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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

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