Talk:Slow-growing hierarchy

I've tried to create comparisons with BEAF and SGH, assuming that \(g_{\varepsilon_0}(n) = n \uparrow\uparrow n\), \(g_{\zeta_0}(n) \approx n \uparrow\uparrow\uparrow n\), \(g_{\eta_0}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\), etc. First two comparisons are correct, but I have doubts about \(g_{\eta_0}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\), although it seems reasonable. Just look at these comparisons past \(\zeta_0\):

\(g_{\zeta_0}(n) \approx n \uparrow\uparrow\uparrow n\)

\(g_{\zeta_0^{\zeta_0}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow 2\)

\(g_{\varepsilon_{\zeta_0+1}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow n\)

\(g_{\varepsilon_{\zeta_0+2}}(n) \approx ((n \uparrow\uparrow\uparrow n) \uparrow\uparrow n) \uparrow\uparrow n \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (2n)\) (using LAPL).

\(g_{\varepsilon_{\zeta_0+3}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (3n)\)

\(g_{\varepsilon_{\zeta_0+\omega}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n^2)\)

\(g_{\varepsilon_{\zeta_0+\omega^\omega}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n^n)\)

\(g_{\varepsilon_{\zeta_0+\varepsilon_0}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow n)\)

\(g_{\varepsilon_{\zeta_0 2}}(n) \approx (n \uparrow\uparrow\uparrow n) \uparrow\uparrow (n \uparrow\uparrow\uparrow n) \approx n \uparrow\uparrow\uparrow (n+1)\)

\(g_{\varepsilon_{\zeta_0 3}}(n) \approx n \uparrow\uparrow\uparrow (n+2)\)

\(g_{\varepsilon_{\zeta_0 \omega}}(n) \approx n \uparrow\uparrow\uparrow (2n)\)

\(g_{\varepsilon_{\zeta_0 \omega 2}}(n) \approx n \uparrow\uparrow\uparrow (3n)\)

\(g_{\varepsilon_{\zeta_0 \omega^2}}(n) \approx n \uparrow\uparrow\uparrow (n^2)\)

\(g_{\varepsilon_{\zeta_0 \omega^\omega}}(n) \approx n \uparrow\uparrow\uparrow (n^n)\)

\(g_{\varepsilon_{\zeta_0 \varepsilon_0}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow n)\)

\(g_{\varepsilon_{\zeta_0^2}}(n) \approx n \uparrow\uparrow\uparrow (n \uparrow\uparrow\uparrow n)\)

I'm not sure, but isn't it that \(g_{\varepsilon_{\zeto_0^\omega}}(n) \approx n \uparrow\uparrow\uparrow\uparrow n\)

Ikosarakt1 (talk ^ contribs) 15:59, April 4, 2013 (UTC)