User blog:Rpakr/Analysis of Taranovsky's C by AAAgoogology (Translation by me and koteitan)

The following is the translation of AAAgoogology's user page "TaranovskyのC表記の解析" by me and koteitan.

Introduction
In this page I will show my analysis results of Taranovsky’s C. There could be mistakes in this page. Also, there are many parts in which the explanation is not very complete, so please comment if you don’t understand something shown in this page.

It is hard to predict the rules to find the fundamental sequence, so I changed them many times by comparing it to the fundamental sequences computed automatically by a program. The current ruleset is verified to be correct for ordinals containing \(\Omega_2\) and up to 9 \(C\)‘s, ordinals containing \(\Omega_3\) and up to 8 \(C\)’s, ordinals containing \(\Omega_4\) and up to 8 \(C\)’s, and ordinals containing \(\Omega_5\) and up to 7 \(C\)’s using the program, but the program can be wrong, and there might be mistakes in ordinals more complicated than the ones that were checked. If you find a mistake, please tell me in the comment.

The three things I think to be significant about the contents of this page are:
 * 1) Made the nested parentheses (which are hard to read) into tree structures which are easier to read.
 * 2) Combined the definition of the definition corresponding to \(n\)-built from below for all \(n\)-th ordinal notation system.
 * 3) Made a conjecture about the method of finding fundamental sequences that is not brute forcing.

For 1., there are people that has tried to express C structure using trees this previously, but I made it easier to understand because I fixed the direction of the edges. Also, we can write tree structures using the method of writing matrices in LaTeX. For 2., now we can understand the notation without having to think about which ordinal notation system it is. The “numbering of nodes” in this article corresponds to \(n\)-built from below. For 3., a brute force method of finding fundamental sequences is already given by Hyp cos, but to better understand the structure of ordinals using Taranovsky’s C, I think it is important to make a method to find fundamental sequences without brute forcing. The analysis results haven’t been checked enough, so will be published under my user page, but you may reprint the contents to other pages. Updates will be posted in the user blog.

Taranovsky’s C
In this page I will introduce my analysis results of the Main Ordinal Notation System published here. I will introduce the Combined System, which combines \(n\)-th ordinal notation system for all n.

Correspondence to tree structure
In this notation there are constants \(\0), \(\Omega_n\) (\(n\ge1\)) and a binary function \(C(a,b)\). In order to make the complicated structure more readable, I will write them in tree structure. \[ C(a,b)= \begin{array}{ccc} a& & \\ C&-&b \end{array} \] \(C\) corresponds to a node with two children. The first argument goes up, and the second argument goes to the right. The root node of the tree is at the bottom left. The following holds for \(\Omega_n\): \[ \Omega_n=C(\Omega_{n+1},0)= \begin{array}{ccc} \Omega_{n+1}& & \\ C&-&0 \end{array} \]

Standard form
Multiple combinations of \(0\), \(\Omega_n\) and \(C\) can equal to the same ordinal. In Taranovsky’s C every ordinal has only 1 standard form. Expressions that are not standard can be turned into standard form expressions using three kinds of transformations: replacement of \(\Omega\), minimizing the second argument and maximizing the first argument. An expression that none of these three can be applied is a standard expression.

Replacement of \(\Omega\)
\[ \begin{array}{ccc} \Omega_{n+1}& & \\ C&-&0 \end{array} \rightarrow\Omega_n \] Do the replacement above.

Minimizing the second argument
If \(C(b,c)\) is standard and \(a>b\), do the replacement below: \[ \begin{array}{ccccc} a& &b& & \\ C&-&C&-&c \end{array} \rightarrow \begin{array}{ccccc} a& & \\ C&-&c \end{array} \]

Comparison
Consider comparing \[ a= \begin{array}{ccccccc} a_m& &a_2& &a_1& & \\ C &\cdots&C &-&C &-&0 \end{array} \] and \[ b= \begin{array}{ccccccc} b_n& &b_2& &b_1& & \\ C &\cdots&C &-&C &-&0 \end{array \] If \(a_1=b_1,\ldots,a_{i-1}=b_{i-1},a_i<b_i\) or \(m<n,a_1=b_1,\ldots,a_m=b_m\), \(a<b\) is true. \(a=0\) corresponds to \(m=0\). If \(m<n\) then \(\Omega_m<\Omega_n\). If two expressions can’t be compared using the rule above, do replacement of \(\Omega\) and compare.

Successor ordinal
Successor ordinal \(a+1\), can be expressed as: \[ a+1=C(0,a)= \begin{array}{ccc} 0& & \\ C&-&a \end{array} \]

Limit ordinals and fundamental sequences 1
Limit ordinals that have fundamental sequences can have different fundamental sequences, but to define fast-growing hierarchy we need to specify one fundamental for every such limit ordinals. \[ \begin{array}{ccc} 0& & \\ C&-&a\\ C&-&b \end{array} \] The above ordinal is a limit ordinal and it has a fundamental sequence. The nth term of its fundamental sequence is \[ \begin{array}{ccccc} a& &a& & \\ C&\cdots(n\ \mathrm{times})\cdots&C&-&b \end{array} \]

If the nth term of the fundamental sequence of \(a\) is \(a_n\), The \(n\)th term of the fundamental sequence of the ordinal below \[ C(a,b)= \begin{array}{ccc} a& & \\ C&-&b \end{array} \] is \[ C(a_n,b)= \begin{array}{ccc} a_n& & \\ C &-&b \end{array} \]

\(\Omega_n\) is a limit ordinal but it does not have a fundamental sequence. The fundamental sequence of \(C(a,b)\) when \(a\) is a limit ordinal that does not have fundamental sequence will be explained later in this page.

