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.

Note that only fundamental sequences for countable limit ordinals need to be defined.

Wainer Hierarchy \((<\varepsilon_0)\) (standard)
The Wainer Hierarchy is the standard method of representing fundamental sequences for ordinals less than \(\varepsion_0\).

(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. \(\alpha_n \neq 0\).

Standard Form
An ordinal \(\gamma<\Gamma_0\) is in standard form iff one of the following hold:


 * \(\gamma = \alpha + \beta\), \(\alpha\omega>\beta\) 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.