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Some of these systems are marked as (standard); this means they are used wiki-wide for fundamental sequences.

In definition of mentioned above hierarchies only countable ordinals are used (i.e. ordinals less than first uncountable ordinal). If $$\alpha$$ is a countable limit ordinal then cofinality of $$\alpha$$ is always equal to $$\omega$$.

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

## Wainer hierarchy $$(\le\varepsilon_0)$$ (standard)

The Wainer Hierarchy is the standard method of representing fundamental sequences for ordinals less than or equal to $$\varepsilon_0$$.[1]

Every nonzero ordinal $$\alpha<\varepsilon_0$$ can be represented in a unique Cantor normal form $$\alpha=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\omega^{\beta_{k}}$$ where $$\alpha>\beta_1\geq\beta_2\geq\cdots\geq\beta_{k-1}\geq\beta_k$$. If $$\beta_k>0$$ then $$\alpha$$ is a limit and we can assign to it a fundamental sequence as follows

$$\alpha[n]=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\left\{\begin{array}{lcr} \omega^\gamma n \text{ if } \beta_k=\gamma+1\\ \omega^{\beta_k[n]} \text{ if } \beta_k \text{ is a limit.}\\ \end{array}\right.$$

Note: $$\omega^0=1$$.

If $$\alpha=\varepsilon_0$$ then $$\alpha[0]=0$$ and $$\alpha[n+1]=\omega^{\alpha[n]}$$.

## Veblen hierarchy $$(<\Gamma_0)$$ (standard)

For case of countable arguments output of any Veblen's function is always a countable ordinal. Namely such case is considered in this section.

### Standard form

An ordinal $$\gamma<\Gamma_0$$ is in standard form iff one of the following hold:

• $$\gamma = \alpha + \beta$$, $$\alpha\omega>\beta$$, $$\alpha$$ is additively indecomposible, and $$\alpha$$ and $$\beta$$ are in standard form.
• $$\gamma = \varphi_\alpha(\beta)$$, $$\alpha<\gamma$$, $$\beta<\gamma$$ and $$\alpha$$ and $$\beta$$ are in standard form.

### Fundamental sequences

All ordinals below are in standard form.

• $$(\varphi_{\beta_1}(\gamma_1) + \cdots + \varphi_{\beta_k}(\gamma_k))[n]=\varphi_{\beta_1}(\gamma_1) + \cdots + (\varphi_{\beta_k}(\gamma_k) [n])$$
• $$\varphi_0(\gamma+1)[n] = \varphi_0(\gamma) \cdot n$$
• $$\varphi_{\beta+1}(0)[n]=\varphi_{\beta}^n(0)$$
• $$\varphi_{\beta+1}(\gamma+1)[n]=\varphi_{\beta}^n(\varphi_{\beta+1}(\gamma)+1)$$
• $$\varphi_{\beta}(\gamma) [n] = \varphi_{\beta}(\gamma [n])$$ for a limit ordinal $$\gamma$$
• $$\varphi_{\beta}(0) [n] = \varphi_{\beta [n]}(0)$$ for a limit ordinal $$\beta$$
• $$\varphi_{\beta}(\gamma+1) [n] = \varphi_{\beta [n]}(\varphi_{\beta}(\gamma)+1)$$ for a limit ordinal $$\beta$$

Where $$\varphi_\delta^n$$ denotes function iteration.

## Veblen function of a finite number of arguments (finitary Veblen function)

### Definition

Let $$z$$ be an empty string or a string consisting of one or more comma-separated zeros $$0,0,...,0$$ and $$s$$ be an empty string or a string consisting of one or more comma-separated ordinals $$\alpha _{1},\alpha _{2},...,\alpha _{n}$$ with $$\alpha _{1}>0$$. The binary function $$\varphi (\beta ,\gamma )$$ can be written as $$\varphi (s,\beta ,z,\gamma )$$ where both $$s$$ and $$z$$ are empty strings. The finitary Veblen functions are defined as follows:

• $$\varphi (\gamma )=\omega ^{\gamma }$$
• $$\varphi (z,s,\gamma )=\varphi (s,\gamma )$$
• if $$\beta >0$$, then $$\varphi (s,\beta ,z,\gamma )$$ denotes the $$(1+\gamma )$$-th common fixed point of the functions $$\xi \mapsto \varphi (s,\delta ,\xi ,z)$$ for each $$\delta <\beta$$

### Normal form

Every non-zero ordinal $$\alpha$$ less than the small Veblen ordinal (SVO) can be uniquely written in normal form for the finitary Veblen function:

$$\alpha =\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})$$

where

• $$k$$ is a positive integer
• $$\varphi (s_{1})\geq \varphi (s_{2})\geq \cdots \geq \varphi (s_{k})$$
• $$s_{m}$$ is a string consisting of one or more comma-separated ordinals $$\alpha _{m,1},\alpha _{m,2},...,\alpha _{m,n_{m}}$$ where $$\alpha _{m,1}>0$$ and each $$\alpha _{m,i}<\varphi (s_{m})$$

### Fundamental sequences for limit ordinals of finitary Veblen function

For limit ordinals $$\alpha <SVO$$, written in normal form for the finitary Veblen function:

