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Pair sequence number is a the output of a program made by BashicuHyudora and posted at a googology-related thread of the Japanese site BBS 2ch.net in 2014.[1][2][3][4], and updated by BashicuHyudora in 2017 on Japanese googology wiki. The algorithm of the program is called the pair sequence system, a weak version of Bashicu's matrix system by the same creator. It is supposed to calculate number comparable to \(f_{\vartheta(\Omega_\omega)+1}(10)\). It is an extension of a system named the primitive sequence system, still by the same author, which generates a number approaching \(f_{\varepsilon_0+1}(10)\).[5]

A pair sequence is a finite sequence of pairs of nonnegative integers, for example (0,0)(1,1)(2,2)(3,3)(3,2). A pair sequence P works as a function from natural numbers to natural numbers, (though we write P[n] rather than P(n)), for example \(n \mapsto (0,0)(1,1)(2,2)(3,3)(3,2)[n]\) is a function. The function P[n] is usually approximated with a function of the form \(H_\alpha\) from the Hardy hierarchy (we note \(P = \alpha\)). For example, \((0,0)(1,1)(2,2)(3,3)(3,2) = \psi(\psi_1(\Omega_2))\).

Original BASIC Code Edit

In the following program, in the loop starting from "for D=0 to 9" and ending at "next", a number close of \(f_{\vartheta(\Omega_\omega)}(C)\) is generated. The program repeats this loop 10 times and finally output a number close of \(f_{\vartheta(\Omega_\omega)+1}(10)\).

dim A[∞],B[∞]:C=9
for D=0 to 9
 for E=0 to C
  A[E]=E:B[E]=E
 next
 for F=C to 0 step -1
  C=C*C
  for G=0 to C
   if A[F]=0 | (A[F-G]<A[F] & (B[F]=0 | B[F-G]<B[F])) then H=G:G=C
  next
  if B[F]=0 then I=0 else I=A[F]-A[F-H]
  for J=1 to C*H
   A[F]=A[F-H]+I:B[F]=B[F-H]:F=F+1
  next
 next
next
print C

Modified BASIC code Edit

The modified version was posted on Japanese version of this page on October 2, 2017 by Bashicu, the author of this program. Bashicu mentions that the original version does not reach \(\vartheta(\Omega_\omega)\) level as expected, with (0,0)(1,1)(2,2)(3,0)(2,1)(3,1)(4,1) as the counterexample, and this version will reach this level.

dim A[∞],B[∞]:C=9
for D=0 to 9
 for E=0 to C
  A[E]=E:B[E]=E
 next
 for F=C to 0 step -1
  C=C*C
  for G=0 to F
   if A[F]=0 | A[F-G]<A[F]-H  then
    if B[F]=0 | B[F-G]<B[F] then I=G:G=C else H=A[F]-A[F-G]
   endif
  next
  for J=1 to C*I
   A[F]=A[F-H]+H:B[F]=B[F-H]:F=F+1
  next
  H=0
 next
next
print C

Verification code Edit

The Bashicu matrix calculator[6] shows the calculation process of pair sequence system. BM1 corresponds to the original version.

Here are some examples of the calculation of some pair sequences. The algorithm is modified so that it always take n=2.

Corresponding ordinals Edit

Up to \(\varepsilon_0\) Edit

When all the values of the second row are 0, it is the same as the primitive sequence system. We have:

\begin{array}{ll} (0,0) &=& 1 \\ (0,0)(0,0) &=& 2 \\ (0,0)(0,0)(0,0) &=& 3 \\ (0,0)(1,0) &=& \omega \\ (0,0)(1,0)(0,0)(0,0) &=& \omega+2 \\ (0,0)(1,0)(0,0)(1,0) &=& \omega \cdot 2 \\ (0,0)(1,0)(1,0) &=& \omega^2 \\ (0,0)(1,0)(1,0)(0,0)(1,0) &=& \omega^2+\omega \\ (0,0)(1,0)(2,0) &=& \omega^\omega \\ (0,0)(1,0)(2,0)(3,0) &=& \omega^{\omega^\omega} \\ (0,0)(1,0)(2,0)(3,0)(4,0) &=& \omega^{\omega^{\omega^\omega}} \\ \end{array} (0,0)(1,1) has fundamental sequence as follows. Here, n is not changed.

