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We will define functions $$\psi, \chi$$ by exclusion $$\varphi, \Phi$$ in the definition of Rathjen's original functions [1]. That simplifies creation of recursive ordinal notation and system of fundamental sequences. Application of this recursive notation to FGH or extended arrows allows to generate large computable numbers.

### Basic Notions

Small greek letters $$\alpha, \beta, \gamma, \delta, \xi, \eta$$ denote ordinals. Each ordinal $$\alpha$$ is identified with the set of its predecessors $$\alpha=\{\beta|\beta<\alpha\}$$. The least ordinal is zero and it is identified with the empty set.

$$\omega$$ is the first transfinite ordinal and the set of all natural numbers.

Every ordinal $$\alpha$$ is either zero, or a successor (if $$\alpha=\beta+1$$), or a limit.

An ordinal $$\alpha$$ is a limit ordinal if for all $$\beta<\alpha$$ there exists an ordinal $$\gamma$$ such that $$\beta<\gamma<\alpha$$

$$S$$ denotes the class of successor ordinals and $$L$$ denotes the class of limit ordinals.

An ordinal $$\alpha$$ is an additive principal number if $$\alpha>0$$ and $$\xi+\eta<\alpha$$ for all $$\xi,\eta<\alpha$$.

$$P=\{\alpha>0|\forall\beta,\gamma<\alpha(\beta+\gamma<\alpha)\}$$ is the class of additive principal numbers.

For every ordinal $$\alpha\notin P\cup\{0\}$$ there exist unique $$\alpha_1,..., \alpha_n\in P$$ such that $$\alpha=\alpha_1+\cdots+\alpha_n$$ and $$\alpha>\alpha _{1}\geq \cdots \geq \alpha _{n}$$

$$\alpha=_{NF}\alpha _{1}+\cdots +\alpha _{n}$$ iff $$\alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha>\alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P$$

The cofinality of a limit ordinal $$\alpha$$ is the least length of increasing sequence such that the limit of this sequence is the ordinal $$\alpha$$.

$$\text{cof}(\alpha)$$ denotes the cofinality of an ordinal $$\alpha$$.

An ordinal $$\alpha$$ is uncountable regular cardinal if it is a limit ordinal larger than $$\omega$$ and $$\text{cof}(\alpha)=\alpha$$.

$$R=\{\alpha\in L|\alpha>\omega\wedge\text{cof}(\alpha)=\alpha\}$$ is the class of uncountable regular cardinals.

The fundamental sequence for a limit ordinal number $$\alpha$$ with cofinality $$\text{cof}(\alpha)=\beta$$ is a strictly increasing sequence $$(\alpha[\eta])_{\eta<\beta}$$ with length $$\beta$$ and with limit $$\alpha$$, where $$\alpha[\eta]$$ is the $$\eta$$-th element of this sequence.

• If $$\alpha$$ is a limit ordinal then $$\alpha\geq\text{cof}(\alpha)\geq\omega$$ and $$\alpha=\sup\{\alpha[\eta]|\eta<\text{cof}(\alpha)\}$$.
• If $$\alpha$$ is a successor ordinal then $$\text{cof}(\alpha)=1$$ and the fundamental sequence has only one element $$\alpha[0]=\alpha-1$$.
• If $$\alpha=0$$ then $$\text{cof}(\alpha)=0$$ and $$\alpha$$ has not fundamental sequence.

$$\kappa$$ is weakly Mahlo iff $$\kappa$$ is a cardinal such that for every function $$f: \kappa\rightarrow\kappa$$ there exists a regular cardinal $$\pi < \kappa$$ such that $$\forall\alpha<\pi(f(\alpha)< \pi)$$.

$$M$$ is the least weakly Mahlo cardinal and $$\varepsilon_{M+1}=\min\{\alpha>M|\alpha=\omega^\alpha\}$$

The variables $$\pi$$, $$\rho$$, $$\kappa$$ are reserved for regular uncountable cardinals less than $$M$$.

Enumeration function $$F$$ of class of ordinals $$X$$ is the unique increasing function such that $$X=\{F(\alpha)|\alpha\in\text{dom}(F)\}$$ where domain of $$F$$, $$\text{dom}(F)$$ is an ordinal number. We use $$\text{Enum}(X)$$ to denote $$F$$.

$$cl(X)$$ is closure of $$X$$

$$cl_M(X)=X\cup\{\alpha<M|\alpha=\sup(X\cap\alpha)\}$$

### Definition of functions $$\chi_\alpha(\beta)$$ and $$\psi_\pi(\gamma)$$

Inductive Definition of functions $$\chi_\alpha: M\rightarrow M$$ for $$\alpha <\varepsilon_{M+1}$$

1. $$\{0,M\}\cup\beta\subset B^n(\alpha, \beta)$$
2. $$\gamma=_{NF}\gamma_1+\cdots+\gamma_k\wedge\gamma_1,...,\gamma_k\in B^n(\alpha, \beta)\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)$$
3. $$\gamma=\chi_\eta(\xi)\wedge\eta,\xi\in B^n(\alpha, \beta)\wedge\eta<\alpha\wedge\xi<M\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)$$
4. $$\gamma=\omega^\delta \wedge M<\delta\wedge\delta\in B^n(\alpha, \beta)\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)$$
5. $$\gamma<\pi\wedge\pi\in B^n(\alpha, \beta)\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)$$
6. $$B(\alpha,\beta)=\bigcup_{n<\omega}B^{n}(\alpha, \beta)$$
7. $$\chi_\alpha=\text{Enum}(cl_M(\{\kappa|\kappa\notin B(\alpha,\kappa)\wedge\alpha\in B(\alpha,\kappa)\}))$$