1. $$(\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k}))[n]=\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})[n]$$
2. $$\varphi (\gamma )[n]=\left\{{\begin{array}{lcr}n\quad {\text{if}}\quad \gamma =1\\\varphi (\gamma -1)\cdot n\quad {\text{if}}\quad \gamma \quad {\text{is a successor ordinal}}\\\varphi (\gamma [n])\quad {\text{if}}\quad \gamma \quad {\text{is a limit ordinal}}\\\end{array}}\right.$$
3. $$\varphi (s,\beta ,z,\gamma )[0]=0$$ and $$\varphi (s,\beta ,z,\gamma )[n+1]=\varphi (s,\beta -1,\varphi (s,\beta ,z,\gamma )[n],z)$$ if $$\gamma =0$$ and $$\beta$$ is a successor ordinal
4. $$\varphi (s,\beta ,z,\gamma )[0]=\varphi (s,\beta ,z,\gamma -1)+1$$ and $$\varphi (s,\beta ,z,\gamma )[n+1]=\varphi (s,\beta -1,\varphi (s,\beta ,z,\gamma )[n],z)$$ if $$\gamma$$ and $$\beta$$ are successor ordinals
5. $$\varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta ,z,\gamma [n])$$ if $$\gamma$$ is a limit ordinal
6. $$\varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta [n],z,\gamma )$$ if $$\gamma =0$$ and $$\beta$$ is a limit ordinal
7. $$\varphi (s,\beta ,z,\gamma )[n]=\varphi (s,\beta [n],\varphi (s,\beta ,z,\gamma -1)+1,z)$$ if $$\gamma$$ is a successor ordinal and $$\beta$$ is a limit ordinal

## Buchholz's hierarchy and its extension

For assignation of fundamental sequences for this hierarchy we should allow fundamental sequences with length greater than $$\omega$$ and to define them as follows:

The fundamental sequence for an 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 successor ordinal then $$\text{cof}(\alpha)=1$$ and the fundamental sequence has only one element $$\alpha[0]=\alpha-1$$. If $$\alpha$$ is a limit ordinal then $$\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}$$.

### Normal form

The normal form for 0 is 0. If $$\alpha$$ is a nonzero ordinal number $$\alpha<\Lambda=\text{min}\{\beta|\psi_\beta(0)=\beta\}$$ then the normal form for $$\alpha$$ is $$\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)$$ where $$k$$ is a positive integer and $$\psi_{\nu_1}(\beta_1)\geq\psi_{\nu_2}(\beta_2)\geq\cdots\geq\psi_{\nu_k}(\beta_k)$$ and each $$\nu_i$$, $$\beta_i$$ are also written in normal form.

### Fundamental sequences

For nonzero ordinals $$\alpha<\Lambda$$, written in normal form, fundamental sequences are defined as follows:

1. If $$\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)$$ where $$k\geq2$$ then $$\text{cof}(\alpha)=\text{cof}(\psi_{\nu_k}(\beta_k))$$ and $$\alpha[\eta]=\psi_{\nu_1}(\beta_1)+\cdots+\psi_{\nu_{k-1}}(\beta_{k-1})+(\psi_{\nu_k}(\beta_k)[\eta])$$,
2. If $$\alpha=\psi_{0}(0)=1$$, then $$\text{cof}(\alpha)=1$$ and $$\alpha[0]=0$$,
3. If $$\alpha=\psi_{\nu+1}(0)$$, then $$\text{cof}(\alpha)=\Omega_{\nu+1}$$ and $$\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta$$,
4. If $$\alpha=\psi_{\nu}(0)$$ and $$\text{cof}(\nu)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}$$, then $$\text{cof}(\alpha)=\text{cof}(\nu)$$ and $$\alpha[\eta]=\psi_{\nu[\eta]}(0)=\Omega_{\nu[\eta]}$$,
5. If $$\alpha=\psi_{\nu}(\beta+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta$$ (and note: $$\psi_\nu(0)=\Omega_\nu$$),
6. If $$\alpha=\psi_{\nu}(\beta)$$ and $$\text{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu<\nu\}$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\psi_{\nu}(\beta[\eta])$$,
7. If $$\alpha=\psi_{\nu}(\beta)$$ and $$\text{cof}(\beta)\in\{\Omega_{\mu+1}|\mu\geq\nu\}$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\nu}(\beta[\gamma[\eta]])$$ where $$\left\{\begin{array}{lcr} \gamma[0]=\Omega_\mu \\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.$$.

If $$\alpha=\Lambda$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=0$$ and $$\alpha[\eta+1]=\psi_{\alpha[\eta]}(0)=\Omega_{\alpha[\eta]}$$.

