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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.


An Example

Lets compare different combinations of Fast-growing hierarchy Functions using nested \(f_{\omega}()\) functions

\(f_{\omega+2}^2(3) = f_{\omega+2}(f_{\omega+2}(3))\)

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

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

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

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

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

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

\(= f_{\omega+2}(f_{\omega}^{f_{\omega}^{f_{\omega}^3(3)+3}(3)}(f_{\omega}^{f_{\omega}^3(3)+3}(3)))\)

\(= f_{\omega+2}(f_{\omega}^{f_{\omega}^{f_{\omega}^3(3)+3}(3)+f_{\omega}^3(3)+3}(3)))\)

\(= f_{\omega+1}^{f_{\omega}^{f_{\omega}^{f_{\omega}^3(3)+3}(3)+f_{\omega}^3(3)+3}(3))}(f_{\omega}^{f_{\omega}^{f_{\omega}^3(3)+3}(3)+f_{\omega}^3(3)+3}(3)))\)

and so on.


An Example up to \(\omega+2\) only

\(f_{\omega}^{ f_{\omega+1}^2(f_{\omega+2}(3)) }( f_{\omega}^{ f_{\omega+1}(f_{\omega+2}(3) ) }( f_{\omega}^{f_{\omega+2}(3)}(f_{\omega}^{f_{\omega+1}^2(3)}(f_{\omega}^{f_{\omega+1}(3)}(f_{\omega+1}(3)))) ) )\)

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

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

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

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

\(= f_{\omega+1}^3(f_{\omega+2}(3))\) which is much smaller than

\(<< f_{\omega+2}(f_{\omega+2}(3)) = f_{\omega+2}^2(3)\)


A General Rule for \(\omega+1\)

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

and

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

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

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

then

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


A General Rule for \(\omega+2\)

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

and

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

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

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

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

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

then

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

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