\begin{array}{ll} (0,0)(1,1)[1] &=& (0,0)(1,0)[1] \\ (0,0)(1,1)[2] &=& (0,0)(1,0)(2,0)[2] \\ (0,0)(1,1)[3] &=& (0,0)(1,0)(2,0)(3,0)[3] \\ (0,0)(1,1)[4] &=& (0,0)(1,0)(2,0)(3,0)(4,0)[4] \\ \end{array} Therefore, \(\{\omega, \omega^\omega, \omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}}, \ldots\}\) and \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ \end{array}

Up to \(\varepsilon_1\) Edit

As for (0,0)(1,1)(1,0), \[(0,0)(1,1)(1,0)[4] = (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)[4]\] and the fundamental sequence is \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ (0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 2 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 3 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 4 \\ (0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1)(0,0)(1,1) &=& \varepsilon_0 \cdot 5 \\ \end{array} Therefore, \[(0,0)(1,1)(1,0) = \varepsilon_0 \cdot \omega\]

\[(0,0)(1,1)(1,0)(1,0)[2] = (0,0)(1,1)(1,0)(0,0)(1,1)(1,0)(0,0)(1,1)(1,0)[2]\] has fundamental sequence of \begin{array}{ll} (0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \\ (0,0)(1,1)(1,0)(0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \cdot 2 \\ (0,0)(1,1)(1,0)(0,0)(1,1)(1,0)(0,0)(1,1)(1,0) &=& \varepsilon_0 \cdot \omega \cdot 3 \\ \end{array}

Therefore, \[(0,0)(1,1)(1,0)(1,0) = \varepsilon_0 \cdot \omega^2 \] In this way, adding (1,0) to the end of the sequence makes the ordinal \(\omega\) times. Adding (1,0)(2,0) to the end of the sequence \[(0,0)(1,1)(1,0)(2,0)[4] = (0,0)(1,1)(1,0)(1,0)(1,0)(1,0)(1,0)[4]\] corresponds to multiplying \(\omega^\omega\) to the ordinal, and therefore \[(0,0)(1,1)(1,0)(2,0) = \varepsilon_0 \cdot \omega^\omega\]

As for (0,0)(1,1)(1,1), \[(0,0)(1,1)(1,1)[4] = (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)[4]\] and the following fundamental sequence is obtained. \begin{array}{ll} (0,0)(1,1) &=& \varepsilon_0 \\ (0,0)(1,1)(1,0)(2,1) &=& \varepsilon_0^2 \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1) &=& \varepsilon_0^{\varepsilon_0} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1) &=& \varepsilon_0^{\varepsilon_0^2} \\ (0,0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1) &=& \varepsilon_0^{\varepsilon_0^{\varepsilon_0}} \\ \end{array} Therefore, \[(0,0)(1,1)(1,1) = \varepsilon_1 = \psi(1) \]

Up to Feferman–Schütte ordinal = \(\Gamma_0\) Edit

Similar calculation results in:

\begin{eqnarray*} (0,0)(1,1)(2,0) &=& \varepsilon_{\omega} = \psi(\omega) \\ (0,0)(1,1)(2,0)(2,0) &=& \varepsilon_{\omega^2} = \psi(\omega^2) \\ (0,0)(1,1)(2,0)(3,0) &=& \varepsilon_{\omega^\omega} = \psi(\omega^\omega) \\ (0,0)(1,1)(2,0)(3,1) &=& \varepsilon_{\varepsilon_0} = \psi(\psi(0)) \\ (0,0)(1,1)(2,0)(3,1)(4,0)(5,1) &=& \varepsilon_{\varepsilon_{\varepsilon_0}} = \psi(\psi(\psi(0))) \\ (0,0)(1,1)(2,1) &=& \zeta_0 = \phi(2,0) = \psi(\Omega) \\ (0,0)(1,1)(2,1)(3,1) &=& \Gamma_0 = \phi(1,0,0) = \psi(\Omega^\Omega) \\ \end{eqnarray*}