## Deedlit's extension of hierarchy of $$\vartheta$$-functions

### With $$\varphi$$, $$\Omega_\alpha$$

#### Definition

• $$C_0(\nu,\alpha,\beta)=\beta\cup\Omega_\nu\cup\{0\}$$
• $$C_{n+1}(\nu,\alpha,\beta)=\{\gamma+\delta,\varphi(\gamma,\delta),\Omega_\gamma,\vartheta_\gamma(\eta):\gamma,\delta,\eta\in C_n(\nu,\alpha,\beta);\eta<\alpha\}$$
• $$C(\nu,\alpha,\beta)=\cup_{n<\omega}C_n(\nu,\alpha,\beta)$$
• $$\vartheta_\nu(\alpha)=\text{min}(\{\beta<\Omega_{\nu+1}:C(\nu,\alpha,\beta)\cap\Omega_{\nu+1}\subseteq\beta\wedge\alpha\in C(\nu,\alpha,\beta)\}\cup\{\Omega_{\nu+1}\})$$

#### Standard form

1. If $$\alpha=0$$, then the standard form for $$\alpha$$ is 0.
2. If $$\alpha$$ is not additively principal, then the standard form for $$\alpha$$ is $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$, where the $$\alpha_i$$ are principal ordinals with $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$$, and the $$\alpha_i$$ are expressed in standard form.
3. If $$\alpha$$ is an additively principal ordinal but not a strongly critical ordinal, then the standard form for $$\alpha$$ is $$\alpha=\varphi(\beta,\gamma)$$ where $$\gamma<\alpha$$ where $$\beta$$ and $$\gamma$$ are expressed in standard form.
4. If $$\alpha$$ is of the form $$\Omega_\beta$$, then $$\Omega_\beta$$ is the standard form for $$\alpha$$.
5. If $$\alpha$$ is a strongly critical ordinal but not of the form $$\Omega_\beta$$, then $$\alpha$$ is expressible in the form $$\vartheta_\nu(\gamma)$$. Then the standard form for $$\alpha$$ is $$\alpha=\vartheta_\nu(\gamma)$$ where \$$\gamma$$ and $$\nu$$ are expressed in standard form.

#### Fundamental sequences

For ordinals $$\alpha<\vartheta(\Omega_{\Omega_{\Omega_\ldots}})$$, written in normal form, fundamental sequences are defined as follows:

1. If $$\alpha=0$$, then $$\text{cof}(\alpha)=0$$ and $$\alpha$$ has fundamental sequence the empty set.
2. If $$\alpha=\varphi(0,0)=1$$ then $$\text{cof}(\alpha)=1$$ and $$\alpha[0]=0$$
3. If $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$, then $$\text{cof}(\alpha)=\text{cof}(\alpha_n)$$ and $$\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])$$
4. If $$\alpha=\varphi(\beta,\gamma)$$ where $$\gamma$$ is a limit ordinal then $$\text{cof}(\alpha)=\text{cof}(\gamma)$$ and $$\alpha[\eta]=\varphi(\beta,\gamma[\eta])$$
5. If $$\alpha=\varphi(0,\gamma+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\varphi(0,\gamma)\cdot\eta$$
6. If $$\alpha=\varphi(\beta+1,0)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=0$$ and $$\alpha[\eta+1]=\varphi(\beta,\alpha[\eta])$$
7. If $$\alpha=\varphi(\beta+1,\gamma+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=\varphi(\beta+1,\gamma)+1$$ and $$\alpha[\eta+1]=\varphi(\beta,\alpha[\eta])$$
8. If $$\alpha=\varphi(\beta,0)$$ where $$\beta$$ is a limit ordinal then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\varphi(\beta[\eta],0)$$
9. If $$\alpha=\varphi(\beta,\gamma+1)$$ where $$\beta$$ is a limit ordinal then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\varphi(\beta[\eta],\varphi(\beta,\gamma)+1)$$
10. If $$\alpha=\Omega_{\beta+1}$$ then $$\text{cof}(\alpha)=\Omega_{\beta+1}$$ and $$\alpha[\eta]=\eta$$
11. If $$\alpha=\Omega_{\beta}$$ where $$\beta$$ is a limit ordinal then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\Omega_{\beta[\eta]}$$
12. If $$\alpha=\vartheta_\nu(\beta+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=\vartheta_\nu(\beta)+1$$ and $$\alpha[\eta+1]=\varphi(\alpha[\eta],0)$$
13. If $$\alpha=\vartheta_\nu(\beta)$$ where $$\omega\le\text{cof}(\beta)\le\Omega_\nu$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\vartheta_\nu(\beta[\eta])$$
14. If $$\alpha=\vartheta_\nu(\beta)$$ where $$\omega\le\text{cof}(\beta)=\Omega_{\mu+1}>\Omega_{\nu}$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\vartheta_\nu(\beta[\gamma[\eta]])$$ with $$\gamma[0]=\Omega_\mu$$ and $$\gamma[\eta+1]=\vartheta_\mu(\beta[\gamma[\eta]])$$

### Without $$\varphi$$, $$\Omega_\alpha$$

#### Definition

• $$C_0(\alpha,\beta)=\beta$$
• $$C_{n+1}(\alpha,\beta)=\{\gamma+\delta,\vartheta_\gamma(\eta):\gamma,\delta,\eta\in C_n(\alpha,\beta);\eta<\alpha\}$$
• $$C(\alpha,\beta)=\cup_{n<\omega}C_n(\alpha,\beta)$$
• $$\vartheta_\nu(\alpha)=\text{min}\{\beta:|\omega\beta|=\Omega_\nu;C(\alpha,\beta)\cap\Omega_{\nu+1}\subseteq\beta;\alpha\in C(\alpha,\beta)\}$$

