User blog comment:Hyp cos/Suggestions about Size Classes of Numbers/@comment-3427444-20170701135427/@comment-28606698-20170701163536

Classification also can be based on FGH+CNF+Ψ-function (FS)

9. ω^1-ω^2 level $$f_{\psi_0(1)}(n) \sim f_{\psi_0(2)}(n)$$ where $$n=10^6$$ (or may be simply 10?)

10. ω^2-ω^4 level $$f_{\psi_0(2)}(n) \sim f_{\psi_0(4)}(n)$$

11. ω^4-ω^ω level $$f_{\psi_0(4)}(n) \sim f_{\psi_0^3(0)}(n)$$ where $$\psi^n$$ denotes function iteration

12. ω^ω-ω^^3 level $$f_{\psi_0^3(0)}(n) \sim f_{\psi_0^4(0)}(n)$$

13. ω^^3-ω^^4 level $$f_{\psi_0^4(0)}(n) \sim f_{\psi_0^5(0)}(n)$$

14.ω^^4-ω^^5 level $$f_{\psi_0^5(0)}(n) \sim f_{\psi_0^6(0)}(n)$$

15.ω^^5-ω^^6 level $$f_{\psi_0^6(0)}(n) \sim f_{\psi_0^7(0)}(n)$$

16.ω^^(7+) level $$f_{\psi_0^7(0)}(n)\sim f_{\psi_0(\Omega)}(n)$$

17.$$ACA_0^{+}$$ level or Ψ(Ω)-Ψ(Ω^2) level $$f_{\psi_0(\Omega)}(n)\sim f_{\psi_0(\Omega^2)}(n)$$

18.$$ATR_0$$ level or Ψ(Ω^2)-Ψ(Ω^Ω) $$f_{\psi_0(\Omega^2)}(n)\sim f_{\psi_0(\Omega^\Omega)}(n)$$

19.$$KP$$ level or Ψ(Ω^Ω)-Ψ(Ω_2) level $$f_{\psi_0(\Omega^\Omega)}(n)\sim f_{\psi_0(\Omega_2)}(n)$$

20.$$\Pi_1^1-CA_0$$ level or Ψ(Ω_2)-Ψ(Ω_ω) level $$f_{\psi_0(\Omega_2)}(n)\sim f_{\psi_0(\Omega_\omega)}(n)$$

21.$$\Pi_1^1-TR_0$$ level or Ψ(Ω_ω)-Ψ(Ω_Ω_...) level $$f_{\psi_0(\Omega_\omega)}(n)\sim f_{\psi_0(\underbrace{\Omega_{\Omega_{..._{\Omega}}}}_{n\quad\Omega's})}(n)$$

22.Beyond $$\Pi_1^1-TR_0$$ level $$>f_{\psi_0(\underbrace{\Omega_{\Omega_{..._{\Omega}}}}_{n\quad\Omega's})}(n)$$ but computable

23. Uncomputable