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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.

## From D(l,0,1) to Epsilon Nought

From a previous blog post it is asserted that each iteration of $$D(l,0,1)$$ , for example

$$D(2,0,1) = D(1,D(2,0,0),D(2,0,0)) >> f_{\omega+2}(3)$$

will progressively increase the ordinal strength of a lower bounded $$f()$$ function. This blog will match this asserted progression against the usual omega to epsilon nought ordinal hierarchy.

## Evaluating omega.2+1

$$D(2,0,1) >> f_{\omega+2}(3)$$

$$D(3,0,1) >> f_{\omega.2}(3)$$

$$D(l,0,1) >> f_{\omega.2}(l)$$

$$D(l,0,2) = D(l-1,D(l,0,1),d(l,0,1)) >> f_{\omega + l-1}(f_{\omega.2}(l))$$

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

$$D(l,0,D(l,0,1)+1) = D(l-1,f_{\omega+l}(f_{\omega.2}(l)),f_{\omega+l}(f_{\omega.2}(l)))$$

$$>> f_{\omega+l-1}(f_{\omega+l}(f_{\omega.2}(l)))$$

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

$$= f_{\omega+l}(f_{\omega+l}(f_{\omega.2}(l))) = f_{\omega+l}^2(f_{\omega.2}(l))$$

## Strong D Functions D(l,m,n) with 3 Parameters

Lets start by comparing $$D(1,0,3)$$ to $$f_{\omega}(3)$$

$$D(1,0,0) = D(0,D(0,1,1),D(0,1,1)) = D(D(1,1),D(1,1)) = D(4,4) >> f_{\omega}(3)$$

$$D(1,0,1) = D(0,D(1,0,0),D(1,0,0)) = D(D(4,4),D(4,4)) >> f_{\omega}(D(4,4))$$

$$>> f_{\omega}(f_{\omega}(3)) >> f_{\omega}^2(3)$$

$$D(1,0,2) = D(0,D(1,0,1),D(1,0,1)) >> f_{\omega}(D(1,0,1)) >> f_{\omega}(f_{\omega}^2(3)) >> f_{\omega}^3(3)$$

First proof: $$D(1,0,n) >> f_{\omega}^4(n)$$

$$D(1,0,3) = D(0,D(1,0,2),D(1,0,2)) >> f_{\omega}(D(1,0,2)) >> f_{\omega}(f_{\omega}^3(3)) = f_{\omega}^4(3)$$

assume

$$D(1,0,n-1) >> f_{\omega}^4(n-1)$$

then

$$D(1,0,n) = D(0,D(1,0,n-1),D(1,0,n-1)) >> f_{\omega}(D(1,0,n-1)) >> f_{\omega}(f_{\omega}^4(n-1))$$

$$= f_{\omega}^4(f_{\omega}(n-1)) >> f_{\omega}^4(n)$$

Next calculation - getting to $$\omega+1$$

$$D(1,0,n+1) = D(0,D(1,0,n),D(1,0,n)) >> f_{\omega}(D(1,0,n)) >> f_{\omega}(f_{\omega}^4(n)) = f_{\omega}^5(n)$$

$$D(1,0,n+p) = D(0,D(1,0,n+p),D(1,0,n+p)) >> f_{\omega}^{p+4}(n)$$

$$D(1,0,n+n-4) >> f_{\omega}^{n-4+4}(n) = f_{\omega}^n(n) = f_{\omega+1}(n)$$

Second proof: $$D(1,m,0) >> f_{\omega}^m(f_{\omega+1}(m))$$

$$D(1,3,0) = D(0,D(1,2,3),D(1,2,3)) >> D(1,0,8) = D(1,0,6+6-4) >> f_{\omega+1}(6)$$

$$= f_{\omega}^6(6) = f_{\omega}^3(f_{\omega}^3(6) >> f_{\omega}^3(f_{\omega}^3(3)) = f_{\omega}^3(f_{\omega+1}(3))$$

assume

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

then

$$D(1,m,0) = D(0,D(1,m-1,m),D(1,m-1,m)) >> f_{\omega}(D(1,m-1,m))$$

$$>> f_{\omega}(f_{\omega}(D(1,m-1,m-1))$$

$$>> f_{\omega}^2(f_{\omega}(D(1,m-1,m-2)) >> f_{\omega}^3(f_{\omega}(D(1,m-1,m-3))$$

