## FANDOM

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This blog has been updated to use the better formula and calculations from another blog that will be easier to follow and understand.

## Strong D Function and $$f_{\epsilon_0}(m)$$

This blog will compare the growth rate of the Strong D Function to the fast growing hierarchy function $$f_{\epsilon_0}(n)$$. Starting with:

$$D(n+1,n-2) >> f_n(n) >> f_{\omega}(n)$$ more information at this link

$$D(1,m,0) >> f_{\omega}^m(f_{\omega+1}(m))$$ more information at this link

$$D(2,m,0) >> f_{\omega+1}^m(f_{\omega+2}(m))$$ more information at this link

and

$$D(l,m,0) >> f_{\phi}^m(f_{\phi+1}(m))$$ when $$D(l-1,m,0) >> f_{\phi-1}^m(f_{\phi}(m))$$ more information at this link

## $$f_{\omega.2}(m)$$

$$D(m,m,0) >> f_{\omega}^m(f_{\omega+m}(m)) = f_{\omega+m}(m) = f_{\omega.2}(m)$$

## $$f_{\omega.2+m}(m)$$

At this point, Strong D Function runs out of steam. Without proof, the approximate size of the Strong D Function required to reach these ordinals are:

$$D(2^m,m,0,0) >> f_{\omega.2+1}(m)$$

$$D(2^{2^m},m,0,0) >> f_{\omega.2+2}(m)$$

$$D(m,m,0,0,0) >> f_{\omega.2+m}(m) = f_{\omega.3}(m)$$

## $$f_{\omega^2}(m)$$ and beyond

Also Without proof, the approximate size of the Strong D Function required to reach these ordinals are:

$$D(m,m,0,...,0) >> f_{\omega.m}(m) >> f_{\omega^2}(m)$$ with approximately $$m$$ zeros.

and

$$D(m,m,0,...,0,...,0,...,0) >> f_{\omega^2+1}(m)$$ with approximately $$2^m$$ zeros.

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## References

Strong D Function and $$f_{\epsilon_0}(m)$$

The following references are outdated because the above proofs and examples will be more reliable. Please keep this in mind if you refer to any of these blogs.