Below we write $$\chi(\alpha,\beta)$$ for $$\chi_\alpha(\beta)$$

Properties of $$\chi$$-functions:

1. $$\chi(\alpha,\beta)<M$$
2. $$\beta>\gamma\geq 0 \Rightarrow \chi(\alpha,\beta)>\chi(\alpha,\gamma)$$
3. $$\alpha>\gamma\geq 0 \Rightarrow \chi(\alpha,\beta)=\chi(\gamma,\chi(\alpha,\beta))$$
4. $$\chi(\alpha,0),\chi(\alpha,\beta+1) \in R$$
5. $$\chi(0,\alpha)=\aleph_{1+\alpha}$$

Relationship between $$\chi$$ and Jäger's function

$$I_\alpha=\text{Enum}(cl_M(\{\kappa|\forall\delta<\alpha(I_\delta(\gamma)=\gamma)\}))$$ is Jäger's function enumerating $$\alpha$$-weakly inaccesibble cardinals and their limits [2]. We write $$I(\alpha,\beta)$$ for $$I_\alpha(\beta)$$. Then, $$\chi(\alpha,\beta)=I(\alpha,\beta)$$ for all $$\alpha<\gamma$$ where $$\gamma=\sup\{\delta(n)|n<\omega\}$$ with $$\delta(0)=0$$ and $$\delta(n+1)=\chi(\delta(n),0)$$

Definition: $$\alpha=_{NF}\chi(\beta,\gamma) \Leftrightarrow\alpha=\chi(\beta,\gamma)\wedge\gamma<\alpha$$

Let $$\Pi$$ be the set of uncountable regular cardinals of the form $$\chi(\alpha,0)$$ or $$\chi(\alpha,\beta+1)$$

$$\Pi=\{\chi(\alpha,0)|\alpha<\varepsilon_{M+1}\}\cup\{\chi(\alpha,\beta+1)|\alpha<\varepsilon_{M+1}\wedge\beta<M\}$$

Definition of $$P_M$$ and $$\alpha^*$$

1. $$P_M(0)=P_M(M)= \emptyset$$
2. $$P_M(\alpha)=\{\alpha\}$$ if $$\alpha < M$$ and $$\alpha \in P$$
3. $$P_M(\alpha)=P_M(\alpha_1)\cup ... \cup P_M(\alpha_n)$$ if $$\alpha = _{NF}\alpha_1+\cdots +\alpha_n$$
4. $$P_M(\alpha)=P_M(\beta)$$ if $$\alpha = \omega^\beta$$ and $$\beta > M$$

$$\alpha^*=\sup(P_M(\alpha)\cup\{0\})$$

Definition of $$\kappa^-$$

$$\kappa^-=\left\{\begin{array}{lcr} \chi(\alpha, \beta) \text{ if }\kappa=\chi(\alpha, \beta+1)\\ \alpha^* \text{ if } \kappa=\chi(\alpha, 0)\\ \end{array}\right.$$

Inductive Definition of functions $$\psi_\pi: M\rightarrow \pi$$ for $$\pi\in \Pi$$

1. $$\kappa^-\cup\{\kappa^-, M\}\subset C_\kappa^n(\alpha)$$
2. $$\gamma=_{NF}\gamma_1+\cdots+\gamma_k\wedge\gamma_1,...,\gamma_k\in C_\kappa^n(\alpha)\Rightarrow\gamma\in C_\kappa^{n+1}(\alpha)$$
3. $$\gamma=\omega^\delta \wedge M<\delta\wedge\delta\in C_\kappa^n(\alpha)\Rightarrow\gamma\in C_\kappa^{n+1}(\alpha)$$
4. $$\pi\in C_\kappa^n(\alpha)\cap\kappa\wedge\gamma<\pi\wedge\pi\in\Pi \Rightarrow\gamma\in C_\kappa^{n+1}(\alpha)$$
5. $$\gamma=_{NF}\chi(\delta, \eta) \wedge\delta,\eta\in C_\kappa^n(\alpha)\Rightarrow\gamma\in C_\kappa^{n+1}(\alpha)$$
6. $$\beta<\alpha\wedge\pi,\beta\in C_\kappa^n(\alpha)\wedge\beta\in C_\pi(\beta)\Rightarrow\psi_\pi(\beta)\in C_\kappa^{n+1}(\alpha)$$
7. $$C_\kappa(\alpha)=\bigcup\{C_\kappa^n(\alpha)|n \in \omega\}$$
8. $$\psi_\kappa(\alpha)=\min\{\xi|\xi\notin C_\kappa(\alpha)\}$$

Definition: $$\alpha=_{NF}\psi_\pi(\beta)\Leftrightarrow\alpha=\psi_\pi(\beta) \wedge\beta\in C_\pi(\beta)$$