Numbering of the nodes
For ordinal \(C(a,b)\), label numbers to the nodes of the tree of \(C(a,b)\) following these rules.
 * 1) The value of every label must be an integer.
 * 2) When you go right or up in an edge, the number must not decrease.
 * 3) Nodes smaller than \(C(a,b)\) and their children do not get labelled.
 * 4) If a node does not have a label in either of its two children, it does not get a label.
 * 5) All other nodes must have a label.
 * 6) If the ordinal corresponding to node \(c\) is the largest in the nodes from the root to \(c\) (including the root and \(c\)), the label of node \(c\) must be bigger than the label of the node below \(c\).
 * 7) If \(\Omega_n\) is the largest ordinal in the nodes from the root to \(\Omega_n\), \(\Omega_n\) has label \(n\).
 * 8) If \(\Omega_n\) is not the largest ordinal in the nodes from the root to \(\Omega_n\), replace \(\Omega_n\) with \(C(\Omega_{n+1},0)\).
 * 9) All nodes must have the largest label it can have without violating any other rules.

Maximizing the first argument
If \(a\) and \(b\) in \\(C(a,b)\) are standard, label the nodes of \(C(a,b)\). If the label of the root is not negative the first argument is maximized. If not, repeat the following procedure to get the first argument maximized.
 * 1) Find the \(\Omega_n\) at the most right and bottom which is determining the label of the root and replace it with \(\Omega_{n+1}\).
 * 2) Remove the nodes with only one child node.

Limit ordinals and fundamental sequences 2
When \(a\) in \(C(a,b)\) is a limit ordinal that doesn’t have a fundamental sequence, label only the root node and the node above it. \(C(a,b)\) does not have a fundamental sequence if the label of the root node is positive, and it has a fundamental sequence if the label of the root node is 0. The fundamental sequence is predicted to be given with the following procedure.
 * 1) Definitions
 * 2) * Let \(a_i\) be the lowermost node with label \(i\). (\(a_0=C(a,b)\), \(a_1=a\).)
 * 3) * Let \(b_{i+1}\) be the result of the application of the same method as the maximization of the first argument to \(a_i\).
 * 4) * Let \(j\) be the minimum number that satisfies \(\Omega_j>a_0\).
 * 5) * Let \(f(x)=C(x,a_0)\). For \(i\le j\), let \(c_i=f^{j-i}(\Omega_j)\). (minimize the second argument as well.)
 * 6) * For \(i>j\), let \(c_i=\Omega_i\).
 * 7) Fundamental sequence type A
 * 8) * If \(b_1\) is included in the root node and the node above it, replace \(b_1\) with \(0\) and repeat the path from the \(a_0\) to \(b_1\) \(n\) times to find the \(n\)th term of the fundamental sequence. (Type A)
 * 9) Fundamental sequence type A’
 * 10) * Otherwise, take the minimum \(i\) that satisfies \(a_{i+1}=b_{i+1}\).
 * 11) * If \(a_{i+1}\ne c_{i+1}\), go to 4.
 * 12) * If \(a_{i+1}=c_{i+1}\), do the same replacement as the maximization of the first argument until \(b_1\) no longer contains descendant nodes greater than or equal to \(a_i\) and less than \(a_{i+1}\), except the descendants of nodes less than \(a_0\) or the descendants of nodes greater than or equal to \(a_{i+1}\).
 * 13) * If the root node and the node above it includes \(b'_1\), replace \(b'_1\) with \(0\) and repeat the path from \(a_0\) to \(b'_1\) \(n\) times to find the \(n\)th term of the fundamental sequence. (Type A’)
 * 14) * Otherwise, go to 4.
 * 15) Replacement of the second argument in fundamental sequence type B (the replacement of the second argument is not done in types A and A’)
 * 16) * If the node under \(a_{i+1}\) (\(x\)) is less than \(a_{i+1}\), consider the ordinal in which the node to the right of \(x\) (\(y)\) is replaced with the node to the right of\(a_0\).
 * 17) * Minimize the second argument assuming infinite repetition between \(a_i\) and \(a_{i+1}\). If it is greater than the original \(y\), replace \(y\) with the node to the right of \(a_0\) in the fundamental sequence. (This is called “replacement of the second argument”).
 * 18) * Otherwise, if the node under \(x\) exists and it is less than \(x\), let it be the new \(x\).
 * 19) * Repeat the following until the replacement of \(y\) is done or the node under \(x\) is greater than \(x\) or \(x\) reaches the root node.
 * 20) * Do the same procedure for \(a_i\).
 * 21) Fundamental sequence type B
 * 22) * Do the replacement of the second argument, replace \(a_{i+1}\) with \(0\) and repeat the path from \(a_i\) to \(a_{i+1}\) \(n\) times, to find the \(n\) th term of the fundamental sequence.

The examples of the procedure will be described later.

Difference between the two node labelling methods
The label of a node may change depending on whether all nodes are labelled or only the nodes that follow above are labelled. Labels will be shown in parentheses. From here, all nodes will be labelled unless stated otherwise because labelling all nodes gives the same labels if the label of the root node is 0 when labeling root nodes and nodes above it in a standard form ordinal.

\[ \begin{array}{ccccc} & &\Omega_4(4)& & \\ & &|& & \\ & &C(3)&-&\Omega_3(3)\\ & &|& & \\ \Omega_3(3)& &C(2)&-&\Omega_3(3)\\ C(1)&-&C(1)&-&0\\ C(0)&-&0& & \end{array} \qquad \begin{array}{ccccc} & &\Omega_4& & \\ & &|& & \\ & &C&-&\Omega_3\\ & &|& & \\ \Omega_3(3)& &C&-&\Omega_3\\ C(2)&-&C&-&0\\ C(1)&-&0& & \end{array} \]

Notation of fundamental sequences
The fundamental sequence will be shown as follows:

\[ \begin{array}{ccccccc} \underline{\Omega_3}& & & &\Omega_3& & \\ \underline{C}&-&\Omega_2& &C&-&\Omega_2\\ C&\rightarrow&0& &C&-&\Omega_2\\ C&\leftarrow&-&-&C&-&0\\ \uparrow& & & & & & \\ C&-&\Omega_3& & & & \end{array} \] The ordinal above \(\uparrow\) has a fundamental sequence. Replace the node to the right of \(\rightarrow\) with the node to the right of \(\leftarrow\), repeat between the two underlined nodes \(n\) times, then replace the upper underlined node above with \(0\) to find the fundamental sequence. The structure between the two underlined nodes will be repeated.

Specifically, The fundamental sequence is as follows. \[ \begin{array}{ccccccccccc} 0& & & &　& & & & & & \\ C&-&\Omega_2& & & & & & & & \\ \vdots& & & & & & & & & & \\ C&-&\Omega_2& &\Omega_3& & & & & & \\ C&-&\Omega_2& &C&-&\Omega_2& &\Omega_3& & \\ C&-&\Omega_2& &C&-&\Omega_2& &C&-&\Omega_2\\ C&-&-&-&C&-&0& &C&-&\Omega_2\\ C&-&-&-&-&-&-&-&C&-&0\\ C&-&\Omega_3& & & & & & & & \end{array} \]

Examples in 0th ordinal notation system
The examples are shown in ascending order.

\[ 1= \begin{array}{ccc} 0& & \\ C&-&0 \end{array} \] \[ 2= \begin{array}{ccccc} 0& &0& & \\ C&-&C&-&0 \end{array} \] \[ \omega= \begin{array}{ccc} 0& & \\ C&-&0\\ C&-&0 \end{array} \] \[ \omega+1= \begin{array}{ccccc} & &0& & \\ & &|& & \\ 0& &C&-&0\\ C&-&C&-&0 \end{array} \] \[ \omega\cdot2= \begin{array}{ccccccc} 0& & & &0& & \\ C&-&0& &C&-&0\\ C&-&-&-&C&-&0 \end{array} \] \[ \omega^2= \begin{array}{ccccc} 0& &0& & \\ C&-&C&-&0\\ C&-&0& & \end{array} \] \[ \omega^\omega= \begin{array}{ccc} 0& & \\ C&-&0\\ C&-&0\\ \uparrow& & \\ C&-&0 \end{array} \]

Replacing the horizontally connected nodes with one node gives a tree structure that corresponds to the hydra in the hydra game.

Examples in 1st ordinal notation system
Label the nodes to find the fundamental sequence of \[ \epsilon_0= \begin{array}{ccc} \Omega_1& & \\ C&-&0 \end{array} \].

\[ \begin{array}{lcc} \Omega_1(1)=a_1\\ C(0)=a_0&-&0 \end{array} \] \[ b_1= \begin{array}{ccc} \Omega_2\\ C&-&0 \end{array} =\Omega_1. \] So, the fundamental sequence repeats between \(a_0\) and \(b_1\) and is \[ \begin{array}{ccc} \underline{\Omega_1}& & \\ \underline{C}&-&0 \end{array} \]. Specifically, \[ \begin{array}{ccc} 0& & \\ C&-&0\\ \vdots& & \\ C&-&0\\ C&-&0 \end{array} \].

\[ \begin{array}{ccc} \Omega_1& & \\ C&-&0\\ C&-&0 \end{array} \] is not standard form. Labelling the nodes gives the following: \[ \begin{array}{ccc} \Omega_1(1)& & \\ C(0)&-&0\\ C(-1)&-&0 \end{array} \] Rewriting into the standard form will be follows. \[ \begin{array}{ccc} \Omega_1& & \\ C&-&0\\ C&-&0 \end{array} \rightarrow \begin{array}{ccc} \Omega_2& & \\ C&-&0\\ C&-&0 \end{array} \rightarrow \begin{array}{ccc} \Omega_1& & \\ C&-&0 \end{array} \]

\[ \epsilon_0\cdot2= \begin{array}{ccccccc} \underline{\Omega_1}& & & & & & \\ \underline{C}&-&0& &\Omega_1& & \\ \uparrow& & & &|& & \\ C&-&-&-&C&-&0 \end{array} \] \[ \epsilon_0\cdot\omega= \begin{array}{ccccccccc} 0& &\Omega_1& & & & & & \\ C&-&C &-&0& &\Omega_1& & \\ C&-&- &-&-&-&C&-&0 \end{array} \] \[ \epsilon_1= \begin{array}{ccccc} \underline{\Omega_1}& &\Omega_1& & \\ \underline{C}&-&C&-&0 \end{array} \]. Specifically the fundamental sequence is as follows; \[ \begin{array}{ccccc} 0& &\Omega_1& & \\ C&-&C&-&0\\ \vdots& & & & \\ C&-&C&-&0\\ C&-&C&-&0 \end{array} \] The procedure which gives the fundamental sequence is as follows; \[ \epsilon_1= \begin{array}{lcccc} \Omega_1(1)=a_1& &\Omega_1& & \\ C(0)=a_0&-&C&-&0 \end{array} \] \[ b_1= \begin{array}{ccccc} \Omega_2& &\Omega_1& & \\ C&-&C&-&0 \end{array} = \begin{array}{ccc} \Omega_2\\ C&-&0 \end{array} =\Omega_1 \] So, the fundamental sequence repeats between \(a_0\) and \(b_1\). For ordinals less than \(\Omega_1\), the procedure which gives \(b_1\) will be omitted because \(b_1=\Omega_1\).
 * & &\Omega_1& & \\
 * & &\Omega_1& & \\
 * & &\Omega_1& & \\

\[ \epsilon_\omega= \begin{array}{ccc} 0& & \\ C&-&\Omega_1\\ C&-&0 \end{array} \] \[ \epsilon_{\epsilon_0}= \begin{array}{ccc} \underline{\Omega_1}& & \\ \underline{C}&-&0\\ \uparrow& & \\ C&-&\Omega_1\\ C&-&0 \end{array} \] \[ \zeta_0= \begin{array}{ccc} \underline{\Omega_1}& & \\ C&-&\Omega_1\\ \underline{C}&-&0 \end{array} \] \[ \epsilon_{\zeta_0+1}= \begin{array}{ccccc} & &\Omega_1& & \\ & &|& & \\ \underline{\Omega_1}& &C&-&\Omega_1\\ \underline{C}&-&C&-&0 \end{array} \] \[ \zeta_1= \begin{array}{ccccccc} \underline{\Omega_1}& & & &\Omega_1& & \\ C&-&\Omega_1& &C&-&\Omega_1\\ \underline{C}&-&-&-&C&-&0 \end{array} \] \[ \phi_3(0)= \begin{array}{ccccc} \underline{\Omega_1}& &\Omega_1& & \\ C&-&C&-&\Omega_1\\ \underline{C}&-&0& & \end{array} \] \[ \phi_\omega(0)= \begin{array}{ccc} 0& & \\ C&-&\Omega_1\\ C&-&\Omega_1\\ \uparrow& & \\ C&-&0 \end{array} \] \[ \phi_{\phi_\omega(0)}(0)= \begin{array}{ccc} 0& & \\ C&-&\Omega_1\\ C&-&\Omega_1\\ \uparrow& & \\ C&-&0\\ C&-&\Omega_1\\ C&-&\Omega_1\\ C&-&0 \end{array} \] \[ \phi(1,0,0)=\Gamma_0= \begin{array}{ccc} \underline{\Omega_1}& & \\ C&-&\Omega_1\\ C&-&\Omega_1\\ \underline{C}&-&0 \end{array} \] \[ \phi(1,0,1)=\Gamma_1= \begin{array}{ccccccc} \underline{\Omega_1}& & & &\Omega_1& & \\ C&-&\Omega_1& &C&-&\Omega_1\\ C&-&\Omega_1& &C&-&\Omega_1\\ \underline{C}&-&-&-&C&-&0 \end{array} \] \[ \phi(1,1,0)= \begin{array}{ccccc} & &\Omega_1& & \\ & &|& & \\ \underline{\Omega_1}& &C&-&\Omega_1\\ C&-&C&-&\Omega_1\\ \underline{C}&-&0& & \end{array} \] \[ \phi(2,0,0)= \begin{array}{ccccccc} \underline{\Omega_1}& & & &\Omega_1& & \\ C&-&\Omega_1& &C&-&\Omega_1\\ C&-&-&-&C&-&\Omega_1\\ \underline{C}&-&0& & & & \end{array} \] \[ \phi(\omega,0,0)= \begin{array}{ccccc} 0& &\Omega_1& & \\ C&-&C&-&\Omega_1\\ C&-&\Omega_1& & \\ \uparrow& & & & \\ C&-&0& & \end{array} \] \[ \phi(1,0,0,0)= \begin{array}{ccccc} \underline{\Omega_1}& &\Omega_1& & \\ C&-&C&-&\Omega_1\\ C&-&\Omega_1& & \\ \underline{C}&-&0& & \end{array} \] \[ \mathrm{small\ Veblen\ ordinal}= \begin{array}{ccc} 0& & \\ C&-&\Omega_1\\ C&-&\Omega_1\\ \uparrow& & \\ C&-&\Omega_1\\ C&-&0 \end{array} \] \[ \mathrm{large\ Veblen\ ordinal}= \begin{array}{ccc} \underline{\Omega_1}& & \\ C&-&\Omega_1\\ C&-&\Omega_1\\ C&-&\Omega_1\\ \underline{C}&-&0 \end{array} \]

The rule of the fundamental sequence of the 1st ordinal notation system is simple: just search down from the upper left \(\Omega_1\) for a node less than \(\Omega_1\), and repeat between the node and the \(\Omega_1\).

Examples in 2nd ordinal notation system
Label the nodes to find the fundamental sequence of \[ \begin{array}{ccc} \Omega_2& & \\ C&-&\Omega_1\\ C&-&0 \end{array} \]. \[ \begin{array}{lcc} \Omega_2(2)=a_2& & \\ C(1)=a_1&-&\Omega_1(1)\\ C(0)=a_0&-&0 \end{array} \] Although \(\Omega_1\) is not the maximum in the path from lower left, the label of the \(\Omega_2\) above after the replacement is 2, so the label of \(\Omega_1\) will be 1. There is no \(b_1=\Omega_1\) in the leftmost column. Label the nodes of \(a_1\), \[ a_1= \begin{array}{ccccc} \Omega_2(2)& &\Omega_2(2)\\ C(1)&-&\Omega_1(1)&-&0 \end{array} \] \[ b_2= \begin{array}{ccccc} \times& &\Omega_3\\ \times&-&\Omega_2&-&0 \end{array} =\Omega_2. \] \(\Omega_1>a_0\), so \(c_2=\Omega_2\). Although we need to consider \(b'_1\) because \(a_2=c_2\), \(b_1=\Omega_1\) doesn’t include ordinals greater than or equal to \(a_1\) and less than \(a_2\), so \(b'_1=b_1\), and there is no \(b'_1\) in the leftmost. For the number less than \(\Omega_1\), \(b'_1=b_1=\Omega_1\), the discussion of \(b'_1\) is omitted. There is no number less than the node to the right of \(a_0\) (which is \(0\)), so we do not do the replacement of the second argument. After here, in similar cases, the discussion of the replacement of the second argument will be omitted. So the fundamental sequence repeats between \(a_1\) and \(a_2\). \[ \psi_0(\Omega_2)=\psi_0(\epsilon_{\Omega_1+1})=\mathrm{Bachmann-Howard\ ordinal}= \begin{array}{ccc} \underline{\Omega_2}& & \\ \underline{C}&-&\Omega_1\\ C&-&0 \end{array} \]

\[ \begin{array}{ccc} \Omega_2& & \\ C&-&\Omega_1\\ C&-&\Omega_1\\ C&-&0 \end{array} \] is not standard form. Labelling the nodes gives the following: \[ \begin{array}{ccc} \Omega_2(2)& & \\ C(1)&-&\Omega_1(1)\\ C(0)&-&\Omega_1(1)\\ C(-1)&-&0 \end{array} \]. Re-writing to standard form is as follows. \[ \begin{array}{ccc} \Omega_2(2)& & \\ C(1)&-&\Omega_1(1)\\ C(0)&-&\Omega_1(1)\\ C(-1)&-&0 \end{array} = \begin{array}{ccccc} \Omega_2(2)& &\Omega_2(2)& & \\ C(1)&-&\Omega_1(1)&-&0\\ C(0)&-&\Omega_1(1)& & \\ C(-1)&-&0& & \end{array} \rightarrow \begin{array}{ccccc} \times& &\Omega_3& & \\ \times&-&\Omega_2&-&0\\ C&-&\Omega_1& & \\ C&-&0& & \end{array} \rightarrow \begin{array}{ccc} \Omega_2& & \\ C&-&\Omega_1\\ C&-&0 \end{array} \].

