

\(\psi\) means UNOCF.


 * \(=0\)
 * \((1)=1=\psi(0)\)
 * \((1,1)=2=\psi(0)\times2\)
 * \((1,2)=\omega=\psi(1)\)
 * \((1,2,1)=\omega+1=\psi(1)+\psi(0)\)
 * \((1,2,1,2)=\omega\times2=\psi(1)\times2\)
 * \((1,2,2)=\omega^2=\psi(2)\)
 * \((1,2,3)=\omega^\omega=\psi(\omega)\)
 * \((1,2,4)=\varepsilon_0=\psi(\Omega)\)
 * \((1,2,4,2)=\varepsilon_0\times\omega=\psi(\Omega+1)\)
 * \((1,2,4,2,3,5)=\varepsilon_0^2=\psi(\Omega+\psi(\Omega))\)
 * \((1,2,4,2,4)=\varepsilon_1=\psi(\Omega\times2)\)
 * \((1,2,4,3)=\varepsilon_\omega=\psi(\Omega\times\omega)\)
 * \((1,2,4,3,4,6)=\varepsilon_{\varepsilon_0}=\psi(\Omega\times\psi(\Omega))\)
 * \((1,2,4,3,5)=\zeta_0=\psi(\Omega^2)\)
 * \((1,2,4,3,5,3,5)=\eta_0=\psi(\Omega^3)\)
 * \((1,2,4,3,5,4)=\varphi(\omega,0)=\psi(\Omega^\omega)\)
 * \((1,2,4,3,5,4,6)=\Gamma_0=\psi(\Omega^\Omega)\)
 * \((1,2,4,4)=\psi(\Omega_2)\)
 * \((1,2,4,4,2,4,4)=\psi(\Omega_2\times2)\)
 * \((1,2,4,4,3)=\psi(\Omega_2\times\omega)\)
 * \((1,2,4,4,3,5,5)=\psi(\Omega_2^2)\)
 * \((1,2,4,4,4)=\psi(\Omega_3)\)
 * \((1,2,4,5)=\psi(\Omega_\omega)\)
 * \((1,2,4,5,4)=\psi(\Omega_{\omega+1})\)
 * \((1,2,4,5,4,5)=\psi(\Omega_{\omega\times2})\)
 * \((1,2,4,5,5)=\psi(\Omega_{\omega^2})\)
 * \((1,2,4,5,6)=\psi(\Omega_{\omega^\omega})\)
 * \((1,2,4,5,6,8)=\psi(\Omega_{\varepsilon_0})\)
 * \((1,2,4,5,7)=\psi(\Omega_\Omega)\)
 * \((1,2,4,5,7,8,10)=\psi(\Omega_{\Omega_\Omega})\)
 * \((1,2,4,6)=\psi(I)=\psi(M)\)
 * \((1,2,4,6,2,4,5,7,9)=\psi(I+\psi_I(I))=\psi(M+\psi_{\psi_M(M)}(M))\)
 * \((1,2,4,6,2,4,6)=\psi(I\times2)=\psi(M+\psi_M(M))\)
 * \((1,2,4,6,3)=\psi(I\times\omega)=\psi(M+\psi_{\psi_M(M+1)}(1))\)
 * \((1,2,4,6,3,5,7)=\psi(I^2)=\psi(M+\psi_{\psi_M(M+1)}(M))\)
 * \((1,2,4,6,4)=\psi(\Omega_{I+1})=\psi(M+\psi_M(M+1))\)
 * \((1,2,4,6,4,5)=\psi(\Omega_{I+\omega})=\psi(M+\psi_M(M+\omega))\)
 * \((1,2,4,6,4,6)=\psi(I_2)=\psi(M\times2)\)
 * \((1,2,4,6,5)=\psi(I_\omega)=\psi(M\times\omega)\)
 * \((1,2,4,6,5,7,9)=\psi(I_I)=\psi(M\times\psi_M(M))\)
 * \((1,2,4,6,6)=\psi(\Omega(2,0))=\psi(M^2)\)
 * \((1,2,4,6,7)=\psi(\Omega(\omega,0))=\psi(M^\omega)\)
 * \((1,2,4,6,7,9,11)=\psi(\Omega(I,0))=\psi(M^{\psi_M(M)})\)
 * \((1,2,4,6,8)=\psi(\Omega(1,0,0))=\psi(M^M)\)
 * \((1,2,4,6,8,9)=\psi(M^{M^\omega})\)
 * \((1,2,4,6,8,10)=\psi(M^{M^M})\)
 * \((1,2,5)=\psi(\Omega_{M+1})\)
 * \((1,2,5,2,5)=\psi(\Omega_{M+1}+\psi_M(\Omega_{M+1}))\)
 * \((1,2,5,3)=\psi(\Omega_{M+1}+\psi_{\psi_M(\Omega_{M+1}+1)}(1))\)
 * \((1,2,5,4)=\psi(\Omega_{M+1}+\psi_M(\Omega_{M+1}+1))\)
 * \((1,2,5,4,6)=\psi(\Omega_{M+1}+M)\)
 * \((1,2,5,4,7)=\psi(\Omega_{M+1}+\psi_{\Omega_{M+1}}(\Omega_{M+1}))\)
 * \((1,2,5,5)=\psi(\Omega_{M+1}\times2)\)
 * \((1,2,5,6)=\psi(\Omega_{M+1}\times\omega)\)
 * \((1,2,5,7)=\psi(\Omega_{M+1}\times M)\)
 * \((1,2,5,7,10)=\psi(\Omega_{M+1}\times\psi_{\Omega_{M+1}}(\Omega_{M+1}))\)
 * \((1,2,5,8)=\psi(\Omega_{M+1}^2)\)
 * \((1,2,5,8,11)=\psi(\Omega_{M+1}^{\Omega_{M+1}})\)
 * \((1,2,6)=\psi(\Omega_{M+2})\)
 * \((1,2,6,6)=\psi(\Omega_{M+2}\times2)\)
 * \((1,2,6,10)=\psi(\Omega_{M+2}^2)\)
 * \((1,2,7)=\psi(\Omega_{M+3})\)
 * \((1,2,8)=\psi(\Omega_{M+4})\)


 * \(f(n)\approx f_{\psi(\Omega_{M+\omega})}(n)\)
 * double primitive sequence number \(\approx f_{\psi(\Omega_{M+\omega})+1}(10)\)

Programs
Jdoodle version without detailed output:

Jdoodle version with detailed output:

Github: