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.

In definition of mentioned above hierarchies only countable ordinals are used (i.e. ordinals less than first uncountable ordinal \(\omega_1\)). If \(\alpha\) is a countable limit ordinal then cofinality of \(\alpha\) is always equal to \(\omega\).

A 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 supremum \(\alpha\), where \(\alpha[n]\) is the n-th element of this sequence.

In this article, we denote by \(\textrm{Lim}\) the class of limit ordinals. A system of fundamental sequences below a countable ordinal \(\alpha\) is a map \([] \colon (\textrm{Lim} \cap \alpha) \times \omega \to \alpha\) satisfying that for any \(\beta \in \textrm{Lim} \cap \alpha\), the map \(\omega \to \alpha, \ n \mapsto \beta[n]\) gives a fundamental sequence of \(\beta\). For convenience, we frequently extend it to \(\{(\beta,n) \in \alpha \times \omega \mid n \in \textrm{cof}(\beta)\}\) to make the definition simpler, or even extend it to \(\alpha \times \omega\) for the same purpose.

Wainer hierarchy \((\le\varepsilon_0)\)
The Wainer Hierarchy is the standard method of representing fundamental sequences for ordinals less than or equal to \(\varepsilon_0\).

Definition
Using the predicate \(=_{\textrm{NF}}\) for Cantor normal form, we define a system \begin{eqnarray*} [] \colon (\varepsilon_0+1) \times \omega & \to & \varepsilon_0+1 \\ (\alpha,n) & \mapsto & \alpha[n] \end{eqnarray*} of fundamental sequences below \(\varepsilon_0+1\) in the following inductive way: By the definition, natural interpretations of \([]\) and \(\textrm{cof}\) into expansion rules and \(\textrm{dom}\) function for the ordinal notation associated to iterated Cantor normal form can be defined by mutual recursion.
 * 1) If \(\alpha =_{\textrm{NF}} 0\), then \(\alpha[n] := 0\). (In that case, we have \(\textrm{cof}(\alpha) = 0\).)
 * 2) Suppose that \(\alpha =_{\textrm{NF}} \omega^{\beta_0} + \cdots + \omega^{\beta_k}\) for unique \(\beta_0,\ldots,\beta_k \in \varepsilon_0\) with \(k \in \omega\).
 * 3) If \(\textrm{cof}(\beta_k) = 0\), then \(\alpha[n] := \omega^{\beta_0} + \cdots + \omega^{\beta_{k-1}}\). (In that case, we have \(\textrm{cof}(\alpha) = 1\). We note that the right hand side for the case \(k = 0\) stands for \(0\).)
 * 4) If \(\textrm{cof}(\beta_k) = 1\), then \(\alpha[n] := \omega^{\beta_0} + \cdots + \omega^{\beta_{k-1}} + \underbrace{\omega^{\beta_k[0]} + \cdots + \omega^{\beta_k[0]}}_n\). (In that case, we have \(\textrm{cof}(\alpha) = \omega\). We note that the right hand side for the case \(k = 0\) stands for \(\omega^{\beta_0[0]} \times n\).)
 * 5) If \(\textrm{cof}(\beta_k) = \omega\), then \(\alpha[n] := \omega^{\beta_0} + \cdots + \omega^{\beta_{k-1}} + \omega^{\beta_k[n]}\). (In that case, we have \(\textrm{cof}(\alpha) = \omega\). We note that the right hand side for the case \(k = 0\) stands for \(\omega^{\beta_0[n]}\).)
 * 6) Suppose \(\alpha = \varepsilon_0\). (In that case, we have \(\textrm{cof}(\alpha) = \omega\).)
 * 7) If \(n = 0\), then \(\alpha[n] := 0\).
 * 8) If \(n \neq 0\), then \(\alpha[n] := \omega^{\alpha[n[0]]}\).

Common misconception
Since Wainer hierarchy is one of the simplest systems of fundamental sequences, beginners tend to study the hierarchy at their first trial to understand fundamental sequences. Then they tend to confound an ordinal and its expression, and hence make mistakes based on this misconception.

One typical misconception is to define [] by partial specialisations, such as: \begin{eqnarray*} (\omega^{\beta_1} + \cdots + \omega^{\beta_k})[n] = \omega^{\beta_1} + \cdots + \omega^{\beta_{k-1}} + \omega^{\beta_k[n]} \end{eqnarray*} , which include massive errors (in certain cases, but more importantly not all cases are accurate). Since the variables are not appropriately quantified with a precise clarification of the range through which they run, the domain of the map [] is ill-defined, and hence [] itself is ill-defined. Even if we define the domain, the definition does not work when \(\textrm{cof}(\beta_k) = 1\). Even if we apply a case classification so that the rule is only applicable to the case \(\textrm{cof}(\beta_k) = \omega\), the partial specialisation does not work because an expression of an ordinal is not unique. For example, we have \(\omega^2 + \omega^{\omega} = \omega^{\omega}\), but the result of the rule is not unique, as we have \begin{eqnarray*} (\omega^2 + \omega^{\omega})[1] & = & \omega^2 + \omega^{\omega[1]} = \omega^2 + \omega \\ \omega^{\omega}[1] & = & \omega^{\omega[1]} = \omega. \end{eqnarray*} This issue clearly explains why we need the predicate \(=_{\textrm{NF}}\) in the definition of [].

Definition
The Veblen hierarchy of functions is defined as follows:
 * \(\varphi_0(\gamma )=\omega ^{\gamma }\)
 * if \(\beta >0\), then \(\varphi_\beta(\gamma )\) denotes the \((1+\gamma)\)-th common fixed point of the functions \(\xi \mapsto \varphi_\delta(\xi)\) for each \(\delta <\beta\)

Veblen Normal Form
We simultaneously define:
 * 1) predicate \(=_{VNF}\) (Veblen Normal Form);
 * 2) set \(T_V\) i.e. countable set of ordinals such that each element of the set can be denoted uniquely using only  the symbols \(0, +,\varphi\) where \(\varphi\)  denotes the hierarchy of Veblen's unary functions;
 * 3) set \(P_V\) (subset of \(T_V\) which includes only additive principal numbers).

Definition of \(=_{VNF}\) Definition of sets \(T_V\) and \(P_V\)
 * 1) \(\alpha=_{VNF}\alpha _{1}+\cdots +\alpha _{n}\) iff \( \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P_V\)
 * 2) \(\alpha=_{VNF}\varphi_\beta(\gamma)\) iff \( \alpha=\varphi_\beta(\gamma)\wedge \beta,\gamma<\alpha\)
 * 1) \(P_V \subset T_V\)
 * 2) \(0 \in T_V\)
 * 3) If \(\alpha=_{VNF}\alpha _{1}+\cdots +\alpha _{n}\) and \(\alpha _{1},\cdots, \alpha _{n}\in P_V\) then \(\alpha \in T_V\)
 * 4) If \(\alpha=_{VNF}\varphi_\beta(\gamma)\) and \( \beta,\gamma\in T_V\) then \(\alpha \in P_V\)

Fundamental sequences
Fundamental sequences for limit ordinals \(\alpha\in T_V\)
 * \(\alpha=\varphi_{\beta_1}(\gamma_1) + \cdots + \varphi_{\beta_k}(\gamma_k)\) then \(\alpha[n]=\varphi_{\beta_1}(\gamma_1) + \cdots + (\varphi_{\beta_k}(\gamma_k) [n])\)
 * \(\alpha=\varphi_0(\gamma+1)\) then \(\alpha[n] = \varphi_0(\gamma) \cdot n\)
 * \(\alpha=\varphi_{\beta+1}(0)\) then \(\alpha[n]=\varphi_{\beta}^n(0)\)
 * \(\alpha=\varphi_{\beta+1}(\gamma+1)\) then \(\alpha[n]=\varphi_{\beta}^n(\varphi_{\beta+1}(\gamma)+1)\)
 * \(\alpha=\varphi_{\beta}(\gamma) \) then \(\alpha[n] = \varphi_{\beta}(\gamma [n])\) for a limit ordinal \(\gamma\)
 * \(\alpha=\varphi_{\beta}(0)\) and \(\beta\) is a limit ordinal, then \(\alpha[n] = \varphi_{\beta [n]}(0)\)
 * \(\alpha=\varphi_{\beta}(\gamma+1)\) and \(\beta\) is a limit ordinal, then \(\alpha[n] = \varphi_{\beta [n]}(\varphi_{\beta}(\gamma)+1)\)

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 for finitary Veblen function
We simultaneously define:
 * 1) predicate \(_{FNF}\) (normal form for finitary Veblen function);
 * 2) set \(T_F\) i.e. countable set of ordinals such that each element of the set can be denoted uniquely using only  the symbols \(0, +,\varphi\) where \(\varphi\)  denotes finitary Veblen function;
 * 3) set \(P_F\) (subset of \(T_F\) which includes only additive principal numbers).

Definition of \(_{FNF}\) Definition of sets \(T_F\) and \(P_F\)
 * 1) \(\alpha=_{FNF}\alpha _{1}+\cdots +\alpha _{n}\) iff \( \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P_F\)
 * 2) \(\alpha=_{FNF}\varphi(\alpha_1,...,\alpha_n)\) iff \( \alpha=\varphi(\alpha_1,...,\alpha_n)\wedge \alpha_1,...,\alpha_n<\alpha\wedge0<\alpha_1\)
 * 1) \(P_F \subset T_F\)
 * 2) \(0 \in T_F\)
 * 3) If \(\alpha=_{FNF}\alpha _{1}+\cdots +\alpha _{n}\) and \(\alpha _{1},\cdots, \alpha _{n}\in P_F\) then \(\alpha \in T_F\)
 * 4) If \(\alpha=_{FNF}\varphi(\alpha_1,...,\alpha_n)\) and \( \alpha_1,...,\alpha_n\in T_F\) then \(\alpha \in P_F\)

Fundamental sequences
Fundamental sequences for limit ordinals \(\alpha\in T_F\) are defined as follows :
 * 1) If \(\alpha=\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})\) then \(\alpha[n]=\varphi (s_{1})+\varphi (s_{2})+\cdots +(\varphi (s_{k})[n])\)
 * 2) If \(\alpha=\varphi (\gamma )\) then \(\alpha[n]=\left\{{\begin{array}{lcr}n\quad {\text{if}}\quad \gamma =1\\\varphi (\gamma -1)\times n\quad {\text{if}}\quad \gamma \quad {\text{is a successor ordinal}}\\\varphi (\gamma [n])\quad {\text{if}}\quad \gamma \quad {\text{is a limit ordinal}}\\\end{array}}\right.\)
 * 3) If \(\alpha=\varphi (s,\beta ,z,\gamma )\) then:
 * 4) \(\alpha[0]=0\) and \(\alpha[n+1]=\varphi (s,\beta -1,\alpha[n],z)\) if \(\gamma =0\) and \(\beta\) is a successor ordinal;
 * 5) \(\alpha[0]=\varphi (s,\beta ,z,\gamma -1)+1\) and \(\alpha[n+1]=\varphi (s,\beta -1,\alpha[n],z)\) if \(\gamma\) and \(\beta\) are successor ordinals;
 * 6) \(\alpha[n]=\varphi (s,\beta ,z,\gamma [n])\) if \(\gamma\) is a limit ordinal;
 * 7) \(\alpha[n]=\varphi (s,\beta [n],z,\gamma )\) if \(\gamma =0\) and \(\beta\) is a limit ordinal;
 * 8) \(\alpha[n]=\varphi (s,\beta [n],\varphi (s,\beta ,z,\gamma -1)+1,z)\) if \(\gamma\) is a successor ordinal and \(\beta\) is a limit ordinal.

Definition of \(\vartheta\)-functions

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

Normal form for \(\vartheta\)-functions
We simultaneously define: Definition of \(_{DNF}\) Definition of sets \(T_D\) and \(P_D\)
 * 1) predicate \(_{DNF}\);
 * 2) set \(T_D\) i.e. countable set of ordinals such that each element of the set can be denoted uniquely using only  the symbols \(0,+,\Omega,\vartheta, \varphi\) where \(\varphi\) denotes binary Veblen function and \(\vartheta\)  denotes the hierarchy of collapsing functions defined in the previous subsection;
 * 3) set \(P_D\) (subset of \(T_D\) which includes only additive principal numbers).
 * 1) \(\alpha=_{DNF}\alpha _{1}+\cdots +\alpha _{n}\) iff \( \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P_D\)
 * 2) \(\alpha=_{DNF}\varphi(\beta,\gamma)\) iff \(\alpha=\varphi(\beta,\gamma)\)  and \(\beta,\gamma<\alpha \)
 * 3) \(\alpha=_{DNF}\vartheta_\nu(\beta)\) iff  \( \alpha=\vartheta_\nu(\beta) \wedge \beta\in C(\nu, \beta, \vartheta_\nu(\beta))\)
 * 1) \(P_D \subset T_D\)
 * 2) \(0 \in T_D\)
 * 3) If \(\alpha=_{DNF}\alpha _{1}+\cdots +\alpha _{n}\) and \(\alpha _{1},\cdots, \alpha _{n}\in P_D\) then \(\alpha \in T_D\)
 * 4) If \(\alpha=_{DNF}\vartheta_\nu(\beta)\) and \( \nu,\beta\in T_D\) then \(\alpha \in P_D\)
 * 5) If \(\alpha=_{DNF}\varphi(\beta,\gamma)\) and \(\beta,\gamma\in T_D\) then \(\alpha \in P_D\)
 * 6) If \(\alpha=\Omega_\beta\) and \(\beta\in T_D\) then \(\alpha \in P_D\)