Consider the fundamental sequence of \[ \begin{array}{ccccccc} \Omega_2& & & & & & \\ C&-&\Omega_1& &　& & \\ C&-&\Omega_1& &\Omega_2& & \\ C&-&-&-&C&-&\Omega_1\\ C&-&0& & & & \end{array} \].

\[ \begin{array}{ccc} \Omega_2& & \\ C&-&\Omega_1\\ C&-&\Omega_1 \end{array} \] in this has a fundamental sequence. Labelling the nodes gives the following: \[ \begin{array}{lcc} \Omega_2(2)=a_2& & \\ C(1)=a_1&-&\Omega_1\\ C(0)=a_0&-&\Omega_1 \end{array} \] \[ a_0= \begin{array}{ccc} \Omega_2(2)& & \\ C(1)&-&\Omega_1\\ C(0)&-&\Omega_1 \end{array} \] \[ b_1= \begin{array}{ccc} \Omega_3& & \\ C&-&\Omega_1\\ C&-&\Omega_1 \end{array} = \begin{array}{ccc} \Omega_2& & \\ C&-&\Omega_1 \end{array} \], so, \[ \begin{array}{ccccccc} \Omega_2& & & & & & \\ \underline{C}&-&\Omega_1& &　& & \\ \underline{C}&-&\Omega_1& &\Omega_2& & \\ \uparrow& & & &|& & \\ C&-&-&-&C&-&\Omega_1\\ C&-&0& & & & \end{array} \] is the fundamental sequence.

Label the nodes to find the fundamental sequence of \[ \begin{array}{ccccccc} \Omega_2& & & & & & \\ C&-&\Omega_1& &\Omega_2& & \\ C&-&-&-&C&-&\Omega_1\\ C&-&0& & & & \end{array} \].

\[ \begin{array}{lcccccc} \Omega_2(2)=a_2& & & & & & \\ C(1)&-&\Omega_1(1)& &\Omega_2(2)& & \\ C(1)=a_1&-&-&-&C(1)&-&\Omega_1(1)\\ C(0)=a_0&-&0& & & & \end{array} \] \[ a_1= \begin{array}{ccccccc} \Omega_2(2)& & & & & & \\ C(1)&-&\Omega_1(1)& &\Omega_2(2)& & \\ C(1)&-&-&-&C(1)&-&\Omega_1(1) \end{array} \] \[ b_2= \begin{array}{ccccccc} \times& & & & & & \\ \times&-&\times& &\times& & \\ \times&-&-&-&\times&-&\Omega_2 \end{array} =\Omega_2 \] Repeat between \(a_1\) and \(a_2\), \[ \begin{array}{ccccccc} \underline{\Omega_2}& & & & & & \\ C&-&\Omega_1& &\Omega_2& & \\ \underline{C}&-&-&-&C&-&\Omega_1\\ C&-&0& & & & \end{array} \] is the fundamental sequence. If \(a_1<\Omega_2\), \(b_2=\Omega_2\) so the procedure to find \(b_2\) is omitted.

\[ \psi_0(\Omega_3)= \begin{array}{ccccc} \underline{\Omega_2}& &\Omega_2& & \\ \underline{C}&-&C&-&\Omega_1\\ C&-&0& & \end{array} \] \[ \psi_0(\Omega_\omega)= \begin{array}{ccc} 0& & \\ C&-&\Omega_2\\ C&-&0\\ \uparrow& & \\ C&-&0 \end{array} \] \[ \psi_0(\Omega_{\omega+1})=\psi_0(\epsilon_{\Omega_\omega+1})= \mathrm{Takeuti-Feferman-Buchholz\ ordinal}= \begin{array}{ccccc} & &0& & \\ & &|& & \\ \underline{\Omega_2}& &C&-&\Omega_2\\ \underline{C}&-&C&-&0\\ C&-&0& & \end{array} \] \[ \psi_0(\Omega_\Omega)= \begin{array}{ccc} \underline{\Omega_1}& & \\ C&-&\Omega_2\\ C&-&0\\ \underline{C}&-&0 \end{array} \]

Label the nodes to find the fundamental sequence of \[ \begin{array}{ccc} \Omega_2& & \\ C&-&\Omega_2\\ C&-&0\\ C&-&0 \end{array} \].

\[ \begin{array}{ccc} \Omega_3(3)& & \\ \Omega_2(2)&-&0\\ C(2)&-&\Omega_2(2)\\ C(1)&-&0\\ C&-&0 \end{array} \] \(b_1=\Omega_1\), \(b_2=\Omega_2\). \[ a_2= \begin{array}{ccc} \Omega_2(2)& & \\ C(2)&-&\Omega_2(2) \end{array}, \] \(b_3=\Omega_3\), so it repeats between \(a_2\) and \(a_3\). \[ \psi_0(\psi_I(0))= \begin{array}{ccc} \underline{\Omega_3}& & \\ C&-&0\\ \underline{C}&-&\Omega_2\\ C&-&0\\ C&-&0 \end{array} \]

To find the fundamental sequence of \[ \begin{array}{ccccccc} \Omega_2& & & & & & \\ C&-&\Omega_2& & & & \\ C&-&0& & & & \\ C&-&\Omega_2& & & & \\ C&-&0& &\Omega_2& & \\ \uparrow& & & &|& & \\ C&-&-&-&C&-&\Omega_2\\ C&-&0& & & & \\ C&-&0& & & & \end{array} \], label the node of the ordinal above \(\uparrow\). \[ \begin{array}{ccc} \Omega_2(2)& & \\ C(2)&-&\Omega_2(2)\\ C(1)&-&0\\ C(1)&-&\Omega_2(2)\\ C(0)&-&0\\ \end{array} \] \[ b_1= \begin{array}{ccc} \times& & \\ \times&-&\Omega_3\\ C&-&0\\ C&-&\Omega_2\\ C&-&0\\ \end{array} = \begin{array}{ccc} \Omega_2& & \\ C&-&\Omega_2\\ C&-&0\\ \end{array} \] so it repeats between \(a_0\) and \(b_1\). \[ \begin{array}{ccccccc} \Omega_2& & & & & & \\ C&-&\Omega_2& & & & \\ \underline{C}&-&0& & & & \\ C&-&\Omega_2& & & & \\ \underline{C}&-&0& &\Omega_2& & \\ \uparrow& & & &|& & \\ C&-&-&-&C&-&\Omega_2\\ C&-&0& & & & \\ C&-&0& & & & \end{array} \]

In the range of 2nd ordinal notation system, the repetition between \(a_0\) and \(b'_1\) (fundamental sequence type A’) seems to be unnecessary.

