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This blog has been replaced. Simpler and more rigorous calculations of the Strong D Function can be found at another blog that can be accessed by this link.


Comparing Fast Growing Hierarchy Functions

This blog is a working page to capture various rules that can be used to compare different combinations of Fast-growing hierarchy Functions.


Basics

1. \(f_b(a) = f_{b-1}^a(a)\)

2. \(f_{b-1}^a(a) = f_b(a)\)

3. \(f_{b-2}^a(a) = f_{b-1}(a)\)

4. \(f_{b-1}^{a-1}(f_{b-2}^a(a)) = f_{b-1}^{a-1}(f_{b-1}(a)) = f_{b-1}^{a}(a) = f_b(a)\)


Basics with \(\omega\)

5. \(f_{\omega}(a) = f_a(a)\)

6. \(f_a(a) = f_{\omega}(a)\)

7. \(f_a^2(a) = f_a(f_a(a)) = f_a(f_{\omega}(a)) << f_{\omega}(f_{\omega}(a)) = f_{\omega}^2(a))\)

8. \(f_{\omega+1}(a) = f_{\omega}^a(a)\)

9. \(f_{\omega+b}^c(a) = f_{\omega+b}^{c-1}(f_{\omega+b}(a))\)


Some Comparisons

10. \(f_b^c(a) = f_{b}^{c-a}(f_{b}^a(a)) = f_{b}^{c-a}(f_{b+1}(a))\) when \(c > a\)

11. \(f_b^c(a) = f_{b}^{c-a}(f_{b+1}(a)) = f_{b}^{c-a-f_{b+1}(a)}(f_b^{f_{b+1}(a)}(f_{b+1}(a)))\) when \(c > f_{b+1}(a)\)

\(= f_{b}^{c-a-f_{b+1}(a)}(f_{b+1}(f_{b+1}(a))) = f_{b}^{c-a-f_{b+1}(a)}(f_{b+1}^2(a))\)

12. \(f_b^{f_{b+1}(a)+a}(a) = f_{b+1}^2(a)\)

then

13. \(f_{b}^{f_{b+1}^2(a).2}(a) = f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}^2(a)}(a)) >> f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}(a)+f_{b+1}(a)}(a)) >> f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}(a)+a}(a))\)

\(= f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}(a)}(f_b^a(a))) = f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}(a)}(f_{b+1}(a))) = f_{b}^{f_{b+1}^2(a)}(f_{b+1}(f_{b+1}(a)))\)

\(= f_{b}^{f_{b+1}^2(a)}(f_{b+1}^2(a)) = f_{b+1}(f_{b+1}^2(a)) = f_{b+1}^3(a)\)

and in general

14. \(f_{b}^{f_{b+1}^n(a).2}(a) >> f_{b+1}^{n+1}(a)\)

15. \(f_n(a+1) = f_{n-1}^{a+1}(a+1) = f_{n-1}(f_{n-1}^{a}(a+1)) >> f_{n-1}(f_{n-1}^{a}(a)) = f_{n-1}(f_{n}(a))\)

16. work in progress


Some Comparisons with \(\omega\)

17. \(f_b^c(a) = f_{b}^{c-1}(f_{b}(a)) = f_{b}^{c-1}(f_{\omega}(a))\) when \(b = a\) when \(c > 0\)

18. \(f_{\omega}(a+1) = f_{a+1}(a+1) = f_{a}^{a+1}(a+1) >> f_{a}^{a+1}(a)\)

19. \(f_{\omega}^{c}(a+1) >> f_{\omega}^{c-1}(f_{a}^{a+1}(a))\)

20. \(f_{\omega}^{c+1}(a) = f_{\omega}^{c}(f_{\omega}(a))\)

21. \(f_{\omega+1}^{c}(a) = f_{\omega+1}^{c-1}(f_{\omega+1}(a)) = f_{\omega+1}^{c-1}(f_{\omega}^a(a))\)

and

22. \(f_{\omega}^{f_{\omega+1}^{a-1}(a).2}(a) >> f_{\omega+1}^{a-1+1}(a) = f_{\omega+1}^a(a) = f_{\omega+2}(a)\)

23. work in progress


Some Comparisons with \(\omega+1\) and \(\omega+2\)

Refer to detailed calculations here

24. \(f_{\omega}^{f_{\omega+1}^n(3).2}(3) >> f_{\omega+1}^{n+1}(3)\)

25. \(f_{\omega}^{f_{\omega+1}^n(f_{\omega+2}(3)).2}(3) >> f_{\omega+1}^{n+1}(f_{\omega+2}(3))\)

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