#### Standard form

1. If $$\alpha=0$$, then the standard form for $$\alpha$$ is 0.
2. If $$\alpha$$ is not additively principal, then the standard form for $$\alpha$$ is $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$, where the $$\alpha_i$$ are principal ordinals with $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$$, and the $$\alpha_i$$ are expressed in standard form.
3. If $$\alpha$$ is additively principal, then $$\alpha$$ is expressible in the form $$\vartheta_\nu(\gamma)$$. Then the standard form for $$\alpha$$ is $$\alpha=\vartheta_\nu(\gamma)$$ where $$\gamma$$ and $$\nu$$ are expressed in standard form.

#### Fundamental sequences

For ordinals $$\alpha<\vartheta(\Omega_{\Omega_{\Omega_\ldots}})$$, written in normal form, fundamental sequences are defined as follows:

1. If $$\alpha=0$$, then $$\text{cof}(\alpha)=0$$ and $$\alpha$$ has fundamental sequence the empty set.
2. If $$\alpha=\vartheta_0(0)=1$$ then $$\text{cof}(\alpha)=1$$ and $$\alpha[0]=0$$
3. If $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$, then $$\text{cof}(\alpha)=\text{cof}(\alpha_n)$$ and $$\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])$$
4. If $$\alpha=\vartheta_{\beta+1}(0)$$ then $$\text{cof}(\alpha)=\Omega_{\beta+1}$$ and $$\alpha[\eta]=\eta$$
5. If $$\alpha=\vartheta_{\beta}(0)$$ where $$\beta$$ is a limit ordinal then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\vartheta_{\beta[\eta]}(0)$$
6. If $$\alpha=\vartheta_\nu(\beta+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\vartheta_\nu(\beta)\eta$$
7. If $$\alpha=\vartheta_\nu(\beta)$$ where $$\omega\le\text{cof}(\beta)\le\Omega_\nu$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\vartheta_\nu(\beta[\eta])$$
8. If $$\alpha=\vartheta_\nu(\beta)$$ where $$\text{cof}(\beta)=\Omega_{\mu+1}>\Omega_{\nu}$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\vartheta_\nu(\beta[\gamma[\eta]])$$ with $$\gamma[0]=\Omega_\mu$$ and $$\gamma[\eta+1]=\vartheta_\mu(\beta[\gamma[\eta]])$$

## Fundamental sequences for the functions collapsing weakly inaccessible cardinals

### Definition

$$\Omega_\alpha$$ with $$\alpha>0$$ is the $$\alpha$$-th uncountable cardinal, $$I_\alpha$$ with $$\alpha>0$$ is the $$\alpha$$-th weakly inaccessible cardinal and for this notation $$I_0=\Omega_0=0$$.

In this section the variables $$\rho$$, $$\pi$$ are reserved for uncountable regular cardinals of the form $$\Omega_{\nu+1}$$ or $$I_{\mu+1}$$.

Then,

• $$C_0(\alpha,\beta) = \beta\cup\{0\}$$
• $$C_{n+1}(\alpha,\beta) = \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\}$$
• $$\cup \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\}$$
• $$\cup \{I_\gamma|\gamma\in C_n(\alpha,\beta)\}$$
• $$\cup \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\}$$
• $$C(\alpha,\beta) = \bigcup_{n<\omega} C_n(\alpha,\beta)$$
• $$\psi_\pi(\alpha) = \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}$$

### Properties

1. $$\psi_{\pi}(0)=1$$
2. $$\psi_{\Omega_1}(\alpha)=\omega^\alpha$$ for $$\alpha<\varepsilon_0$$
3. $$\psi_{\Omega_{\nu+1}}(\alpha)=\omega^{\Omega_\nu+\alpha}$$ for $$1\le\alpha<\varepsilon_{\Omega_\nu+1}$$ and $$\nu>0$$

### Standard form for ordinals $$\alpha<\beta=\text{min}\{\xi|I_\xi=\xi\}$$

1. The standard form for 0 is 0
2. If $$\alpha$$ is of the form $$\Omega_\beta$$, then the standard form for $$\alpha$$ is $$\alpha= \Omega_\beta$$ where $$\beta<\alpha$$ and $$\beta$$ is expressed in standard form
3. If $$\alpha$$ is of the form $$I_\beta$$, then the standard form for $$\alpha$$ is $$\alpha= I_\beta$$ where $$\beta<\alpha$$ and $$\beta$$ is expressed in standard form
4. If $$\alpha$$ is not additively principal and $$\alpha>0$$, then the standard form for $$\alpha$$ is $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$, where the $$\alpha_i$$ are principal ordinals with $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$$, and the $$\alpha_i$$ are expressed in standard form
5. If $$\alpha$$ is an additively principal ordinal but not of the form $$\Omega_\beta$$ or $$I_\gamma$$, then $$\alpha$$ is expressible in the form $$\psi_\pi(\delta)$$. Then the standard form for $$\alpha$$ is $$\alpha=\psi_\pi(\delta)$$ where $$\pi$$ and $$\delta$$ are expressed in standard form

### Fundamental sequences

The fundamental sequence for an 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.

