User blog:P進大好きbot/FGH along my OCF with weakly 1-Mahlo

This is an English translation of a Japanese blog post submitted to a Japanese googological event.

= Notation =

I define a set \(S\) of strings in the following recursive way: I put \(\textrm{一} := \textrm{桜}(\textrm{花}(\textrm{閃},\textrm{閃}),\textrm{閃})\) and \(\textrm{王} := \textrm{桜}(\textrm{花}(\textrm{閃},\textrm{閃}),\textrm{一})\).
 * 1) \(\textrm{閃},\textrm{華} \in S\).
 * 2) For all \(a,b \in S\), \(a b \in S\).Here \(a b\) denotes the concatenation of \(a\) and \(b\).
 * 3) For all \(a,b \in S\), \(\textrm{花}(a,b) \in S\).
 * 4) For all \(b \in S\), \(\textrm{散}(b) \in S\).
 * 5) For all \(a,b \in S\), \(\textrm{桜}(a,b) \in S\).

I define a subset \(PS \subset S\) in the following way:
 * 1) \(\textrm{閃},\textrm{華} \in PS\).
 * 2) For all \(a,b \in S\), \(\textrm{花}(a,b) \in PS\).
 * 3) For all \(b \in S\), \(\textrm{散}(b) \in PS\).
 * 4) For all \(a,b \in S\), \(\textrm{桜}(a,b) \in PS\).

= Preparation =

I define a map \begin{eqnarray*} \partial \colon S & \to & S \\ a & \mapsto & \partial(a), \end{eqnarray*} \(2\)-ary relations \(a \leq b\) and \(a < b\) on \(a,b \in S\), and a \(3\)-ary relation \(a \triangleleft_b c\) on \(a,b,c \in S\) simultaneously in the foolowing recursive way:
 * Definition of \(\partial(a)\)


 * 1) If \(a = \textrm{華}\), then \(\partial(a) := \textrm{閃}\).
 * 2) If \(a = \textrm{花}(b,\textrm{閃})\) for some \(b \in S\), then \(\partial(a) := b\).
 * 3) If \(a = \textrm{花}(b,\textrm{一})\) for some \(b \in S\), then \(\partial(a) := \textrm{花}(b,\textrm{閃})\).
 * 4) Suppose \(a = \textrm{花}(b,c \textrm{一})\) for some \(b,c \in S\).
 * 5) If \(c < \textrm{花}(b,c)\), then \(\partial(a) := \textrm{花}(b,c)\).
 * 6) If \(c < \textrm{花}(b,c)\) does not hold, then \(\partial(a) := c\).
 * 7) If \(a = \textrm{散}(\textrm{閃})\), then \(\partial(a) := \textrm{閃}\).
 * 8) If \(a = \textrm{散}(b \textrm{一})\) for some \(b \in S\), then \(\partial(a) := \textrm{散}(b)\).
 * 9) Otherwise, \(\partial(a) := \textrm{閃}\).
 * Definition of \(\leq\) and \(<\)