### A system of fundamental sequences

Inductive definition of $$T$$

1. $$0 \in T$$ and $$M \in T$$
2. $$\alpha=_{NF}\alpha_1+\cdots+\alpha_k\wedge\alpha_1,...,\alpha_k\in T\Rightarrow\alpha\in T$$
3. $$\alpha=_{NF}\chi(\beta,\gamma)\wedge\beta,\gamma\in T\Rightarrow\alpha\in T$$
4. $$\alpha=_{NF}\psi_\pi(\beta)\wedge\pi,\beta\in T \wedge\beta<M \Rightarrow\alpha\in T$$
5. $$\alpha=\omega^\beta\wedge M<\beta<\varepsilon_{M+1}\wedge\beta\in T\Rightarrow\alpha\in T$$

Definition of fundamental sequences for non-zero ordinals $$\alpha\in T$$:

1. If $$\alpha=\alpha_1+\cdots+\alpha_k$$ then $$\text{cof} (\alpha)= \text{cof} (\alpha_k) \wedge \alpha[\eta]=\alpha_1+\cdots+(\alpha_k[\eta])$$
2. If $$\alpha=0$$ then $$\text{cof}(\alpha)=0$$
3. If $$\alpha=\psi_{\chi(0,0)}(0)$$ then $$\text{cof}(\alpha)=\alpha=1 \wedge \alpha[0]=0$$
4. If $$\alpha=\psi _{\chi(0,\beta+1)}(0)$$ then $$\text{cof}(\alpha)=\omega \wedge \alpha[n]=\chi(0,\beta)\times n$$
5. If $$\alpha=\psi_{ \chi(0,\beta)}(\gamma+1)$$ then $$\text{cof}(\alpha)=\omega \wedge \alpha[n]=\psi_{\chi(0,\beta)}(\gamma)\times n$$
6. If $$\alpha=\psi _{\chi(\beta+1,0)}(0)$$ then $$\text{cof}(\alpha)=\omega \wedge \alpha[0]=0 \wedge \alpha[n+1]=\chi(\beta,\alpha[n])$$
7. If $$\alpha=\psi _{\chi(\beta+1,\gamma+1)}(0)$$ then $$\text{cof}(\alpha)=\omega \wedge \alpha[0]=\chi(\beta+1,\gamma)+1 \wedge \alpha[n+1]=\chi(\beta,\alpha[n])$$
8. If $$\alpha=\psi_{\chi(\beta+1,\gamma)}(\delta+1)$$ then $$\text{cof}(\alpha)=\omega \wedge \alpha[0]= \psi_{\chi(\beta+1,\gamma)}(\delta)+1 \wedge \alpha[n+1]=\chi(\beta,\alpha[n])$$
9. If $$\alpha=\psi _{\chi(\beta,0)}(0) \wedge M>\text{cof}(\beta)\geq\omega$$ then $$\text{cof} (\alpha)= \text{cof} (\beta) \wedge \alpha[\eta]=\chi(\beta[\eta],0)$$
10. If $$\alpha=\psi_{ \chi(\beta,\gamma+1)}(0) \wedge M>\text{cof}(\beta)\geq\omega$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)\wedge \alpha[\eta]=\chi(\beta[\eta],\chi(\beta,\gamma)+1)$$
11. If $$\alpha=\psi_{ \chi(\beta,\gamma)}(\delta+1) \wedge M>\text{cof} (\beta)\geq\omega$$ then $$\text{cof}(\alpha)=\text{cof}(\beta) \wedge \alpha[\eta]=\chi(\beta[\eta],\psi_{\chi(\beta,\gamma)}(\delta)+1)$$
12. If $$\alpha=\psi_{\chi(\beta,0)}(0) \wedge \text{cof}(\beta)=M$$ then $$\text{cof}(\alpha)= \omega \wedge \alpha[0]=0 \wedge \alpha[n+1]=\chi(\beta[\alpha[n]],0)$$
13. If $$\alpha=\psi_{ \chi(\beta,\gamma+1)}(0) \wedge \text{cof} (\beta)=M$$ then $$\text{cof} (\alpha)= \omega \wedge \alpha[0]=\chi(\beta,\gamma)+1 \wedge \alpha[n+1]=\chi(\beta[\alpha[n]],0)$$
14. If $$\alpha=\psi_{\chi(\beta,\gamma)}(\delta+1) \wedge \text{cof} (\beta)=M$$ then $$\text{cof} (\alpha)= \omega \wedge \alpha[0]= \psi_{ \chi(\beta,\gamma)}(\delta)+1 \wedge \alpha[n+1]=\chi(\beta[\alpha[n]],0)$$
15. If $$\alpha=\omega^\beta\wedge \beta = M+1$$ then $$\text{cof}(\alpha)= \omega \wedge \alpha[n] = M \times n$$
16. If $$\alpha=\omega^\beta \wedge \beta=\gamma +1 \wedge M<\gamma$$ then $$\text{cof}(\alpha) = \omega\wedge \alpha[n]=\omega^\gamma\times n$$
17. If $$\alpha=\omega^\beta\wedge M<\beta\wedge \omega\le \text{cof}(\beta) \le M$$ then $$\text{cof}(\alpha)= \text{cof}(\beta)\wedge \alpha[\eta]=\omega^{\beta[\eta]}$$
18. If $$\alpha=\chi(\beta,0) \vee \alpha=\chi(\beta,\gamma+1) \vee \alpha=M$$ then $$\text{cof}(\alpha)=\alpha \wedge \alpha[\eta]=\eta$$
19. If $$\alpha=\chi(\beta,\gamma) \wedge M>\text{cof}(\gamma)\geq\omega$$ then $$\text{cof} (\alpha)=\text{cof}(\gamma)\wedge \alpha[\eta]=\chi(\beta,\gamma[\eta])$$
20. If $$\alpha=\psi_\pi(\beta) \wedge \pi>\text{cof}(\beta)\geq\omega$$ then $$\text{cof} (\alpha)= \text{cof}(\beta) \wedge \alpha[\eta]=\psi_\pi(\beta[\eta])$$
21. If $$\alpha=\psi_\pi(\beta) \wedge \text{cof}(\beta)=\rho\geq\pi$$ then $$\text{cof} (\alpha)=\omega \wedge \alpha[n]=\psi _\pi(\beta[\gamma[n]])$$ where $$\gamma[0]=\psi_\rho(0)$$ and $$\gamma[k+1]=\psi_\rho(\beta[\gamma[k]])$$

