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 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 \((<\varepsilon_0)\) (standard)
The Wainer Hierarchy is the standard method of representing fundamental sequences for ordinals less than \(\varepsilon_0\).

(Iterated) Cantor Normal Form
Every ordinal \(\alpha\) can be represented uniquely in the form \(\omega^{\alpha_0} + \cdots + \omega^{\alpha_n}\), with \(\alpha_0 \geq \cdots \geq \alpha_n\); this is called Cantor's normal form.

This can be done iteratively for any ordinal \(\alpha<\varepsilon_0\): either \(\alpha=0\), or \(\alpha = \omega^\beta + \gamma\) where \(\beta\) and \(\gamma\) are ordinals that can also be represented in this way and \(\beta\) is maximal. This is iterated Cantor's normal form, and will be used in the definitions below.

Fundamental Sequences
The two following definitions are equivalent. The first one is easier to read, and the second one is more similar to the notations presented further down the page.


 * \((\alpha+\omega^\beta)[n] = \alpha+\omega^\beta[n]\) if \alpha+\omega^\beta is in CNF
 * (Equivalently: \((\omega^\alpha+\beta)[n] = \omega^\alpha+\beta[n]\) if \(\omega^{\alpha+1}>\beta\))
 * \(\omega^{\alpha+1}[n] = \omega^\alpha n\)
 * \(\omega^\lambda[n] = \omega^{\lambda[n]}\) for limit ordinals \(\lambda\)

Equivalently, with all ordinals represented in ICNF with trailing \(+0\)s removed:


 * \((\omega^{\alpha+1})[0] = 0\)
 * \(\omega^{\alpha+\omega^0}[n+1] = \omega^{\alpha+1}[n] + \omega^\alpha\)
 * \((\omega^\alpha+\beta)[n] = \omega^\alpha+\beta[n]\) if \(\beta \neq 0\)
 * \(\omega^\lambda[n] = \omega^{\lambda[n]}\) if \(\lambda = \omega^{\alpha_0} + \cdots + \omega^{\alpha_n}\) is a nonzero limit ordinal i.e. \(\alpha_n \neq 0\).

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.

Where \(\varphi_\delta^n\) denotes function iteration.
 * \((\varphi_{\beta_1}(\gamma_1) + \cdots + \varphi_{\beta_k}(\gamma_k))[n]=\varphi_{\beta_1}(\gamma_1) + \cdots + (\varphi_{\beta_k}(\gamma_k) [n])\)
 * \(\varphi_0(\gamma+1)[n] = \varphi_0(\gamma) \cdot n\)
 * \(\varphi_{\beta+1}(0)[n]=\varphi_{\beta}^n(0)\)
 * \(\varphi_{\beta+1}(\gamma+1)[n]=\varphi_{\beta}^n(\varphi_{\beta+1}(\gamma)+1)\)
 * \(\varphi_{\beta}(\gamma) [n] = \varphi_{\beta}(\gamma [n])\) 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 \(\beta\)

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 \(\alpha=\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n\), 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 \(\alpha<\vartheta(\Omega_{\Omega_{\Omega_{...}}})\), 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 \(\gamma[\eta+1]=\vartheta_\mu(\beta[\gamma[\eta]])\)

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 \(\alpha[n]=\max\{\beta|\beta<\alpha\land L(\beta)\le L(\alpha)+n\}\).