Fundamental sequences
Fundamental sequences for non-zero ordinals \(\alpha\in T_D\) were defined by Deedlit 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 \(\gamma[\eta+1]=\vartheta_\mu(\beta[\gamma[\eta]])\)

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 \(\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}\).

Buchholz's Normal Form
We simultaneously define:
 * 1) predicate \(_{BNF}\) (Buchholz's Normal Form);
 * 2) set \(T_B\) i.e. countable set of ordinals such that each element of the set can be denoted uniquely using only  the symbols \(0, +,\psi\) where \(\psi\)  denotes the hierarchy of extended Buchholz's functions;
 * 3) set \(P_B\) (subset of \(T_B\) which includes only additive principal numbers).

Definition of \(_{BNF}\) Definition of sets \(T_B\) and \(P_B\)
 * 1) \(\alpha=_{BNF}\alpha _{1}+\cdots +\alpha _{n}\) iff \( \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P_B\)
 * 2) \(\alpha=_{BNF}\psi_\beta(\gamma)\) iff \( \alpha=\psi_\beta(\gamma)\wedge \gamma\in C_\beta(\gamma)\)
 * 1) \(P_B \subset T_B\)
 * 2) \(0 \in T_B\)
 * 3) If \(\alpha=_{BNF}\alpha _{1}+\cdots +\alpha _{n}\) and \(\alpha _{1},\cdots, \alpha _{n}\in P_B\) then \(\alpha \in T_B\)
 * 4) If \(\alpha=_{BNF}\psi_\beta(\gamma)\) and \( \beta,\gamma\in T_B\) then \(\alpha \in P_B\)

Fundamental sequences
Fundamental sequences for non-zero ordinals \(\alpha\in T_B\) 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
\(\Omega_\alpha\) 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,


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

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 \(\nu>0\)

Normal form for the functions collapsing weakly inaccessible cardinals
We simultaneously define:
 * 1) predicate \(_{WNF}\);
 * 2) set \(T_W\) i.e. countable set of ordinals such that each element of the set can be denoted uniquely using only  the symbols \(0,+,I, \Omega,\psi\) where \(\psi\)  denotes the hierarchy of functions collapsing weakly inaccessible cardinals;
 * 3) set \(P_W\) (subset of \(T_W\) which includes only additive principal numbers).

Definition of \(_{WNF}\) Definition of sets \(T_W\) and \(P_W\)
 * 1) \(\alpha=_{WNF}\alpha _{1}+\cdots +\alpha _{n}\) iff \( \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P_B\)
 * 2) \(\alpha=_{WNF}\psi_\pi(\gamma)\) iff \( \alpha=\psi_\pi(\gamma)\wedge \gamma\in C(\gamma, \psi_\pi(\gamma))\)
 * 3) \(\alpha=_{WNF}\Omega_\beta\) iff \(\alpha=\Omega_\beta\wedge \beta<\alpha\)
 * 4) \(\alpha=_{WNF}I_\beta\) iff \(\alpha=I_\beta\wedge \beta<\alpha\)
 * 1) \(P_W \subset T_W\)
 * 2) \(0 \in T_W\)
 * 3) If \(\alpha=_{WNF}\alpha _{1}+\cdots +\alpha _{n}\) and \(\alpha _{1},\cdots, \alpha _{n}\in P_W\) then \(\alpha \in T_W\)
 * 4) If \(\alpha=_{WNF}\psi_\pi(\gamma)\) and \( \pi,\gamma\in T_W\) then \(\alpha \in P_W\)
 * 5) If \(\alpha=_{WNF}\Omega_\beta\) and \(\beta\in T_W\) then \(\alpha \in P_W\)
 * 6) If \(\alpha=_{WNF}I_\beta\) and \(\beta\in T_W\) then \(\alpha \in P_W\)