### Ordinal notation

We define a recursive set $$\mathcal{T}$$ and a recursive relation $$\triangleleft$$ such that $$(\mathcal{T},\triangleleft)$$ is isomorphic to $$(T,<)$$. Below we simultaneously define:

1. Sets of terms $$\mathcal{T}, \mathcal{P}, \mathcal{R}, \mathcal{L}$$
2. Operation $$a\widetilde{\times}n$$ for $$a\in \mathcal{P}$$ and $$n\in \mathbb{N}$$
3. Functions $$^{\star}:\mathcal{T}\rightarrow \mathcal{T}, ^{\ominus}:\mathcal{R}\rightarrow \mathcal{T}, V:T\rightarrow \mathcal{T}, o:\mathcal{T}\rightarrow T, G:\mathcal{T}\times\mathcal{T}\rightarrow \mathbb{N}$$
4. Coefficient sets $$K_r(a)$$ for $$r\in \mathcal{R}$$ and $$a\in \mathcal{T}$$
5. A linear ordering $$\triangleleft$$ on $$\mathcal{T}\times\mathcal{T}$$
6. Cofinality type $$\text{tp}(a)$$ and fundamental sequence $$(a[x])_{x\triangleleft \text{tp}(a)}$$ for $$a\in \mathcal{T}$$

We assume that $$\langle ... \rangle$$ is a primitive recursive coding function on finite sequences of natural numbers.

1. Definition of $$V:T\rightarrow \mathcal{T}$$

1. $$V(0) = 0$$
2. $$V(M) = \langle 1,0\rangle$$
3. $$V(\alpha_1+\cdots+ \alpha_n) = \langle 2,V(\alpha_1),...,V(\alpha_n) \rangle$$
4. $$V(\psi_\rho(\alpha)) =\langle 3,V(\rho),V(\alpha)\rangle$$
5. $$V(\chi(\alpha,\beta)) = \langle 4,V(\alpha),V(\beta)\rangle$$
6. $$V(\omega^\alpha) = \langle 5,V(\alpha)\rangle$$

2. Definition of $$o:\mathcal{T}\rightarrow T$$

1. $$o(0) = 0$$
2. $$o(\langle 1,0\rangle ) = M$$
3. $$o(\langle 2,a_1,...,a_n\rangle ) = o(a_1)+\cdots+o(a_n)$$
4. $$o(\langle 3,r,a\rangle ) = \psi_{o(r)}(o(a))$$
5. $$o(\langle 4,a,b\rangle ) = \chi(o(a),o(b))$$
6. $$o(\langle 5,a\rangle ) = \omega^{o(a)}$$

3. Abbreviations and conventions

 Element of $$\mathcal{T}$$ Abbreviation $$\langle 1,0\rangle$$ $$\widetilde{M}$$ $$\langle 2,a_1,...,a_n\rangle$$ $$a_1\widetilde{+}\cdots\widetilde{+}a_n$$ $$\langle 3,r,a\rangle$$ $$\widetilde{\psi}_r(a)$$ $$\langle 4,a,b\rangle$$ $$\widetilde{\chi}(a,b)$$ $$\langle 5,a\rangle$$ $$\widetilde{\omega}^{a}$$ $$\widetilde{\psi}_{ \widetilde{\chi}(0,0)}(0)$$ $$\widetilde{1}$$ $$\widetilde{\psi}_{ \widetilde{\chi}(0,0)}(\widetilde{1})$$ $$\widetilde{\omega}$$

We also use the following abbreviations and conventions:

1. Small Latin letters $$a, b, c, d, e, x$$ range over elements of $$\mathcal{T}$$
2. The letters $$p, r$$ are reserved for elements of $$\mathcal{R}$$
3. $$\mathbb{N}=\{0,1,2,3,...\}$$ is the set of natural numbers
4. The letters $$i,k,m,n$$ denote elements of $$\mathbb{N}$$
5. $$a = b$$ iff $$a$$ and $$b$$ are the same strings of symbols
6. $$a \trianglelefteq b :\Leftrightarrow a \triangleleft b$$ or $$a = b$$
7. If $$A$$ is a subset of $$\mathcal{T}$$ then $$A\triangleleft a :\Leftrightarrow \forall x\in A(x\triangleleft a)$$

4. Definition of $$a\widetilde{\times}n$$ for $$a\in \mathcal{P}$$ and $$n\in \mathbb{N}$$

1. $$a\widetilde{\times}0 = 0$$
2. $$a\widetilde{\times}1 = a$$
3. If $$n \geq 2$$ then $$a\widetilde{\times}n = \underbrace{a\widetilde{+}\cdots\widetilde{+}a}_{n\text{ copies of }a}$$

We use $$\widetilde{n}$$ as an abbreviation for $$\widetilde{1}\widetilde{\times}n$$

5. Definition of sets $$\mathcal{T}, \mathcal{P}, \mathcal{R}, \mathcal{L}$$

1. $$\mathcal{R}\subset \mathcal{P}\subset \mathcal{T}$$
2. $$\mathcal{R}\subset \mathcal{L}\subset \mathcal{T}$$
3. $$0 \in \mathcal{T}$$
4. $$\widetilde{1}\in \mathcal{P}$$
5. $$\widetilde{M}\in \mathcal{P}\cap\mathcal{L}$$
6. If $$a_1,...,a_n \in \mathcal{P} \wedge n\geq 2 \wedge a_n \trianglelefteq \cdots \trianglelefteq a_1 \wedge a_n =\widetilde{1}$$ then $$a_1\widetilde{+}\cdots\widetilde{+}a_n \in \mathcal{T}$$
7. If $$a_1,...,a_n \in P \wedge n\geq 2 \wedge a_n \trianglelefteq \cdots \trianglelefteq a_1 \wedge \widetilde{1} \triangleleft a_n$$ then $$a_1\widetilde{+}\cdots\widetilde{+}a_n \in \mathcal{L}$$
8. If $$a,b \in \mathcal{T} \wedge G(a,b)=1 \wedge b \triangleleft \widetilde{M}\wedge b \in \mathcal{L}$$ then $$\widetilde{\chi}(a,b ) \in \mathcal{P} \cap \mathcal{L}$$
9. If $$a,b \in \mathcal{T} \wedge G(a,b)=1 \wedge b \triangleleft \widetilde{M}\wedge b \notin \mathcal{L}$$ then $$\widetilde{\chi}(a,b )\in \mathcal{R}$$
10. If $$a\in \mathcal{T} \wedge p \in \mathcal{R} \wedge K_p (a) \triangleleft a \wedge a \triangleleft \widetilde{M}\wedge \neg(p=\widetilde{\chi}(0,0)\wedge a=0)$$ then $$\widetilde{\psi}_p(a)\in \mathcal{P} \cap \mathcal{L}$$
11. If $$a\in \mathcal{T} \wedge \widetilde{M}\triangleleft a$$ then $$\widetilde{\omega}^{a}\in \mathcal{P} \cap \mathcal{L}$$

6. Definition of $$r^{\ominus}$$ for $$r\in \mathcal{R}$$

1. if $$r= \widetilde{\chi}(a,0)$$ then $$r^{\ominus} = a^{\star}$$
2. if $$r= \widetilde{\chi}(a, b)$$ and $$\text{tp}(b)=\widetilde{1}$$ then $$\left\{\begin{array}{lcr} r^{\ominus} = \widetilde{\chi}(a, b[0]) \text{ if } G(a,b[0])=1 \\ r^{\ominus} = b[0] \text{ if } G(a,b[0])=0 \\ \end{array}\right.$$

7. Definition of $$K_r(a)$$ for $$r\in \mathcal{R}$$ and $$a\in \mathcal{T}$$

1. $$K_r(0)= K_r(\widetilde{M})=\emptyset$$
2. $$K_r (b_1\widetilde{+}\cdots\widetilde{+} b_n) = K_r(b_1) \cup ... \cup K_r(b_n)$$
3. $$K_r (\widetilde{\chi}(b, c )) = K_r(b) \cup K_r(c )$$
4. $$K_r (\widetilde{\psi}_p (b)) = \left\{\begin{array}{lcr} \emptyset \text{ if } \widetilde{\psi}_p (b) \trianglelefteq r^{\ominus}\\ K_r(p) \text{ if } r^{\ominus} \triangleleft \widetilde{\psi}_p (b) \wedge p \triangleleft r \\ \{b\} \cup K_r(p) \cup K_r(b) \text{ if } r^{\ominus} \triangleleft \widetilde{\psi}_p (b) \wedge r \trianglelefteq p \\ \end{array}\right.$$
5. $$K_r (\widetilde{\omega}^{b}) = K_r(b)$$

$$K_ r(c ) \triangleleft e \Leftrightarrow o(c )\in C_{o(r)}(o(e ))$$

8. Definition of function $$a^{\star}$$ for $$a\in \mathcal{T}$$

1. $$0^{\star} =\widetilde{M}^{\star} = 0$$
2. $$a^{\star}= \max(a_1^{\star},...,a_n^{\star})$$ if $$a= a_1\widetilde{+}\cdots\widetilde{+}a_n$$
3. $$a^{\star}= b^{\star}$$ if $$a= \widetilde{\omega}^{b}$$
4. $$a^{\star}= a$$ if $$a\in \mathcal{P}$$ and $$a \triangleleft \widetilde{M}$$