 * 1) For all \(a,b \in S\), \(a \leq b\) is equivalent to \(a < b\) or \(a = b\).
 * 2) For all \(a \in S\), \(a < \textrm{閃}\) never holds.
 * 3) For all \(b \in S\), if \(b \neq \textrm{閃}\), then \(\textrm{閃} < b\).
 * 4) For all \(a,c \in PS\) and \(b,d \in S\), \(a b < c d\) is equivalent to that either one of the following holds:
 * 5) \(a = c\) and \(b < d\).
 * 6) \(a < c\).
 * 7) For all \(a,c \in PS\) and \(b \in S\), \(a b < c\) is equivalent to \(a < c\).
 * 8) For all \(a,b \in PS\) and \(c \in S\), \(a < b c\) is equivalent to \(a \leq b\).
 * 9) For all \(a \in PS\), if \(a \neq \textrm{華}\), then \(a < \textrm{華}\).
 * 10) For all \(b \in PS\), \(\textrm{華} < b\) never holds.
 * 11) For all \(a,b,c,d \in S\), \(\textrm{花}(a,b) < \textrm{花}(c,d)\) is equivalent to that either one of the following holds:
 * 12) \(a = c\) and \(b < d\).
 * 13) \(a < c\) and \(b \leq \textrm{花}(c,d)\).
 * 14) \(c < a\) and \(\textrm{花}(a,b) < d\).
 * 15) For all \(a,b,d \in S\), \(\textrm{花}(a,b) < \textrm{散}(d)\) is equivalent to \(a < \textrm{散}(d)\) and \(b \leq \textrm{散}(d)\).
 * 16) For all \(b,c,d \in S\), \(\textrm{散}(b) < \textrm{花}(c,d)\) is equivalent to \(\textrm{散}(b) \leq c\) or \(\textrm{散}(b) < d\).
 * 17) For all \(a,b,c,d \in S\), \(\textrm{花}(a,b) < \textrm{桜}(c,d)\) is equivalent to \(\textrm{花}(a,b) < c\) and \(\textrm{花}(a,b) \triangleleft_c d\).
 * 18) For all \(a,b,c,d \in S\), \(\textrm{桜}(a,b) < \textrm{花}(c,d)\) is equivalent to that either one of the following holds:
 * 19) \(a \leq \textrm{花}(c,d)\).
 * 20) \(\textrm{花}(c,d) \triangleleft_a b\) does not hold.
 * 21) For all \(b,d \in S\), \(\textrm{散}(b) < \textrm{散}(d)\) is equivalent to \(b < d\).
 * 22) For all \(b,c,d \in S\), \(\textrm{散}(b) < \textrm{桜}(c,d)\) is equivalent to \(\textrm{散}(b) < c\) and \(\textrm{散}(b) \triangleleft_c d\).
 * 23) For all \(a,b,d \in S\), \(\textrm{桜}(a,b) < \textrm{散}(d)\) is equivalent to that either one of the following holds:
 * 24) \(a \leq \textrm{散}(d)\).
 * 25) \(\textrm{散}(d) \triangleleft_a b\) does not holds.
 * 26) For all \(a,b,c,d \in S\), \(\textrm{桜}(a,b) < \textrm{桜}(c,d)\) is equivalent to that either one of the following holds:
 * 27) \(a = c\) and \(b < d\).
 * 28) \(a < c\) and \(a < \textrm{桜}(c,d)\).
 * 29) \(c < a\) and \(\textrm{桜}(a,b) < c\).
 * Definition of \(\triangleleft\)


 * 1) For all \(c,d \in S\), \(\textrm{閃} \triangleleft_c d\).
 * 2) For all \(c,d \in S\), \(\textrm{華} \triangleleft_c d\).
 * 3) For all \(a \in PS\) and \(b,c,d \in S\), \(a b \triangleleft_c d\) is equivalent to \(a \triangleleft_c d\) and \(b \triangleleft_c d\).
 * 4) For all \(a,b,c,d \in S\), \(\textrm{花}(a,b) \triangleleft_c d\) is equivalent to \(a \triangleleft_c d\) and \(b \triangleleft_c d\).
 * 5) For all \(b,c,d \in S\), \(\textrm{散}(b) \triangleleft_c d\) is equivalent to \(b \triangleleft_c d\).
 * 6) For all \(a,b,c,d \in S\), \(\textrm{桜}(a,b) \triangleleft_c d\) is equivalent to that either one of the following holds:
 * 7) \(\textrm{桜}(a,b) \leq \partial(c)\).
 * 8) \(a < c\) and \(a \triangleleft_c d\).
 * 9) \(c \leq a\), \(b < d\), \(a \triangleleft_c d\), and \(b \triangleleft_c d\).

I define a map \begin{eqnarray*} \otimes \colon \mathbb{N} \times S & \to & S \\ (n,a) & \mapsto & \otimes(n,a) \end{eqnarray*} in the following recursive way:
 * 1) If \(n = 0\), then \(\otimes(n,a) := \textrm{閃}\).
 * 2) If \(n = 1\), then \(\otimes(n,a) := a\).
 * 3) If \(n > 1\), then \(\otimes(n,a) := \otimes(n-1,a) a\).

