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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

Here, \(\alpha[n]\) denotes the \(n\)th term of a fixed fundamental sequence assigned to ordinal \(\alpha\). A system of fundamental sequences for limit ordinals below a given supremum is not unique, and the Hardy hierarchy heavily depends on the choice of such a system. In particular, the Hardy hierarchy is ill-defined unless a specific choice of a system of fundamental sequences is explicitly fixed in the context. One hierarchy, the Wainer hierarchy, is explained at the article for the fast-growing hierarchy.

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

Functions

Below is the list of comparisons between the 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 the Hardy Hierarchy relates to the Fast-growing hierarchy by the following way: \(H_{\omega^\alpha}(n) = f_\alpha(n)\), where \(\alpha < \varepsilon_0\).

See also

Ordinals, ordinal analysis, and set theory

Basics: cardinal numbers · normal function · ordinal numbers · limit ordinals · fundamental sequence · ordinal notation · transfinite induction
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (the countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions Veblen function · ordinal collapsing function · Madore's function · Buchholz's function · Jäger's function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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