Examples in 3rd ordinal notation system or further
Label the nodes to find the fundamental sequence of \[ \begin{array}{ccc} \Omega_3& & \\ C&-&\Omega_2\\ C&-&0\\ C&-&0 \end{array} \].

\[ \begin{array}{ccc} \Omega_3(3)& & \\ C(2)&-&\Omega_2(2)\\ C(1)&-&0\\ C(0)&-&0 \end{array} \] \(b_1=\Omega_1\), \(b_2=\Omega_2\), \(b_3=\Omega_3\) so it repeats between \(a_2\) and \(a_3\). \[ \begin{array}{ccc} \underline{\Omega_3}& & \\ \underline{C}&-&\Omega_2\\ C&-&0\\ C&-&0 \end{array} \]

The upper bound of recursive ordinals expressible in the \(n\)-th ordinal notation system is \(f^{n}(C(\Omega_{n+1},\Omega_n))\) where \(f(x)=C(x,0)\).

Example of the replacement of second argument 1
\[ \begin{array}{ccccccc} \underline{\Omega_3}& & & &\Omega_3& & \\ \underline{C}&-&\Omega_2& &C&-&\Omega_2\\ C&\rightarrow&0& &C&-&\Omega_2\\ C&\leftarrow&-&-&C&-&0 \end{array} \] The labels of the nodes are as follows: \[ \begin{array}{ccccccc} \Omega_3(3)& & & &\Omega_3& & \\ C(2)&-&\Omega_2(2)& &C&-&\Omega_2\\ C(1)&-&0& &C&-&\Omega_2\\ C(0)&-&-&-&C&-&0 \end{array} \] \[ b_1=b'_1= \begin{array}{ccccc} & &\Omega_3& & \\ & &|& & \\ & &C&-&\Omega_2\\ & &|& & \\ \Omega_2& &C&-&\Omega_2\\ C&-&C&-&0 \end{array} \] \(b_2=\Omega_2\), \(b_3=c_3=\Omega_3\) so it repeats between \(a_2\) and \(a_3\). \[ \begin{array}{ccc} \Omega_3& & \\ C&-&\Omega_2 \end{array} > \begin{array}{ccc} \Omega_3& & \\ C&-&\Omega_2\\ C&-&0 \end{array} \], so we need to consider the replacement of the second argument. \[ \begin{array}{ccccccc} \vdots\\ C&-&\Omega_2& &\Omega_3& & \\ C&-&\Omega_2& &C&-&\Omega_2\\ C&-&\Omega_2& &C&-&\Omega_2\\ C&-&-&-&C&-&0 \end{array} \] The second argument is minimum, so the replacement is needed because it is greater than the original \(0\).

Example of the replacement of second argument 2
\[ \begin{array}{ccccccccccc} \underline{\Omega_3}& & & & & & & & & & \\ \underline{C}&-&\Omega_2& &\Omega_3& & & &\Omega_3& & \\ C&\rightarrow&0& &C&-&\Omega_2& &C&-&\Omega_2\\ C&-&-&-&C&-&0& &C&-&\Omega_2\\ C&\leftarrow&-&-&-&-&-&-&C&-&0 \end{array} \] The labels of the nodes are as follows: \[ \begin{array}{ccccccccccc} \Omega_3(3)& & & & & & & & & & \\ C(2)&-&\Omega_2(2)& &\Omega_3(3)& & & &\Omega_3& & \\ C(1)&-&0& &C(2)&-&\Omega_2(2)& &C&-&\Omega_2\\ C(1)&-&-&-&C(1)&-&0& &C&-&\Omega_2\\ C(0)&-&-&-&-&-&-&-&C&-&0 \end{array} \] \[ b_1=b'_1= \begin{array}{ccccc} & &\Omega_3& & \\ & &|& & \\ & &C&-&\Omega_2\\ & &|& & \\ \Omega_2& &C&-&\Omega_2\\ C&-&C&-&0 \end{array} \] \(b_2=\Omega_2\), \(b_3=c_3=\Omega_3\) so it repeats between \(a_2\) and \(a_3\). \[ \begin{array}{ccc} \Omega_3& & \\ C&-&\Omega_2 \end{array} > \begin{array}{ccc} \Omega_3& & \\ C&-&\Omega_2\\ C&-&0 \end{array} \], so we need to consider the replacement of the second argument. \[ \begin{array}{ccccccc} \vdots\\ C&-&\Omega_2& &\Omega_3& & \\ C&-&\Omega_2& &C&-&\Omega_2\\ C&-&\Omega_2& &C&-&\Omega_2\\ C&-&-&-&C&-&0 \end{array} \] The second argument is minimum, so the replacement is needed because it is greater than \(0\).

