The Hardy hierarchy, named after G. H. Hardy, is a family of functions \((H_\alpha:\mathbb N\rightarrow\mathbb N)_{\alpha<\mu}\) where \(\mu\) is a large countable ordinal such that a fundamental sequence is assigned for each limit ordinal less than \(\mu\). The Hardy hierarchy is defined as follows: - \(H_0(n)=n\) - \(H_{\alpha+1}(n)=H_\alpha(n+1)\) - \(H_\alpha(n)=H_{\alpha\[n\]}(n)\) if and only if \(\alpha\) is a limit ordinal, where \(\alpha\[n\]\) denotes the \(n\)th element of the fundamental sequence assigned to the limit ordinal \(\alpha\) Every nonzero ordinal \(\alpha<\varepsilon_0=\min\{\beta\|\beta=\omega^\beta\}\) 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.\) If \(\alpha=\varepsilon_0\) then \(\alpha\[0\]=0\) and \(\alpha\[n+1\]=\omega^{\alpha\[n\]}\). Using this system of fundamental sequences we can define the Hardy hierarchy up to \(\varepsilon_0\). For \(\alpha<\varepsilon_0\) the Hardy Hierarchy relates to the [](Fast-growing_hierarchy.md) as follows \(H_{\omega^\alpha}(n)=f_\alpha(n)\) and at \(\varepsilon_0\) the Hardy hierarchy "catches up" to the fast-growing hierarchy i.e. \(f_{\varepsilon_0}(n-1) ≤ H_{\varepsilon_0}(n) ≤ f_{\varepsilon_0}(n+1)\) for all \(n ≥ 1\). There are much stronger systems of fundamental sequences you can see on the following pages: - <a href="http://googology.wikia.com/wiki/List_of_systems_of_fundamental_sequences" class="external text">List of systems of fundamental sequences</a> - [](Madore's_ψ_function.md) - [](Buchholz's_ψ_functions.md) - [](Jäger's_collapsing_functions_and_ρ-inaccessible_ordinals.md) - [Collapsing functions based on a weakly Mahlo cardinal](User_blog:Denis_Maksudov/Ordinal_functions_collapsing_the_least_weakly_Mahlo_cardinal;_a_system_of_fundamental_sequences "User blog:Denis Maksudov/Ordinal functions collapsing the least weakly Mahlo cardinal; a system of fundamental sequences") The Hardy hierarchy has the following property \(H_{\alpha+\beta}(n)=H_\alpha(H_\beta(n))\). ## See also [](Fast-growing_hierarchy.md) [](Slow-growing_hierarchy.md) ## References Hardy,G.H. A theorem concerning the infinite cardinal numbers. Quarterly Journal of Mathematics (1904) vol.35 pp.87–94