Let $$S=\{\alpha|\text{cof}(\alpha)=1\}$$ and $$L=\{\alpha|\text{cof}(\alpha)\geq\omega\}$$ where $$S$$ denotes the set of successor ordinals and $$L$$ denotes the set of limit ordinals.

For non-zero ordinals written in standard form fundamental sequences defined as follows:

1. If $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$ with $$n\geq 2$$ then $$\text{cof}(\alpha)=\text{cof}(\alpha_n)$$ and $$\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])$$
2. If $$\alpha=\psi_{\pi}(0)$$ then $$\alpha=\text{cof}(\alpha)=1$$ and $$\alpha[0]=0$$
3. If $$\alpha=\psi_{\Omega_{\nu+1}}(1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\left\{\begin{array}{lcr} \alpha[\eta]=\Omega_{\nu}\cdot\eta \text{ if }\nu>0 \\ \alpha[\eta]=\eta \text{ if }\nu=0\\ \end{array}\right.$$
4. If $$\alpha=\psi_{\Omega_{\nu+1}}(\beta+1)$$ and $$\beta\geq 1$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\Omega_{\nu+1}}(\beta)\cdot\eta$$
5. If $$\alpha=\psi_{ I_{\nu+1}}(1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=I_\nu+1$$ and $$\alpha[\eta+1]=\Omega_{\alpha[\eta]}$$
6. If $$\alpha=\psi_{ I_{\nu+1}}(\beta+1)$$ and $$\beta\geq 1$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=\psi_{ I_{\nu+1}}(\beta)+1$$ and $$\alpha[\eta+1]=\Omega_{\alpha[\eta]}$$
7. If $$\alpha=\pi$$ then $$\text{cof}(\alpha)=\pi$$ and $$\alpha[\eta]=\eta$$
8. If $$\alpha=\Omega_\nu$$ and $$\nu\in L$$ then $$\text{cof}(\alpha)=\text{cof}(\nu)$$ and $$\alpha[\eta]=\Omega_{\nu[\eta]}$$
9. If $$\alpha=I_\nu$$ and $$\nu\in L$$ then $$\text{cof}(\alpha)=\text{cof}(\nu)$$ and $$\alpha[\eta]=I_{\nu[\eta]}$$
10. If $$\alpha=\psi_\pi(\beta)$$ and $$\omega\le\text{cof}(\beta)<\pi$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\psi_\pi(\beta[\eta])$$
11. If $$\alpha=\psi_\pi(\beta)$$ and $$\text{cof}(\beta)=\rho\geq\pi$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])$$ with $$\gamma[0]=1$$ and $$\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])$$

Limit of this notation is $$\lambda$$. If $$\alpha=\lambda$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=1$$ and $$\alpha[\eta+1]=I_{\alpha[\eta]}$$.

## Fundamental sequences for the functions collapsing $$\alpha$$-weakly inaccessible cardinals

### Definition

An ordinal is $$\alpha$$-weakly inaccessible if it's an uncountable regular cardinal and it's a limit of $$\gamma$$-weakly inaccessible cardinals for all $$\gamma<\alpha$$.

Let $$I(\alpha,\beta)$$ be the $$(1+\beta)$$th $$\alpha$$-weakly inaccessible cardinal if $$\beta=0$$ or $$\beta=\gamma+1$$, and $$I(\alpha,\beta)=\text{sup}\{I(\alpha, \xi)|\xi<\beta\}$$ if $$\beta$$ is a limit ordinal.

In this section the variables $$\rho$$, $$\pi$$ are reserved for uncountable regular cardinals of the form $$I(\alpha,0)$$ or $$I(\alpha,\beta+1)$$.

Then,

• $$C_0(\alpha,\beta) = \beta\cup\{0\}$$
• $$C_{n+1}(\alpha,\beta) = \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\}$$
• $$\cup \{I(\gamma,\delta)|\gamma,\delta\in C_n(\alpha,\beta)\}$$
• $$\cup \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\}$$
• $$C(\alpha,\beta) = \bigcup_{n<\omega} C_n(\alpha,\beta)$$
• $$\psi_\pi(\alpha) = \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}$$

### Properties

1. $$I(0,\alpha)=\Omega_{1+\alpha}=\aleph_{1+\alpha}$$
2. $$I(1,\alpha)=I_{1+\alpha}$$
3. $$\psi_{I(0,0)}(\alpha)=\omega^\alpha$$ for $$\alpha<\varepsilon_0$$
4. $$\psi_{I(0,\alpha+1)}(\beta)=\omega^{I(0,\alpha)+1+\beta}$$ for $$\beta<\varepsilon_{I(0,\alpha)+1}$$