9. Definition of $$G(a,b)$$ for $$a\in \mathcal{T}$$ and $$b\triangleleft \widetilde{M}$$

$$G(a,b)=1$$ iff one of the following cases holds:

1. $$b\notin \mathcal{P}$$,
2. $$b = \widetilde{\chi}(c, d) \wedge (c \trianglelefteq a \vee b\trianglelefteq a^{\star})$$,
3. $$b = \widetilde{\psi}_r(e ) \wedge r = \widetilde{\chi}(c ,d ) \wedge (c \trianglelefteq a \vee r \trianglelefteq a^{\star} \vee \exists x \in K_r (a) (e \trianglelefteq x) )$$,

otherwise $$G(a,b)=0$$

10. Definition of $$a\triangleleft b$$ for $$a,b\in \mathcal{T}$$

1. $$0 \triangleleft a$$ iff $$a\neq 0$$.
2. $$\widetilde{M}\triangleleft c \Rightarrow \widetilde{M}\triangleleft \widetilde{\omega}^{c }$$
3. $$\widetilde{M}\triangleleft a\triangleleft b \Rightarrow \widetilde{\omega}^{a}\triangleleft \widetilde{\omega}^{b}$$
4. $$\widetilde{M}\trianglelefteq a\Rightarrow \widetilde{\psi}_p (b) \triangleleft a$$ and $$\widetilde{\chi}(b,c ) \triangleleft a$$
5. $$\widetilde{\chi}(a,b) \triangleleft \widetilde{\chi}(c ,d )$$ iff one of the following cases holds:
1. $$a \triangleleft c$$ and $$b \triangleleft \widetilde{\chi}(c, d )$$ and $$a^{\star}\triangleleft \widetilde{\chi}(c ,d )$$,
2. $$a = c$$ and $$b \triangleleft d$$,
3. $$c \triangleleft a$$ and $$(\widetilde{\chi}(a,b) \trianglelefteq d \vee \widetilde{\chi}(a,b) \triangleleft c ^{\star})$$.
6. $$\widetilde{\chi}(a,b) \triangleleft \widetilde{\psi}_r(e )$$ iff $$\widetilde{\chi}(a,b) \triangleleft r$$ and $$K_ r(\widetilde{\chi}(a,b)) \triangleleft e$$, otherwise $$\widetilde{\psi}_r(e ) \triangleleft \widetilde{\chi}(a,b)$$.
7. $$\widetilde{\psi}_p (b) \triangleleft \widetilde{\psi}_r(a)$$ iff one of the following cases holds:
1. $$p \triangleleft r$$ and $$p \triangleleft \widetilde{\psi}_r(a)$$,
2. $$p = r$$ and $$b \triangleleft a$$,
3. $$r \triangleleft p$$ and $$\widetilde{\psi}_p (b) \triangleleft r$$.
8. $$a_1\widetilde{+}\cdots\widetilde{+}a_m \triangleleft b_1\widetilde{+}\cdots\widetilde{+}b_n (2 \le m, 2 \le n)$$ iff one of the following cases holds:
1. $$m < n$$ and $$a_i = b_i$$ for $$1 \le i \le m$$,
2. there exists a $$k$$ such that $$1 \le k \le \min(m,n)$$ and $$a_k \triangleleft b_k$$ and $$a_i = b_i$$ for $$1 \le i < k$$.
9. Let $$b \in \mathcal{P}$$ and $$2 \le n$$. Then:
1. $$a_1\widetilde{+}\cdots\widetilde{+}a_n \triangleleft b$$ if $$a_1 \triangleleft b$$,
2. $$b \triangleleft a_1\widetilde{+}\cdots\widetilde{+}a_n$$ if $$b \trianglelefteq a_i$$ for some $$1 \le i \le n$$.
10. If no one of rules 10.1-10.9 can be applied, then use the property $$a \trianglelefteq b \Leftrightarrow\neg ( b \triangleleft a)$$