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

Fundamental sequences for non-zero ordinals \(\alpha\in T_W\) 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 \(\alpha=\text{cof}(\alpha)=1\) 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 \(I(\alpha, 0)\) be the first \(\alpha\)-weakly inaccessible cardinal, \(I(\alpha, \beta+1)\) be the next \(\alpha\)-weakly inaccessible cardinal after \(I(\alpha,\beta)\), and \(I(\alpha,\beta)=\sup\{I(\alpha,\gamma)|\gamma<\beta\}\) for limit ordinal \(\beta\).

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,


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

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 \(\beta<\varepsilon_{I(0,\alpha)+1}\)

Normal form for the functions collapsing \(\alpha\)-weakly inaccessible cardinals
We simultaneously define:
 * 1) predicate \(_{ANF}\);
 * 2) set \(T_A\) i.e. countable set of ordinals such that each element of the set can be denoted uniquely using only  the symbols \(0,+,I, \psi\) where \(\psi\)  denotes the hierarchy of functions collapsing \(\alpha\)-weakly inaccessible cardinals;
 * 3) set \(P_A\) (subset of \(T_A\) which includes only additive principal numbers).

Definition of \(_{ANF}\) Definition of sets \(T_A\) and \(P_A\)
 * 1) \(\alpha=_{ANF}\alpha _{1}+\cdots +\alpha _{n}\) iff \( \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P_A\)
 * 2) \(\alpha=_{ANF}\psi_\pi(\gamma)\) iff \( \alpha=\psi_\pi(\gamma)\wedge \gamma\in C(\gamma, \psi_\pi(\gamma))\)
 * 3) \(\alpha=_{ANF}I(\beta,\gamma)\) iff \(\alpha=I(\beta, \gamma)\wedge \beta,\gamma<\alpha\)
 * 1) \(P_A \subset T_A\)
 * 2) \(0 \in T_A\)
 * 3) If \(\alpha=_{ANF}\alpha _{1}+\cdots +\alpha _{n}\) and \(\alpha _{1},\cdots, \alpha _{n}\in P_A\) then \(\alpha \in T_A\)
 * 4) If \(\alpha=_{ANF}\psi_\pi(\gamma)\) and \( \pi,\gamma\in T_A\) then \(\alpha \in P_A\)
 * 5) If \(\alpha=_{ANF}I(\beta,\gamma)\) and \(\beta,\gamma\in T_A\) then \(\alpha \in P_A\)

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

Fundamental sequences for non-zero ordinals \(\alpha\in T_A\) 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 \(\alpha=\psi_{I(0,0)}(0)\) 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 \(\alpha=I(\beta,\gamma)\) and \(\gamma\notin L\) then \(\text{cof}(\alpha)=\alpha\) 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 \(M_{\gamma+1}\).

Fundamental sequences
For the function, collapsing weakly Mahlo cardinals to countable ordinals, the fundamental sequences can be defined as follows: where \(R\) denotes class of all uncountable regular cardinals and \(W\) denotes class of all weakly Mahlos cardinals.
 * \(C_0= \{0,1\}\)
 * \(C_{n+1}= \{\alpha+\beta,M_\gamma,\chi_\delta(\epsilon),\psi_\zeta(\eta)|\alpha,\beta,\gamma,\delta,\epsilon,\zeta,\eta\in C_n\wedge\delta\in W\wedge\zeta\in R\}\)
 * \(L(\alpha)=\text{min}\{n<\omega|\alpha\in C_n\}\)
 * \(\alpha[n]=\text{max}\{\beta<\alpha|L(\beta)\le L(\alpha)+n\}\)

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:

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

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

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

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

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

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

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

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*}

This system of fundamental sequences can be defined from a proof-theoretic concept known as a "norm".

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 \(\lambda\) and \(\zeta\)
See the system of fundamental sequences associated to Klev's \(\mathcal{O}^+\) and \(\mathcal{O}^{++}\).