Hardy hierarchy

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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 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:

The Hardy hierarchy has the following property \(H_{\alpha+\beta}(n)=H_\alpha(H_\beta(n))\).

See also

Fast-growing hierarchy

Slow-growing hierarchy

References

Hardy,G.H. A theorem concerning the infinite cardinal numbers. Quarterly Journal of Mathematics (1904) vol.35 pp.87–94