### Standard form for ordinals $$\alpha<\psi_{I(1,0,0)}(0)=\text{min}\{\xi|I(\xi,0)=\xi\}$$

1. The standard form for 0 is 0
2. If $$\alpha$$ is of the form $$I(\beta,\gamma)$$, then the standard form for $$\alpha$$ is $$\alpha=I(\beta,\gamma)$$ where $$\beta,\gamma<\alpha$$ and $$\beta,\gamma$$ are expressed in standard form
3. If $$\alpha$$ is not additively principal and $$\alpha>0$$, then the standard form for $$\alpha$$ is $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$, where the $$\alpha_i$$ are principal ordinals with $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$$, and the $$\alpha_i$$ are expressed in standard form
4. If $$\alpha$$ is an additively principal ordinal but not of the form $$I(\beta,\gamma)$$, then $$\alpha$$ is expressible in the form $$\psi_\pi(\delta)$$. Then the standard form for $$\alpha$$ is $$\alpha=\psi_\pi(\delta)$$ where $$\pi$$ and $$\delta$$ are expressed in standard form

### Fundamental sequences

The fundamental sequence for an 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.

Let $$S=\{\alpha|\text{cof}(\alpha)=1\}$$ and $$L=\{\alpha|\text{cof}(\alpha)\geq\omega\}$$ where $$S$$ denotes the set of successor ordinals and $$L$$ denotes the set of limit ordinals.

For non-zero ordinals $$\alpha<\psi_{I(1,0,0)}(0)$$ written in standard form fundamental sequences defined as follows:[2]

1. If $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$ with $$n\geq 2$$ then $$\text{cof}(\alpha)=\text{cof}(\alpha_n)$$ and $$\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])$$
2. If $$\alpha=\psi_{I(0,0)}(0)$$ then $$\alpha=\text{cof}(\alpha)=1$$ and $$\alpha[0]=0$$
3. If $$\alpha=\psi_{I(0,\beta+1)}(0)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=I(0,\beta)\cdot\eta$$
4. If $$\alpha=\psi_{I(0,\beta)}(\gamma+1)$$ and $$\beta\in\{0\}\cup S$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{I(0,\beta)}(\gamma)\cdot\eta$$
5. If $$\alpha=\psi_{I(\beta+1,0)}(0)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=0$$ and $$\alpha[\eta+1]=I(\beta,\alpha[\eta])$$
6. If $$\alpha=\psi_{I(\beta+1,\gamma+1)}(0)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=I(\beta+1,\gamma)+1$$ and $$\alpha[\eta+1]=I(\beta,\alpha[\eta])$$
7. If $$\alpha=\psi_{I(\beta+1,\gamma)}(\delta+1)$$ and $$\gamma\in\{0\}\cup S$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=\psi_{I(\beta+1,\gamma)}(\delta)+1$$ and $$\alpha[\eta+1]=I(\beta,\alpha[\eta])$$
8. if $$\alpha=\psi_{I(\beta,0)}(0)$$ and $$\beta\in L$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=I(\beta[\eta],0)$$
9. if $$\alpha=\psi_{I(\beta,\gamma+1)}(0)$$ and $$\beta\in L$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=I(\beta[\eta],I(\beta,\gamma)+1)$$
10. if $$\alpha=\psi_{I(\beta,\gamma)}(\delta+1)$$ and $$\beta\in L$$ and $$\gamma\in \{0\}\cup S$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=I(\beta[\eta],\psi_{I(\beta,\gamma)}(\delta)+1)$$
11. If $$\alpha=\pi$$ then $$\text{cof}(\alpha)=\pi$$ and $$\alpha[\eta]=\eta$$
12. If $$\alpha=I(\beta,\gamma)$$ and $$\gamma\in L$$ then $$\text{cof}(\alpha)=\text{cof}(\gamma)$$ and $$\alpha[\eta]=I(\beta,\gamma[\eta])$$
13. If $$\alpha=\psi_\pi(\beta)$$ and $$\omega\le\text{cof}(\beta)<\pi$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\psi_\pi(\beta[\eta])$$
14. If $$\alpha=\psi_\pi(\beta)$$ and $$\text{cof}(\beta)=\rho\geq\pi$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])$$ with $$\gamma[0]=1$$ and $$\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])$$

Limit of this notation $$\psi_{I(1,0,0)}(0)$$. If $$\alpha=\psi_{I(1,0,0)}(0)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=0$$ and $$\alpha[\eta+1]=I(\alpha[\eta],0)$$

## The functions collapsing weakly Mahlo cardinals

### Definition

An ordinal is weakly Mahlo if it's an uncountable regular cardinal, and regular cardinals in it (in another word, less than it) are stationary.

Let $$M_0=0$$, $$M_{\alpha+1}$$ be the next weakly Mahlo cardinal after $$M_\alpha$$, and $$M_\alpha=\sup\{M_\beta|\beta<\alpha\}$$ for limit ordinal $$\alpha$$. Then, \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{M_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\wedge\pi\text{ is weakly Mahlo}\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\wedge\pi\text{ is uncountable regular}\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ \chi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\text{ is uncountable regular}\} \\ \psi_\pi(\alpha) &=& \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \end{eqnarray*}

In this section the variables $$\rho, \pi$$ are reserved for uncountable regular cardinals of the form $$\chi_\alpha(\beta)$$ or $$M_{\gamma+1}$$.

