10,843 Pages

## REPLACED

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