Note: $$a\triangleleft b$$ and $$a,b\in \mathcal{T} \Leftrightarrow o(a)<o(b)$$

11. Definition of $$\text{tp}(a)$$ and $$a[x]$$ for $$a \in \mathcal{T}$$ and $$x \triangleleft \text{tp}(a)$$

1. If $$a=a_1\widetilde{+}\cdots\widetilde{+}a_n \wedge a_n[x]=0$$ then $$\text{tp}(a)= \text{tp}(a_n) \wedge a[x]= \left\{\begin{array}{lcr} a_1 \text{ if } n=2\\ a_1\widetilde{+}\cdots\widetilde{+}a_{n-1}\text{ if } n>2\\ \end{array}\right.$$
2. If $$a=a_1\widetilde{+}\cdots\widetilde{+}a_n \wedge 0\triangleleft a_n[x]$$ then $$\text{tp}(a)= \text{tp}(a_n) \wedge a[x]= \left\{\begin{array}{lcr} a_1\widetilde{+}(a_2[x]) \text{ if } n=2\\ a_1\widetilde{+}\cdots\widetilde{+}a_{n-1}\widetilde{+}(a_n[x])\text{ if } n>2\\ \end{array}\right.$$
3. If $$a=0$$ then $$\text{tp}(a)=0$$
4. If $$a=\widetilde{1}$$ then $$\text{tp} (a)=\widetilde{1} \wedge a[0]=0$$
5. If $$a=\widetilde{\psi} _{\widetilde{\chi}(0,b)}(0) \wedge \text{tp}(b)=\widetilde{1}$$ then $$\text{tp}(a)=\widetilde{\omega} \wedge a[\widetilde{n}]= \widetilde{\chi}(0,b)^{\ominus}\widetilde{\times}n$$
6. If $$a=\widetilde{\psi}_{\widetilde{\chi}(0,b)}(c)\wedge \text{tp}(c)=\widetilde{1}$$ then $$\text{tp}(a)=\widetilde{\omega} \wedge a[\widetilde{n}]=\widetilde{\psi}_{\widetilde{\chi}(0,b)}(c[0]) \widetilde{\times}n$$
7. If $$a=\widetilde{\psi} _{\widetilde{\chi}(b,0)}(0) \wedge \text{tp}(b)=\widetilde{1}$$ then $$\text{tp}(a)=\widetilde{\omega} \wedge a[0]=0 \wedge a[\widetilde{n}]=\widetilde{\chi}(b[0],a[\widetilde{n}[0]])$$ for $$n>0$$
8. If $$a=\widetilde{\psi} _{\widetilde{\chi}(b,c)}(0)\wedge \text{tp}(b)=\text{tp}(c)=\widetilde{1}$$ then $$\text{tp}(a)=\widetilde{\omega} \wedge a[0]=\widetilde{\chi}(b,c)^{\ominus}\widetilde{+}\widetilde{1} \wedge a[\widetilde{n}]=\widetilde{\chi}(b[0],a[\widetilde{n}[0]])$$ for $$n>0$$
9. If $$a=\widetilde{\psi}_{\widetilde{\chi}(b,c)}(d) \wedge \text{tp}(b)=\text{tp}(d)=\widetilde{1}$$ then $$\text{tp}(a)=\widetilde{\omega} \wedge a[0]= \widetilde{\psi}_{\widetilde{\chi}(b,c)}(d[0])\widetilde{+}\widetilde{1} \wedge a[\widetilde{n}]=\widetilde{\chi}(b[0],a[\widetilde{n}[0]])$$ for $$n>0$$
10. If $$a=\widetilde{\psi} _{\widetilde{\chi}(b,0)}(0) \wedge \widetilde{\omega} \trianglelefteq \text{tp}(b) \triangleleft \widetilde{M}$$ then $$\text{tp} (a)= \text{tp} (b) \wedge a[x]=\widetilde{\chi}(b[x],0)$$
11. If $$a=\widetilde{\psi}_{ \widetilde{\chi}(b,c)}(0) \wedge \text{tp}(c)=\widetilde{1}\wedge \widetilde{\omega} \trianglelefteq \text{tp}(b) \triangleleft \widetilde{M}$$ then $$\text{tp}(a)=\text{tp}(b)\wedge a[x]=\widetilde{\chi}(b[x],\widetilde{\chi}(b,c)^{\ominus}\widetilde{+}\widetilde{1})$$
12. If $$a=\widetilde{\psi}_{ \widetilde{\chi}(b,c)}(d)\wedge \text{tp}(d)=\widetilde{1} \wedge \widetilde{\omega} \trianglelefteq \text{tp}(b) \triangleleft \widetilde{M}$$ then $$\text{tp}(a)=\text{tp}(b) \wedge a[x]=\widetilde{\chi}(b[x],\widetilde{\psi}_{\widetilde{\chi}(b,c)}(d[0])\widetilde{+}\widetilde{1})$$
13. If $$a=\widetilde{\psi}_{\widetilde{\chi}(b,0)}(0) \wedge \text{tp}(b)=\widetilde{M}$$ then $$\text{tp}(a)= \widetilde{\omega} \wedge a[0]=0 \wedge a[\widetilde{n}]=\widetilde{\chi}(b[a[\widetilde{n}[0]]],0)$$ for $$n>0$$
14. If $$a=\widetilde{\psi}_{ \widetilde{\chi}(b,c)}(0) \wedge \text{tp}(c)=\widetilde{1}\wedge \text{tp} (b)=\widetilde{M}$$ then $$\text{tp} (a)= \widetilde{\omega} \wedge a[0]=\widetilde{\chi}(b,c)^{\ominus}\widetilde{+}\widetilde{1} \wedge a[\widetilde{n}]=\widetilde{\chi}(b[a[\widetilde{n}[0]]],0)$$ for $$n>0$$
15. If $$a=\widetilde{\psi}_{\widetilde{\chi}(b,c)}(d) \wedge \text{tp}(d)=\widetilde{1}\wedge \text{tp} (b)=\widetilde{M}$$ then $$\text{tp} (a)= \widetilde{\omega} \wedge a[0]= \widetilde{\psi}_{ \widetilde{\chi}(b,c)}(d[0])\widetilde{+}\widetilde{1} \wedge a[\widetilde{n}]=\widetilde{\chi}(b[a[\widetilde{n}[0]]],0)$$ for $$n>0$$
16. If $$a=\widetilde{\omega}^{b}\wedge b=\widetilde{M}\widetilde{+}\widetilde{1}$$ then $$\text{tp}(a)= \widetilde{\omega}\wedge a[\widetilde{n}]=\widetilde{M}\widetilde{\times}n$$
17. If $$a=\widetilde{\omega}^{b}\wedge b=c\widetilde{+}\widetilde{1} \wedge \widetilde{M}\triangleleft c$$ then $$\text{tp}(a)= \widetilde{\omega}\wedge a[\widetilde{n}]=\widetilde{\omega}^{c}\widetilde{\times}n$$
18. If $$a=\widetilde{\omega}^{b}\wedge \widetilde{M}\triangleleft b\wedge \widetilde{\omega}\trianglelefteq \text{tp}(b) \trianglelefteq \widetilde{M}$$ then $$\text{tp}(a)= \text{tp}(b)\wedge a[x]=\widetilde{\omega}^{b[x]}$$
19. If $$(a=\widetilde{\chi}(b, c) \wedge \text{tp}(c)\in \{0,\widetilde{1}\}) \vee a=\widetilde{M}$$ then $$\text{tp} (a)=a \wedge a[x]=x$$
20. If $$a=\widetilde{\chi}(b,c) \wedge \widetilde{\omega} \trianglelefteq \text{tp}(c)\triangleleft\widetilde{M}$$ then $$\text{tp} (a)=\text{tp}(c)\wedge a[x]=\widetilde{\chi}(b,c[x])$$
21. If $$a=\widetilde{\psi}_p(b) \wedge \widetilde{\omega} \trianglelefteq \text{tp}(b) \triangleleft p$$ then $$\text{tp} (a)= \text{tp}(b) \wedge a[x]=\widetilde{\psi}_p(b[x])$$
22. If $$a=\widetilde{\psi}_p(b) \wedge p \trianglelefteq \text{tp}(b)=r$$ then $$\text{tp} (a)=\widetilde{\omega} \wedge a[\widetilde{n}]=\widetilde{\psi} _p(b[c_n])$$ where $$c_0 =\widetilde{\psi}_r (0)$$ and $$c_{m+1}=\widetilde{\psi}_r(b[c_m])$$ for $$m\in \mathbb{N}$$