### Standard form for ordinals $$\alpha<\text{min}\{\xi|M_\xi=\xi\}$$

1. The standard form for 0 is 0
2. If $$\alpha$$ is a weakly Mahlo cardinal, then the standard form for $$\alpha$$ is $$\alpha= M_\beta$$ where $$\beta<\alpha$$ and $$\beta$$ is expressed in standard form
3. If $$\alpha$$ is an uncountable regular cardinal of the form $$\chi_\pi(\beta)$$, then the standard form for $$\alpha$$ is $$\alpha= \chi_\pi(\beta)$$ where $$\pi$$ is a weakly Mahlo cardinal and $$\pi,\beta$$ are expressed in standard form
4. If $$\alpha$$ is not additively principal and $$\alpha>0$$, then the standard form for $$\alpha$$ is $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$, where the $$\alpha_i$$ are principal ordinals with $$\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n$$, and the $$\alpha_i$$ are expressed in standard form
5. If $$\alpha$$ is an additively principal ordinal but not of the form $$M_\beta$$ or $$\chi_\rho(\gamma)$$, then $$\alpha$$ is expressible in the form $$\psi_\pi(\delta)$$. Then the standard form for $$\alpha$$ is $$\alpha=\psi_\pi(\delta)$$ where $$\pi$$ is an uncountable regular cardinal and $$\pi, \delta$$ are expressed in standard form

### Fundamental sequences for the functions collapsing weakly Mahlo cardinals

The fundamental sequence for an 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.

Let $$L=\{\alpha|\text{cof}(\alpha)\geq\omega\}$$ denotes the set of all limit ordinals.

For non-zero ordinals $$\alpha<\text{min}\{\xi|M_\xi=\xi\}$$ written in the standard form fundamental sequences are defined as follows:

1. If $$\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n$$ with $$n\geq 2$$ then $$\text{cof}(\alpha)=\text{cof}(\alpha_n)$$ and $$\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])$$
2. If $$\alpha=\psi_\pi(0)$$ then $$\text{cof}(\alpha)=\alpha=1$$ and $$\alpha[0]=0$$
3. If $$\alpha=\psi_{\chi_\pi(\beta)}(\gamma+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\left\{\begin{array}{lcr} \chi_\pi(\gamma)\cdot \eta \text{ if }0\le\gamma<\beta\\ \psi_{\chi_\pi(\beta)}(\gamma)\cdot \eta \text{ if }\gamma\geq\beta\\ \end{array}\right.$$
4. If $$\alpha=\psi_{M_\beta}(\gamma+1)$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\chi_{M_\beta}(\gamma)\cdot \eta$$
5. If $$\alpha=\pi$$ then $$\text{cof}(\alpha)=\pi$$ and $$\alpha[\eta]=\eta$$
6. If $$\alpha=M_\beta$$ and $$\beta\in L$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=M_{\beta[\eta]}$$
7. If $$\alpha=\psi_\pi(\beta)$$ and $$\omega\le\text{cof}(\beta)<\pi$$ then $$\text{cof}(\alpha)=\text{cof}(\beta)$$ and $$\alpha[\eta]=\psi_\pi(\beta[\eta])$$
8. If $$\alpha=\psi_\pi(\beta)$$ where $$\text{cof}(\beta)=\rho\geq\pi$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])$$ with $$\gamma[0]=1$$ and $$\gamma[\eta+1]=\left\{\begin{array}{lcr} \psi_{\rho}(\beta[\gamma[\eta]])\text{ if }\rho=\chi_{M_{\delta+1}}(\epsilon)\\ \chi_{\rho}(\beta[\gamma[\eta]])\text{ if }\rho= M_{\delta+1}\\ \end{array}\right.$$

Limit of this notation is $$\nu$$. If $$\alpha=\nu$$ then $$\text{cof}(\alpha)=\omega$$ and $$\alpha[0]=1$$ and $$\alpha[\eta+1]=M_{\alpha[\eta]}$$

### Another system of fundamental sequences

For the function, collapsing weakly Mahlo cardinals to countable ordinals, the fundamental sequences also can be defined as follows:

• $$C_0= \{0,1\}$$
• $$C_{n+1}= \{\alpha+\beta,M_\gamma,\chi_\delta(\epsilon),\psi_\zeta(\eta)|\alpha,\beta,\gamma,\delta,\epsilon,\zeta,\eta\in C_n\wedge\delta\in W\wedge\zeta\in R\}$$
• $$L(\alpha)=\text{min}\{n<\omega|\alpha\in C_n\}$$
• $$\alpha[n]=\text{max}\{\beta<\alpha|L(\beta)\le L(\alpha)+n\}$$

where $$R$$ denotes set of all uncountable regular cardinals and $$W$$ denotes set of all weakly Mahlos cardinals.

## Fundamental sequences for the function collapsing $$\alpha$$-weakly Mahlo cardinals

### Definition of $$M(\alpha ;\beta )$$

If $$\beta =0$$ or $$\beta =\beta '+1$$ then $$M(\alpha ;\beta )$$ is an $$\alpha$$-weakly Mahlo cardinal.