Example of the replacement of second argument 3
\[ \begin{array}{ccccccc} \underline{\Omega_4}& & & &\Omega_4& & \\ \underline{C}&-&\Omega_3& &C&-&\Omega_3\\ C&-&0& &C&-&\Omega_3\\ C&\rightarrow&0& &C&-&0\\ C&\leftarrow&-&-&C&-&0 \end{array} \] The labels of the nodes are as follows: \[ \begin{array}{ccccccc} \Omega_4(4)& & & &\Omega_4& & \\ C(3)&-&\Omega_3(3)& &C&-&\Omega_3\\ C(2)&-&0& &C&-&\Omega_3\\ C(1)&-&0& &C&-&0\\ C(0)&-&-&-&C&-&0 \end{array} \] \[ b_1=b'_1= \begin{array}{ccccc} & &\Omega_4& & \\ & &|& & \\ & &C&-&\Omega_3\\ & &|& & \\ & &C&-&\Omega_3\\ & &|& & \\ \Omega_2& &C&-&0\\ C&-&C&-&0 \end{array} \] \(b_2=\Omega_2\), \(b_3=\Omega_3\), \(b_4=c_4=\Omega_4\) so it repeats between \(a_3\) and \(a_4\). \[ \begin{array}{ccc} \Omega_4& & \\ C&-&\Omega_3 \end{array} > \begin{array}{ccc} \Omega_4& & \\ C&-&\Omega_3\\ C&-&0 \end{array} \] so we need to consider the replacement of the second argument. \[ \begin{array}{ccccccc} \vdots\\ C&-&\Omega_3& &\Omega_4& & \\ C&-&\Omega_3& &C&-&\Omega_3\\ C&-&\Omega_3& &C&-&\Omega_3\\ C&-&\Omega_3& &C&-&0\\ C&-&-&-&C&-&0 \end{array} = \begin{array}{ccc} \vdots\\ C&-&\Omega_3\\ C&-&0 \end{array} \] Since it becomes \(0\) in the minimization of the second argument, the replacement is not done here. \[ \begin{array}{ccc} \Omega_4& & \\ C&-&\Omega_3\\ C&-&0 \end{array} > \begin{array}{ccc} \Omega_4& & \\ C&-&\Omega_3\\ C&-&0\\ C&-&0 \end{array} \], so consider the replacement of the second argument below. \[ \begin{array}{ccccccc} \vdots\\ C&-&\Omega_3& &\Omega_4& & \\ C&-&\Omega_3& &C&-&\Omega_3\\ C&-&\Omega_3& &C&-&\Omega_3\\ C&-&0& &C&-&0\\ C&-&-&-&C&-&0 \end{array} \] so the second argument is minimum and greater than \(0\), so the replacement is needed.

Example of the replacement of second argument 4
\[ \begin{array}{ccccccccccc} \underline{\Omega_4}& & & &\Omega_4& & & & & & \\ \underline{C}&-&\Omega_3& &C&-&\Omega_3& &\Omega_4& & \\ C&\rightarrow&0& &C&-&\Omega_3& &C&-&\Omega_3\\ C&-&-&-&C&-&0& &C&-&\Omega_3\\ C&\leftarrow&-&-&-&-&-&-&C&-&0 \end{array} \]

Example of the replacement of second argument 5
\[ \begin{array}{ccccccccccccc} \underline{\Omega_3}& & & & & & & & & & & & \\ C&\rightarrow&0& &0& &\Omega_2& & & & & & \\ \underline{C}&-&\Omega_2& &C&-&C&-&\Omega_2& &\Omega_2& & \\ C&-&-&-&C&-&0& & & &C&-&\Omega_2\\ C&\leftarrow&-&-&-&-&-&-&-&-&C&-&0 \end{array} \]

Example of the replacement of second argument 6
\[ \begin{array}{ccccccccccc} \underline{\Omega_4}& & & & & & & & & & \\ C&-&0& & & & & & & & \\ C&\rightarrow&0& &\Omega_2& & & & & & \\ \underline{C}&-&\Omega_3& &C&-&\Omega_3& &\Omega_2& & \\ C&-&-&-&C&-&0& &C&-&\Omega_3\\ C&-&\Omega_2& & & & & &C&-&0\\ C&\leftarrow&-&-&-&-&-&-&C&-&0 \end{array} \]

Example of the replacement of second argument 7
\[ \begin{array}{ccccccccccc} \underline{\Omega_4}& & & & & & & & & & \\ C&-&0& &\Omega_3& & & & & & \\ C&\rightarrow&\Omega_2& &C&-&\Omega_2& &\Omega_3& & \\ \underline{C}&-&\Omega_3& &C&-&\Omega_3& &C&-&\Omega_2\\ C&-&-&-&C&-&0& &C&-&\Omega_3\\ C&-&\Omega_2& & & & & &C&-&0\\ C&\leftarrow&-&-&-&-&-&-&C&-&\Omega_2 \end{array} \]

Example of the replacement of second argument 8
\[ \begin{array}{ccccccc} \underline{\Omega_3}& & & &\Omega_2& & \\ C&\rightarrow&0& &C&-&\Omega_2\\ \underline{C}&-&\Omega_2& &C&-&0\\ C&\rightarrow&0& &C&-&\Omega_2\\ C&\leftarrow&-&-&C&-&0 \end{array} \]

Example of repetition between \(a_0\) and \(b'_1\) (fundamental sequence type A’)
\[ \begin{array}{ccccccccccc} \underline{\Omega_2}& & & &\Omega_2& & & & & & \\ C&-&\Omega_3& &C&-&\Omega_3& &\Omega_2& & \\ C&-&-&-&C&-&0& &C&-&\Omega_3\\ C&-&-&-&-&-&-&-&C&-&0\\ \underline{C}&-&0& & & & & & & & \end{array} \] The labels of the nodes are as follows: \[ \begin{array}{ccccccccccc} \Omega_4(4)& & & & & & & & & & \\ \Omega_3(3)&-&0& & & & & & & & \\ \Omega_2(3)&-&0& &\Omega_2(3)& & & & & & \\ C(3)&-&\Omega_3(3)& &C(3)&-&\Omega_3(3)& &\Omega_2(3)& & \\ C(2)&-&-&-&C(2)&-&0& &C(3)&-&\Omega_3(3)\\ C(1)&-&-&-&-&-&-&-&C(2)&-&0\\ C(0)&-&0& & & & & & & & \end{array} \] \[ b_1= \begin{array}{ccccc} & &\Omega_2& & \\ & &|& & \\ \Omega_3& &C&-&\Omega_3\\ C&-&C&-&0\\ C&-&0& & \end{array} \] \[ b_2= \begin{array}{ccccc} & &\Omega_2& & \\ & &|& & \\ \Omega_3& &C&-&\Omega_3\\ C&-&C&-&0 \end{array} \] \(b_3=\Omega_3\), \(b_4=c_4=\Omega_4\) so \(a_4=b_4=c_4\). In \[ b_1= \begin{array}{ccccc} & &\Omega_2(3)& & \\ & &|& & \\ \Omega_3(3)& &C(3)*&-&\Omega_3(3)\\ C(2)&-&C(2)&-&0\\ C(1)&-&0& & \end{array} \], The node labelled with \(*\) is greater than or equal \(a_3\) and less than \(a_4\). Since \(b'_1=\Omega_2\), it repeats between \(a_0\) and \(b'_1\).