Note: if $$a \in \mathcal{T}$$ and $$x \triangleleft \text{tp}(a)$$ then $$a[x]\in \mathcal{T}$$ and $$a[x]\triangleleft a$$

12. Definition of $$f(a,n)$$ for $$a\in \mathcal{T}$$ and $$n\in \mathbb{N}$$

1. If $$\text{tp} (a)=0$$ then $$f(a,n)=n+1$$
2. If $$\text{tp} (a)=\widetilde{1}$$ then $$f(a,n)= s_n$$ where $$s_0=n$$ and $$s_{m+1}= f(a[0],s_m)$$ for $$m\in \mathbb{N}$$
3. If $$\text{tp} (a)=\widetilde{\omega}$$ then $$f(a,n)= f(a[\widetilde{n}],n)$$

If $$a = V(\beta)$$ then for the function of fast-growing hierarchy $$f_\beta(n) = f(a,n)$$

13. Definition of $$F(n)$$ for $$n\in \mathbb{N}$$

$$F(n) = f(a_n,n)$$ where $$a_n =\widetilde{\psi}_{\widetilde{\chi}(0,0)}(\widetilde{\chi}(b_n,0))$$ with $$b_0=\widetilde{M}\widetilde{+}\widetilde{1}$$ and $$b_{m+1}=\widetilde{\omega}^{b_m}$$ for $$m\in \mathbb{N}$$

14. Definition of $$h(n,a,k)$$ for $$a\in \mathcal{T}$$ and $$n,k\in \mathbb{N}\backslash \{0\}$$

1. If $$\text{tp} (a)\in \{0,\widetilde{1}\}$$ then $$h(n,a,1)=n$$
2. If $$\text{tp} (a)=0$$ then $$h(n,a,k+1)=h(n,a,k)+n$$
3. If $$\text{tp} (a)=\widetilde{1}$$ then $$h(n,a,k+1)=h(n,a[0],h(n,a,k))$$
4. If $$\text{tp} (a)=\widetilde{\omega}$$ then $$h(n,a,k)= h(n,a[\widetilde{k}],n)$$

If $$a = V(\beta)$$ then for extended arrows $$n\uparrow^{\beta}k = h(n,a,k)$$

References

1. M. Rathjen (1990). Ordinal Notations Based on a Weakly Mahlo Cardinal. Arch. Math. Logic (1990) 29: 249-263
2. G. Jäger (1984). ρ-inaccessible ordinals, collapsing functions and a recursive notation system. Arch. Math. Logik Grundlagenforsch, Vol. 24, 49-62
3. M. Rathjen (1991). Proof-theoretic analysis of KPM. Arch. Math. Logic (1991) 30: 377-403