List of systems of fundamental sequences

This article is an (incomplete) list of systems of fundamental sequences, which are vital for defining the fast-growing, Hardy and slow-growing hierarchies.

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 \(\text{cof}(\alpha)=\omega\) 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 \(\varepsilon_0\).

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:


 * \(\gamma = \alpha + \beta\) with \(\alpha\omega>\beta\) and \(\alpha\) is additively indecomposible, and \(\alpha\) and \(\beta\) are in standard form.
 * \(\gamma = \varphi_\alpha(\beta)\) with \(\alpha<\gamma\) and \(\beta<\gamma\) 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 of \(\varphi_\delta\).

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 \(\alpha[\eta+1]=\psi_{\alpha[\eta]}(0)=\Omega_{\alpha[\eta]}\).

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 \(\gamma[\eta+1]=\vartheta_\mu(\beta[\gamma[\eta]])\)

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 \(\gamma[\eta+1]=\vartheta_\mu(\beta[\gamma[\eta]])\)

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 \(\alpha[\eta+1]=I_{\alpha[\eta]}\).

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:


 * 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 \(\alpha[\eta+1]=I(\alpha[\eta],0)\)

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(dead link).

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: where \(R\) denotes set of all uncountable regular cardinals and \(W\) denotes set of all weakly Mahlos cardinals.
 * \(C_0= \{0,1\}\)
 * \(C_{n+1}= \{\alpha+\beta,M_\gamma,\chi_\delta(\epsilon),\psi_\zeta(\eta)|\alpha,\beta,\gamma,\delta,\epsilon,\zeta,\eta\in C_n\wedge\delta\in W\wedge\zeta\in R\}\)
 * \(L(\alpha)=\text{min}\{n<\omega|\alpha\in C_n\}\)
 * \(\alpha[n]=\text{max}\{\beta<\alpha|L(\beta)\le L(\alpha)+n\}\)

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:

\(C_{0}=\{0\}\)

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

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

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

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

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

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

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. Let \(L(\alpha)\) be the amount of C’s in standard representation of \(\alpha\), then \(\alpha[n]=\max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\land\beta\textrm{ is standard}\}\).

Fundamental sequences up to Church-Kleene ordinal
See the system of fundamental sequences associated to Kleene's \(\mathcal{O}\).

Fundamental sequences up to limits of ITTM ordinals λ and ζ
See the system of fundamental sequences associated to Klev's \(\mathcal{O}^+\) and \(\mathcal{O}^{++}\).