Example of similar ordinals having different fundamental sequences 1.1
\[ \begin{array}{ccccccccccc} \Omega_4& & & &\Omega_4& & & &\Omega_4& & \\ C&-&\Omega_3& &C&-&\Omega_3& &C&-&\Omega_3\\ \underline{C}&-&0& &C&-&\Omega_3& &C&-&\Omega_3\\ \underline{C}&-&-&-&C&-&0& &C&-&\Omega_3\\ C&-&-&-&-&-&-&-&C&-&0 \end{array} \]

Example of similar ordinals having different fundamental sequences 1.2
\[ \begin{array}{ccccccccccc} \underline{\Omega_4}& & & &\Omega_4& & & & & & \\ \underline{C}&-&\Omega_3& &C&-&\Omega_3& &\Omega_4& & \\ C&\rightarrow&0& &C&-&\Omega_3& &C&-&\Omega_3\\ C&-&-&-&C&-&0& &C&-&\Omega_3\\ C&\leftarrow&-&-&-&-&-&-&C&-&0 \end{array} \]

The ordinal which has the same fundamental sequence as the one in example 1.1 is the following. \[ \begin{array}{ccccccccccccc} & &\Omega_4& & & & & & & & & & \\ & &|& & & & & & & & & & \\ & &C&-&\Omega_3& &\Omega_4& & & & & & \\ & &|& & & &|& & & & & & \\ \Omega_3& &C&-&\Omega_3& &C&-&\Omega_3& &\Omega_4& & \\ C&-&C&-&0& &C&-&\Omega_3& &C&-&\Omega_3\\ \underline{C}&-&-&-&-&-&C&-&0& &C&-&\Omega_3\\ \underline{C}&-&-&-&-&-&-&-&-&-&C&-&0 \end{array} \]

Example of similar ordinals having different fundamental sequences 2.1
\[ \begin{array}{ccccccccccccccc} \underline{\Omega_3}& & & &\Omega_3& & & & & & & & & & \\ C&-&\Omega_4& &C&-&\Omega_4& &\Omega_3& & & & & & \\ C&-&-&-&C&-&0& &C&-&\Omega_4& &\Omega_3& & \\ C&-&-&-&-&-&-&-&C&-&0& &C&-&\Omega_4\\ C&-&-&-&-&-&-&-&-&-&-&-&C&-&0\\ \underline{C}&-&0& & & & & & & & & & & & \end{array} \]

Example of similar ordinals having different fundamental sequences 2.2
\[ \begin{array}{ccccccccccccccc} \underline{\Omega_5}& & & & & & & & & & & & & & \\ C&-&0& & & & & & & & & & & & \\ C&-&0& &\Omega_3& & & & & & & & & & \\ \underline{C}&-&\Omega_4& &C&-&\Omega_4& &\Omega_3& & & & & & \\ C&-&-&-&C&-&0& &C&-&\Omega_4& &\Omega_2& & \\ C&-&-&-&-&-&-&-&C&-&0& &C&-&\Omega_4\\ C&-&-&-&-&-&-&-&-&-&-&-&C&-&0\\ C&-&0& & & & & & & & & & & & \end{array} \]

The ordinal which has the same fundamental sequence as the one in example 2.1 is the following.

\[ \begin{array}{ccccccccccccccc} & &\Omega_2& & & & & & & & & & & & \\ & &|& & & & & & & & & & & & \\ \Omega_4& &C&-&\Omega_4& & & & & & & & & & \\ C&-&C&-&0& & & & & & & & & & \\ \underline{C}&-&0& &\Omega_3& & & & & & & & & & \\ C&-&\Omega_4& &C&-&\Omega_4& &\Omega_3& & & & & & \\ C&-&-&-&C&-&0& &C&-&\Omega_4& &\Omega_2& & \\ C&-&-&-&-&-&-&-&C&-&0& &C&-&\Omega_4\\ C&-&-&-&-&-&-&-&-&-&-&-&C&-&0\\ \underline{C}&-&0& & & & & & & & & & & & \end{array} \]

Example of similar ordinals having different fundamental sequences 2.3
\[ \begin{array}{ccccccccccccccc} \underline{\Omega_5}& & & & & & & & & & & & & & \\ C&-&0& & & & & & & & & & & & \\ C&-&0& &\Omega_3& & & & & & & & & & \\ \underline{C}&-&\Omega_4& &C&-&\Omega_4& &\Omega_2& & & & & & \\ C&-&-&-&C&-&0& &C&-&\Omega_4& &\Omega_2& & \\ C&-&-&-&-&-&-&-&C&-&0& &C&-&\Omega_4\\ C&-&-&-&-&-&-&-&-&-&-&-&C&-&0\\ C&-&0& & & & & & & & & & & & \end{array} \]

The ordinal which has the same fundamental sequence as the one in example2.1 is the following. \[ \begin{array}{ccccccccccccccc} & &\Omega_2& & & & & & & & & & & & \\ & &|& & & & & & & & & & & & \\ \Omega_4& &C&-&\Omega_4& &\Omega_2& & & & & & & & \\ C&-&C&-&0& &C&-&\Omega_4& & & & & & \\ C&-&-&-&-&-&C&-&0& & & & & & \\ \underline{C}&-&0& &\Omega_3& & & & & & & & & & \\ C&-&\Omega_4& &C&-&\Omega_4& &\Omega_2& & & & & & \\ C&-&-&-&C&-&0& &C&-&\Omega_4& &\Omega_2& & \\ C&-&-&-&-&-&-&-&C&-&0& &C&-&\Omega_4\\ C&-&-&-&-&-&-&-&-&-&-&-&C&-&0\\ \underline{C}&-&0& & & & & & & & & & & & \end{array} \]