The Hardy hierarchy is a certain hierarchy mapping ordinals \(\alpha\) to functions \(H_\alpha: \mathbb{N} \rightarrow \mathbb{N}\). For large ordinals \(\alpha\), \(H_\alpha\) grows extremely fast. The Hardy hierarchy is named after G. H. Hardy, who first described it in his 1904 paper "A theorem concerning the infinite cardinal numbers". It is less known than its speedier cousin, the fast-growing hierarchy. However, it can on occasion be more useful than the fast-growing hierarchy; for example, it can more easily be related to the numbers resulting from Goodstein sequences.

The functions are defined as follows:

  • \(H_0(n) = n\)
  • \(H_{\alpha+1}(n) = H_\alpha(n+1)\)
  • \(H_\alpha(n) = H_{\alpha[n]}(n)\) if \(\alpha\) is a limit ordinal

\(\alpha[n]\) denotes the \(n\)th term of fundamental sequence assigned to ordinal \(\alpha\). Definitions of \(\alpha[n]\) can vary, giving different versions of the Hardy hierarchy. One hierarchy, the Wainer hierarchy, is explained at the article for the fast-growing hierarchy.

It can be shown that Hardy hierarchy has the property: \(H_{\alpha+\beta}(n) = H_{\alpha}(H_{\beta}(n))\).


Below is the list of comparisons between Hardy Hierarchy and other googological notations.

\(H_0(n) = n\)

\(H_1(n) = n+1\)

\(H_2(n) = n+2\)

\(H_m(n) = n+m\)

\(H_\omega(n) = 2n\)

\(H_{\omega+1}(n) = 2(n+1) = 2n+2\)

\(H_{\omega+2}(n) = 2(n+2)\)

\(H_{\omega+m}(n) = 2(n+m)\)

\(H_{\omega 2}(n) = 4n\)

\(H_{(\omega 2)+m}(n) = 4(n+m)\)

\(H_{\omega 3}(n) = 8n\)

\(H_{(\omega 3)+m}(n) = 8(n+m)\)

\(H_{\omega m}(n) = (2^m)n\)

\(H_{\omega m+x}(n) = 2^m(n+x)\)

\(H_{\omega^2}(n) = 2^n*n\)

\(H_{\omega^2+1}(n) = 2^{n+1}*(n+1)\)

\(H_{\omega^2+2}(n) = 2^{n+2}*(n+2)\)

\(H_{\omega^2+\omega}(n) = 2^{2n}*(2n)\)

\(H_{\omega^2+\omega 2}(n) = 2^{4n}*(4n)\)

\(H_{(\omega^2) 2}(n) = 2^{2^n*n}*(2^n*n)\)

\(H_{(\omega^2) 3}(n) = 2^{2^{2^n*n}*(2^n*n)}*(2^{2^n*n}*(2^n*n))\)

\(H_{(\omega^2) m}(n) = 2^{H_{(\omega^2) m-1}(n)}*(H_{(\omega^2) m-1}(n))\)

\(H_{\omega^3}(n) \approx n \uparrow\uparrow n\)

\(H_{(\omega^3) 2}(n) \approx n \uparrow\uparrow (n \uparrow\uparrow n)\)

\(H_{\omega^4}(n) \approx n \uparrow\uparrow\uparrow n\)

\(H_{\omega^5}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\)

\(H_{\omega^m}(n) \approx \{n,n,m-1\}\)

\(H_{\omega^\omega}(n) \approx \{n,n,n-1\}\)

\(H_{\omega^\omega+1}(n) \approx \{n,n,n\}\)

\(H_{\omega^\omega+2}(n) \approx \{n,n,n+1\}\)

\(H_{\omega^\omega+\omega}(n) \approx \{n,n,2n\}\)

\(H_{\omega^\omega+\omega^2}(n) \approx \{n,n,2^n*n\}\)

\(H_{(\omega^\omega) 2}(n) \approx \{n,n,\{n,n,n\}\}\)

\(H_{(\omega^\omega) 3}(n) \approx \{n,n,\{n,n,\{n,n,n\}\}\}\)

\(H_{\omega^{\omega+1}}(n)\) or \(H_{(\omega^\omega) \omega}(n) \approx \{n,n,1,2\}\)

\(H_{\omega^{\omega+2}}(n) \approx \{n,n,2,2\}\)

\(H_{\omega^{\omega+m}}(n) \approx \{n,n,m,2\}\)

\(H_{\omega^{\omega 2}}(n) \approx \{n,n,n,2\}\)

\(H_{\omega^{\omega 2+1}}(n) \approx \{n,n,1,3\}\)

\(H_{\omega^{\omega 2+2}}(n) \approx \{n,n,2,3\}\)

\(H_{\omega^{\omega m+2}}(n) \approx \{n,n,m,3\}\)

\(H_{\omega^{\omega m}}(n) \approx \{n,n,n,m\}\)

\(H_{\omega^{\omega m+x}}(n) \approx \{n,n,x,m+1\}\)

\(H_{\omega^{\omega^2}}(n) \approx \{n,n,n,n\}\)

\(H_{\omega^{\omega^3}}(n) \approx \{n,n,n,n,n\}\)

\(H_{\omega^{\omega^{m-2}}}(n) \approx \{n,m (1) 2\} = m \& n\)

\(H_{\omega^{\omega^m}}(n) \approx \{n,m+2 (1) 2\}\)

\(H_{\omega^{\omega^\omega}}(n) \approx \{n,n+2 (1) 2\}\)

We can note that Hardy Hierarchy relates to the Fast-growing hierarchy by the following way: \(H_{\omega^\alpha}(n) = f_\alpha(n)\), where \(\alpha < \varepsilon_0\).

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