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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.


Questions and Comments

As always, any feedback on the assumptions made on this page, will be greatly appreciated.

B1mb0w


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The following references are outdated because the above proofs and examples will be more reliable.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.

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