User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620

1) What's the difference between 16 and 17 in computable? 2) Your scale contains a gigantic gap between 23 and 24. Defining upper bounds for functions such as DAN or Loader's using “24” it will be a big inaccuracy.

I propose another computable scale based solely on PTO of some theory.

Θ(t) = hPTO(t)(n), where h - Hardy hierarchy; n - googol; t - theory; another definition: Θ(t) - some function f(n), that dominates every provably total computable function in t-theory.

1. n 2. Θ(RFA) = f2(n)

3. Θ(EFA) = f3(n)

4. Θ(PRA) = Θ(RCA0) = fω(n)

5. Θ(П1-CA0) = fωω(n)

6. Θ(П2-CA0) = fωω ω(n)

7. Θ(PA) = Θ(ACA0) = Θ(KP-ω)= fε 0 (n)

8. Θ(ACA) = fε ε 0 (n)

9. Θ(Δ11-CR) ~ φ(ω,0)

10. Θ(Δ11-CA) ~ φ(ε0,0)

11. Θ(ATR0) ~ φ(1,0,0)

12. Θ(KPω) = Θ(ACA+BI) ~ ψ(εΩ+1)

13. Θ(KPω1CK) = ψ(εΩ 2+1 ) 14. Θ(П11-CA0) = Θ(Δ12-CA0) ~ ψ(Ωω)

15. Θ(П11-CA+BI) = Θ(KPl) = ψ(εΩ ω+1 ) 16. Θ(Δ12-CR) ~ ψ(Ωωω)

17. Θ(Δ12-CA) ~ ψ(Ωε 0 )

18. Θ(П11-TR0) ~ ψ(ψI(0))

19. Θ(Δ12-CA+BI) = Θ(KPi) ~ ψ(εI+1)

20. Θ(KPM) ~ ψ(εM+1)

21. Θ(KP+П3-ref) ~ ψ(εK+1) 22. Θ(KP+Пn-ref)

23. Θ(KP+Пω-ref) = Θ(KPi+∃σ(Lσ≺1Lσ+1)) = Θ(KP+∃ (+1)-stable-ordinal) ~ ψ(εΞ+1)

24. Θ(KP+∀n∃σ≥n(Lσ≺1Lσ+n)) = Θ(KP+∃ n((+n)-stable-ordinal)) ~ ψ(εϒ+1) 25. Θ(KP+П11-ref) = Θ(KP+∃ (σ+)-stable-ordinal[zoo 2.10]) 

26. Θ(KP+∃ inaccessible-stable-ordinal[zoo 2.11]) 27. Θ(KP+∃ Mahlo-stable-ordinal[zoo 2.11])

28. Θ(KP+∃ doubly(+1)-stable-ordinal[zoo 2.13])

29. Θ(П12-CA0) = Θ(Δ13-CA0) = Θ(KP+∃ nonprojectable-ordinal[zoo 2.15]) 30. Θ(П12-CA+BI) = Θ(KP+Σ1-sep)

31. Θ(Δ13-CR)

32. Θ(Δ13-CA)

33. Θ(П12-TR0)

34. Θ(KPi+П2-reflecting on (β<σ|Lβ≺1Lσ))

35. Θ(KPi+П3-reflecting on (β<σ|Lβ≺1Lσ)) 36. Θ(KPi+Пn-reflecting on (β<σ|Lβ≺1Lσ))

37. Θ(KPi+Пω-reflecting on (β<σ|Lβ≺1Lσ)) = Θ(KP+∃ 2-(+1)-stable-ordinal) 38. Θ(П13-CA0) = Θ(Δ14-CA0) 39. Θ(П13-CA+BI) = Θ(KP+Σ2-sep) 40. Θ(Z2) = Θ(ZFC-P(ω)) = Θ(KP+∃β|(Lβ/Lβ+1)∩ω1=∅[zoo 2.17]) 41. Θ(KP+∃β|(Lβ/Lβ+2)∩ω1=∅[zoo 2.18]) 42. Θ(KP+∃β|(Lβ/Lβ+β)∩ω1=∅[zoo 2.19]) 43. Θ(KP+∃ω1) = Θ(KP+∃γ admissible on β|(Lβ/Lγ)∩ω1=∅[zoo 2.19]) 44. Θ(Z3) = Θ(ZFC-P(ω)+∃ω1) 45. Θ(Zn) = Θ(ZFC-P(ω)+∀n∃ωn) 46. Θ(ZFC-P(ω)+∃ωω) 47. Θ(ZFC-P(ω)+∃omega fixed point) 48. Θ(ZFC-P(ω)+∃power-admissible-cardinal) 49. Θ(ZFC-P(ω)+∃Σ2-extendible (β|Vβ≺2Vγ,undefined where γ - inaccessible cardinal)) 50. Θ(ZFC-P(ω)+∃Σ3-extendible (β|Vβ≺3Vγ,undefined where γ - inaccessible cardinal)) 51. Θ(ZFC) = Θ(ZFC-P(ω)+∃β|(Vβ/Vβ+1)∩Vγ=∅,undefined where γ - inaccessible cardinal) 52. Θ(ZFC+∀n∃n-inaccessible cardinal) 53. Θ(ZFC+∀n∃n-Mahlo cardinal) 54. Θ(ZFC+∀n∃n-Weakly compact cardinal) 55. Θ(ZFC+∀n∃П1n-indescribable cardinal) 56. Θ(ZFC+∀n∃n-subtle cardinal) 57. Θ(ZFC+∀n∃n-measurable cardinal) 58. Θ(Z2+PD) = Θ(ZFC+∀n∃n-Woodin cardinal) 59. Θ(ZFC+∀n∃n-huge cardinal) 60. Θ(ZFC+∃I0-cardinal)

zoo - D.A.Madore - A zoo of ordinals - 2017