= Fundamental Sequence =

I define maps \begin{eqnarray*} \textrm{dom} \colon S & \to & S \\ a & \mapsto & \textrm{dom}(a) \end{eqnarray*} and \begin{eqnarray*} [ \, \ ] \colon S \times S & \to & S \\ (a,n) & \mapsto & [a,n] \end{eqnarray*} simultaneously in the following recursive way:
 * 1) Suppose \(a = \textrm{閃}\).
 * 2) \(\textrm{dom}(a) := \textrm{閃}\).
 * 3) \([a,n] := \textrm{閃}\).
 * 4) Suppose \(a = \textrm{華}\), \(a = \textrm{花}(b,\textrm{閃})\) for some \(b \in S\), \(a = \textrm{花}(b,c \textrm{一})\) for some \(b,c \in S\), or \(a = \textrm{散}(b)\) for some \(b \in S\).
 * 5) \(\textrm{dom}(a) := a\).
 * 6) \([a,n] := n\).
 * 7) Suppose \(a = b c\) for some \(b \in PS\) and \(c \in S\).
 * 8) \(\textrm{dom}(a) = \textrm{dom}(c)\).
 * 9) If \([c,n] = \textrm{閃}\), then \([a,n] := b\).
 * 10) If \([c,n] \neq \textrm{閃}\), then \([a,n] := b [c,n]\).
 * 11) Suppose that none of the conditions above holds.
 * 12) Suppose \(a = \textrm{花}(b,c)\) for some \(b,c \in S\).
 * 13) \(\textrm{dom}(a) := \textrm{dom}(c)\).
 * 14) \([a,n] := \textrm{花}(b,[c,n])\).
 * 15) Suppose \(a = \textrm{散}(b)\) for some \(b \in S\).
 * 16) \(\textrm{dom}(a) := \textrm{dom}(b)\).
 * 17) \([a,n] := \textrm{散}([b,n])\).
 * 18) Suppose \(a = \textrm{桜}(b,c)\) for some \(b,c \in S\).
 * 19) Suppose \(\textrm{dom}(c) = \textrm{閃}\).
 * 20) Suppose \(b = \textrm{花}(d,e)\) for some \(d,e \in S\).
 * 21) Suppose \(\textrm{dom}(d) = \textrm{閃}\).
 * 22) Suppose \(\textrm{dom}(e) = \textrm{閃}\).
 * 23) \(\textrm{dom}(a) := \textrm{一}\).
 * 24) \([a,n] := \textrm{閃}\).
 * 25) Suppose \(\textrm{dom}(e) \neq \textrm{閃}\).
 * 26) \(\textrm{dom}(a) := \textrm{王}\).
 * 27) Suppose \(n = \otimes(m,\textrm{一})\) for some \(m \in \mathbb{N}\).
 * 28) If \([e,\textrm{閃}] < \textrm{花}(\textrm{閃},[e,\textrm{閃}])\), then \([a,n] := \otimes(m,\textrm{花}(\textrm{閃},[e,\textrm{閃}]))\).
 * 29) If \([e,\textrm{閃}] < \textrm{花}(\textrm{閃},[e,\textrm{閃}])\) does not hold, then \([a,n] := \otimes(m,[e,\textrm{閃}])\).
 * 30) If \(n\) is not expressed as \(\otimes(m,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \textrm{閃}\).
 * 31) Suppose \(\textrm{dom}(d) = \textrm{一}\).
 * 32) \(\textrm{dom}(a) := \textrm{王}\).
 * 33) If \(n = \otimes(m+1,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \textrm{花}([d,\textrm{閃}],[a,\otimes(m,\textrm{一})])\).
 * 34) Suppose \(n\) is not expressed as \(\otimes(m+1,\textrm{一})\) for some \(m \in \mathbb{N}\).
 * 35) If \(\textrm{dom}(e) = \textrm{閃}\), then \([a,n] := \textrm{閃}\).
 * 36) If \(\textrm{dom}(e) \neq \textrm{閃}\), then \([a,n] := \textrm{花}(d,[e,\textrm{閃}]) \textrm{一}\).
 * 37) Suppose \(\textrm{dom}(d) \neq \textrm{閃},\textrm{一}\).
 * 38) \(\textrm{dom}(a) := \textrm{dom}(d)\).
 * 39) If \(\textrm{dom}(e) = \textrm{閃}\), then \([a,n] := \textrm{桜}(\textrm{花}([d,n],\textrm{閃}),\textrm{閃})\).
 * 40) If \(\textrm{dom}(e) \neq \textrm{閃}\), then \([a,n] := \textrm{桜}(\textrm{花}([d,n],\textrm{花}(d,[e,\textrm{閃}]) \textrm{一}),\textrm{閃})\).
 * 41) Suppose \(b = \textrm{散}(d)\) for some \(d \in S\).
 * 42) \(\textrm{dom}(a) := \textrm{王}\).
 * 43) If \(n = \otimes(m+1,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \textrm{花}([a,\otimes(m,\textrm{一})],\textrm{閃})\).
 * 44) Suppose that \(n\) is not expressed as \(\otimes(m+1,\textrm{一})\) for some \(m \in \mathbb{N}\).
 * 45) If \(\textrm{dom}(d) = \textrm{閃}\), then \([a,n] := \textrm{閃}\).
 * 46) If \(\textrm{dom}(d) \neq \textrm{閃}\), then \([a,n] := \textrm{散}([d,\textrm{閃}])\).
 * 47) Suppose \(b = \textrm{華}\).
 * 48) \(\textrm{dom}(a) := \textrm{王}\).
 * 49) If \(n = \otimes(m+1,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \textrm{散}([a,\otimes(m,\textrm{一})],\textrm{閃}))\).
 * 50) If \(n\) is not expressed as \(\otimes(m+1,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \textrm{閃})\).
 * 51) Suppose that none of the conditions above holds.
 * 52) \(\textrm{dom}(a) := \textrm{閃}\).
 * 53) \([a,n] := \textrm{閃}\).
 * 54) Suppose \(\textrm{dom}(c) = \textrm{一}\).
 * 55) \(\textrm{dom}(a) := \textrm{王}\).
 * 56) If \(n = \otimes(m,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \otimes(m,\textrm{桜}(b,[c,\textrm{閃}]))]\).
 * 57) If \(n\) is not expressed as \(\otimes(m,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \textrm{閃}\).
 * 58) Suppose \(\textrm{dom}(c) \neq \textrm{閃},\textrm{一}\).
 * 59) Suppose \(\textrm{dom}(c) < b\).
 * 60) \(\textrm{dom}(a) := \textrm{dom}(c)\).
 * 61) \([a,n] := \textrm{桜}(b,[c,n])\).
 * 62) Suppose that \(\textrm{dom}(c) < b\) does not hold.
 * 63) \(\textrm{dom}(a) := \textrm{王}\).
 * 64) For each \(m \in \mathbb{N}\), I define \(c_m \in S\) in the following recursive way:
 * 65) If \(m = \textrm{閃}\), then \(c_m := \textrm{桜}(\textrm{dom}(c),\textrm{閃})\).
 * 66) If \(m \neq \textrm{閃}\), then \(c_m := \textrm{桜}(\textrm{dom}(c),[c,c_{m-1}])\).
 * 67) If \(n = \otimes(m,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \textrm{桜}(b,[c,c_m])\).
 * 68) If \(n\) is not expressed as \(\otimes(m,\textrm{一})\) for some \(m \in \mathbb{N}\), then \([a,n] := \textrm{閃}\).