Up to Large Veblen ordinal = \(\psi(\Omega^{\Omega^\Omega})\) Edit

\begin{eqnarray*} (0,0)(1,1)(2,1)(3,1)(4,0) &=& ψ(\Omega^{\Omega \cdot \omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1) &=& ψ(\Omega^{\Omega \cdot \Omega}) = ψ(\Omega^{\Omega^2}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(4,0) &=& ψ(\Omega^{\Omega^2 \cdot \omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(4,1) &=& ψ(\Omega^{\Omega^2 \cdot \Omega}) = ψ(\Omega^{\Omega^3}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,0) &=& ψ(\Omega^{\Omega^\omega}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1) &=& ψ(\Omega^{\Omega^\Omega}) \\ \end{eqnarray*}

Up to \(\vartheta(\Omega_\omega)\) Edit

\begin{eqnarray*} (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1) &=& ψ(\Omega^{\Omega^{\Omega^2}}) \\ (0,0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)(7,1) &=& ψ(\Omega^{\Omega^{\Omega^\Omega}}) \\ (0,0)(1,1)(2,2) &=& \psi(\varepsilon_{\Omega+1}) = \psi(\psi_1(0)) \\ (0,0)(1,1)(2,2)(0,0) &=& \psi(\psi_1(0))+1 \\ (0,0)(1,1)(2,2)(1,0) &=& \psi(\psi_1(0)) \omega \\ (0,0)(1,1)(2,2)(2,0) &=& \psi(\psi_1(0) \omega) \\ (0,0)(1,1)(2,2)(3,0) &=& \psi(\psi_1(\omega)) \\ (0,0)(1,1)(2,2)(3,0)(4,0) &=& \psi(\psi_1(\omega^\omega)) \\ (0,0)(1,1)(2,2)(3,0)(4,1) &=& \psi(\psi_1(\psi(0)))=\psi(\psi_1(\varepsilon_0)) \\ (0,0)(1,1)(2,2)(3,1) &=& \psi(\psi_1(\Omega)) \\ (0,0)(1,1)(2,2)(3,2) &=& \psi(\psi_1(\Omega_2)) \\ (0,0)(1,1)(2,2)(3,3) &=& \psi(\psi_1(\psi_2(0))) \\ (0,0)(1,1)(2,2)(3,3)(4,4) &=& \psi(\psi_1(\psi_2(\psi_3(0)))) \\ (0,0)(1,1)(2,2)(3,3)...(9,9) &=& \psi(\psi_1(\psi_2(\psi_3(\psi_4(\psi_5(\psi_6(\psi_7(\psi_8(0))))))))) \\ \end{eqnarray*}

By defining \(Pair(n) = (0,0)(1,1) \ldots (n,n)[n]\), one has \[Pair(n) \approx f_{\vartheta(\Omega_\omega)}(n)\]

Sources Edit

  1. First version of BASIC program
  2. Modified version of BASIC program
  3. Name of the system and the number
  4. Blog post which introduces pair sequence system
  5. BASIC program of \(\varepsilon_0\) level and a blog post which introduces it
  6. Bashicu matrix calculator

See also Edit

Googology in Asia

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Pair sequence number · Bashicu matrix system
Indian counting system: Lakh · Crore · Uppala · Bodhisattva
Chinese and Japanese counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Buddhist text: Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra
Other: Taro's multivariable Ackermann function · Sushi Kokuuhen

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