Let on this page $$W(\alpha )$$ be the set of all $$\alpha$$-weakly Mahlo cardinals, $$R$$ be the set of all uncountable regular cardinals less than $$\text{min}\{\xi |\xi =M(\xi ;0)\}$$ and the variable $$\pi$$ be reserved for uncountable regular cardinals of the forms $$M(\alpha ;\beta +1)$$ or $$M(\alpha ;0)$$.

If $$\alpha =0$$ then $$W(\alpha )=R$$

If $$\alpha >0$$ then $$W(\alpha )=\{\gamma \in R|\forall \delta <\alpha :W(\delta )\cap \gamma {\text{ is stationary in }}\gamma \}$$

If $$\beta =0$$ then $$M(\alpha ;\beta )=\text{min}W(\alpha )$$

If $$\beta =\beta '+1$$ then $$M(\alpha ;\beta )=\text{min}\{\gamma \in W(\alpha )|\gamma >M(\alpha ,\beta ')\}$$

If $$\beta$$ is a limit ordinal then $$M(\alpha ;\beta )=\sup\{M(\alpha ;\gamma )|\gamma <\beta \}$$

### Definition of $$\psi _{\pi }(\alpha )$$

$$C_{0}(\alpha ,\beta )=\beta \cup \{0\}$$

$$C_{n+1}(\alpha ,\beta )=\{\gamma +\delta |\gamma ,\delta \in C_{n}(\alpha ,\beta )\}$$

$$\cup \{M(\gamma ;\delta )|\gamma ,\delta \in C_{n}(\alpha ,\beta )\}$$

$$\cup \{\chi _{\pi }^{\gamma }(\delta )|\pi ,\gamma ,\delta \in C_{n}(\alpha ,\beta )\wedge \delta <\alpha \}$$

$$\cup \{\psi _{\pi }(\gamma )|\pi ,\gamma \in C_{n}(\alpha ,\beta )\wedge \gamma <\alpha \}$$

$$C(\alpha ,\beta )=\bigcup _{n<\omega }C_{n}(\alpha ,\beta )$$

$$\chi _{\pi }^{\gamma }(\alpha )=\min(\{\beta <\pi |C(\alpha ,\beta )\cap \pi \subseteq \beta \wedge \beta \in W(\gamma )\}\cup \{\pi \})$$

$$\psi _{\pi }(\alpha )=\min(\{\beta <\pi |C(\alpha ,\beta )\cap \pi \subseteq \beta \}\cup \{\pi \})$$

### Fundamental sequences

The fundamental sequence for a countable limit ordinal $$\alpha$$ is defined as follows:

$$C_{0}=\{0\}$$

$$C_{n+1}=\{\gamma +\delta |\gamma ,\delta \in C_{n}\}$$

$$\cup \{M(\gamma ;\delta )|\gamma ,\delta \in C_{n}\}$$

$$\cup \{\chi _{\pi }^{\gamma }(\delta )|\pi ,\gamma ,\delta \in C_{n}\}$$

$$\cup \{\psi _{\pi }(\gamma )|\pi ,\gamma \in C_{n}\}$$

$$L(\alpha )={\text{min}}\{n<\omega |\alpha \in C_{n}\}$$

$$\alpha [n]={\text{max}}\{\beta <\alpha |L(\beta )\leq L(\alpha )+n\}$$

## Fundamental sequences for notation using weakly compact cardinal

### Definition

Let $$K$$ denote the weakly compact cardinal, $$\Omega_0=0$$ and $$\Omega_\alpha$$ is the $$\alpha$$-th uncountable cardinal. Then, \begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0,K\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\xi,\gamma)|\pi,\xi,\gamma\in C_n(\alpha,\beta)\wedge\xi<\alpha\wedge\gamma<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ A(\alpha) &=& \{\beta<K|C(\alpha,\beta)\cap K\subseteq\beta\wedge\beta\text{ is uncountable regular} \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)A(\xi)\text{ is stationary in }\beta\} \\ \chi_\pi(\xi,\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A(\xi)\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*}

### Fundamental sequences

For notation using weakly compact cardinal fundamental sequences defined as follows:

\begin{eqnarray*} C_0 &=& \{0,K\} \\ C_{n+1} &=& \{\alpha+\beta|\alpha,\beta\in C_n\} \\ &\cup& \{\Omega_\alpha|\alpha\in C_n\} \\ &\cup& \{\chi_\pi(\xi,\alpha)|\pi,\xi,\alpha\in C_n\} \\ &\cup& \{\psi_\pi(\alpha)|\pi,\alpha\in C_n\} \\ L(\alpha) &=& \min\{n<\omega|\alpha\in C_n\} \\ \alpha[n] &=& \max\{\beta<\alpha|L(\beta)\le L(\alpha)+n\} \end{eqnarray*}

## Taranovsky's C

The fundamental sequences of Taranovsky’s notation can be easily defined.[3] Let $$L(\alpha)$$ be the amount of C’s in standard representation of $$\alpha$$, then $$\alpha[n]=\max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}$$.