= Large Number =

I define a computable partial function \begin{eqnarray*} \textrm{剣伎} \colon (a,m,n) \mapsto \textrm{剣伎「} a,m,n \textrm{」} \end{eqnarray*} on \(S \times \mathbb{N} \times \mathbb{N}\) in the following recursive way:
 * 1) If \(m = 0\), then \(\textrm{剣伎「} a,m,n \textrm{」} := n\).
 * 2) Suppose \(m = 1\).
 * 3) If \([a,\textrm{閃}] = a\), then \(\textrm{剣伎「} a,m,n \textrm{」} := n+1\).
 * 4) If \([a,\textrm{閃}] \neq a\), then \(\textrm{剣伎「} a,m,n \textrm{」} := \textrm{剣伎「} [a,\otimes(n,\textrm{一})],n,n \textrm{」}\).
 * 5) If \(m > 1\), then \(\textrm{剣伎「} a,m,n \textrm{」} = \textrm{剣伎「} a,m-1,\textrm{剣伎「} a,1,n \textrm{」} \textrm{」}\).

I name the computable large number \(\textrm{剣伎「} \textrm{桜}(\textrm{花}(\textrm{閃},\textrm{閃}),\textrm{華} \textrm{華} \textrm{華}),3,3) \textrm{」}\) as "剣伎「桜花閃々三華数」".

= Aalysis =

I denote by \(\psi\) the OCF with weakly \(1\)-Mahlo, and \(f\) the FGH with respect to the canonical recursive system of fundamental sequences introduced this Japanese blog post.

The computable large number 剣伎「桜花閃々三華数」is greater than \(f_{\psi_{\Omega}(M_1 \times 4)}(3)\) with respect to the OCF and the corresponding system of fundamental sequences.

The limit of this notation is \(\psi_{\Omega}(M_1 \times \omega)\), and hence is greater than almost all other well-defined computable notations. I note that many of "notations" in this wiki lacks the well-definedness or the computability. For example, a simple use of an OCF does not yield a computable notation or a computable large number, even though the creators do not understand reasons by the lack of the knowledge of the definition of the computability or the notion of a Turing machine. One of the aim of this blog post is to have others to understand the difference of an OCF and a computable notation.

Readers can understand that the computation process never uses actual ordinals. Using ordinals or \(\in\)-relation causes uncomputability, even if it is defined in an inductive way. That is why mathematicians usually create ordinal notation systems instead of actual ordinals. Well, unfortunately, many googologists do not even know the definition of the notion of an ordinal notation by confounding it with an OCF and by the lack of the knowledge of the definition of the notion of a primitive recursive well-ordering. For the detailed explanation of the difference, see my blog post.

Therefore this computable notation is not my strongest one. For a stronger computable notation, see my other blog. The growth rate of the limit function (restricted to its domain) of that notation goes beyond \(\textrm{PTO}(T)\), where \(T\) is the formal theory we are working on (usually \(\textrm{ZFC}\) set theory). Therefore I guess that it yields the largest computable large number among ever defined well-defined one.