Question[]
Is this number equal to
\(f_{\theta(\Omega_{\omega})}(10)\) ? ARsygo (talk) 23:28, June 26, 2017 (UTC)
Obviously yes,
\(f_{\theta(\Omega_\omega)}(10)=f_{\theta(\Omega_{10})}(10)\)
although I did not assign fundamental sequences for theta-function since I could not get enoughly short ruleset for this function but I wrote rules for psi-function. That is why, as you can see on the page of sourse, I gave second definition for numbers of this seria via psi-function:
\(f_{\theta(\Omega_i)}(10)=f_{\psi_0(\Omega_i^{\Omega_i})}(10)=f_{\psi_0(\psi_i^3(0))}(10)\) where i is a positive integer.
Family of Psi-functions is simplier version ot theta-functions and allows to express each non-zero ordinal less than omega fixed point in terms of psi and zero. On the page of sourse you can see rules 1-6 for psi-functions which allow to assign fundamental sequences even for cases if subscript is a limit ordinal, for example:
\(\psi_0(\Omega_\omega^{\Omega_\omega})[n]=\psi_0(\psi_\omega^3(0))[n]=\psi_0(\psi_{\psi_0(1)}^3(0))[n]=\) \(=\psi_0(\psi_{\psi_0(1)}^2(\psi_{\psi_0(1)[n]}(0)))=\psi_0(\psi_{\psi_0(1)}^2(\psi_{n}(0)))=\psi_0(\Omega_\omega^{\Omega_n})\)
--Denis Maksudov (talk) 17:01, June 27, 2017 (UTC)