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

$$>> f_{\omega}^{m+1}(f_{\omega}^{m-1}(f_{\omega+1}(m-1))) = f_{\omega}^{m.2}(f_{\omega}^{m-1}(m-1))$$

$$= f_{\omega}^{m.2+m-2}(f_{\omega}(m-1)) >> f_{\omega}^{m.2+m-2}(m) >> f_{\omega}^{m.2-2}(f_{\omega}^m(m))$$

$$>> f_{\omega}^m(f_{\omega}^m(m)) = f_{\omega}^m(f_{\omega+1}(m))$$

Next calculation - general formula for $$D(1,m,n)$$

$$D(1,m,1) = D(D(1,m,0),D(1,m,0)) >> f_{f_{\omega}(D(1,m,0))}(f_{\omega}(D(1,m,0))) = f_{\omega}(D(1,m,0))$$

$$>> f_{\omega}(f_{\omega}^m(f_{\omega+1}(m))) = f_{\omega}^{m+1}(f_{\omega+1}(m))$$

$$D(1,m,n) >> f_{\omega}^{m+n}(f_{\omega+1}(m))$$

Next calculation - general formula for $$D(1,n,n)$$

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

Next calculation - $$D(2,0,0)$$

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

$$>> f_{\omega+1}(D(1,2,2)) >> f_{\omega+1}(f_{\omega}^{2.2}(f_{\omega+1}(2))))$$

$$= f_{\omega+1}(f_{\omega}^4(f_{\omega}^2(2))) = f_{\omega+1}(f_{\omega}^5(f_{\omega}(2))))$$

$$>> f_{\omega+1}(f_{\omega}^5(5)) = f_{\omega+1}(f_{\omega+1}(5)) = f_{\omega+1}^2(5)$$

Third proof: $$D(2,0,n) >> f_{\omega+1}^2(n)$$

$$D(2,0,5) >> D(2,0,0) >> f_{\omega+1}^2(5)$$

assume

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

then

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

$$>> f_{\omega+1}(f_{\omega+1}^2(n-1)) = f_{\omega+1}^2(f_{\omega+1}(n-1))$$

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

Next calculation - getting to $$\omega+2$$

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

$$>> f_{\omega+1}(D(2,0,n))) >> f_{\omega+1}(f_{\omega+1}^2(n))) = f_{\omega+1}^3(n))$$

$$D(2,0,n+p) = D(1,D(2,0,n+p),D(2,0,n+p)) >> f_{\omega+1}^{p+2}(n))$$

$$D(2,0,n+n-2) >> f_{\omega+1}^{n-2+2}(n) = f_{\omega+2}(n)$$

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

$$D(2,3,0) = D(1,D(2,2,3),D(2,2,3)) >> D(1,D(2,0,8),D(2,0,8))$$

$$>> f_{\omega}^{D(2,0,8).2}(f_{\omega+1}(D(2,0,8))) >> f_{\omega+1}(D(2,0,8))$$

$$= f_{\omega+1}(D(2,0,5+5-2)) >> f_{\omega+1}(f_{\omega+2}(5))$$

$$= f_{\omega+1}(f_{\omega+1}^5(5)) = f_{\omega+1}^3(f_{\omega+1}^3(5)) >> f_{\omega+1}(f_{\omega+1}^3(3))$$

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

assume

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

then

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

$$>> f_{\omega+1}(D(2,m-1,m)) >> f_{\omega+1}(f_{\omega+1}(D(2,m-1,m-1))$$

$$>> f_{\omega+1}^2(f_{\omega+1}(D(2,m-1,m-2)) >> f_{\omega+1}^3(f_{\omega+1}(D(2,m-1,m-3))$$

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

$$>> f_{\omega+1}^{m+1}(f_{\omega+1}^{m-1}(f_{\omega+1}^{m-1}(f_{\omega+2}(m-1))))$$

$$>> f_{\omega+1}^{m.2}(f_{\omega+1}^{m-1}(f_{\omega+2}(m-1))) = f_{\omega+1}^{m.3-1}(f_{\omega+1}^{m-1}(m-1))$$

$$= f_{\omega+1}^{m.4-3}(f_{\omega+1}(m-1)) >> f_{\omega+1}^{m.4-3}(m) = f_{\omega+1}^{m.3-3}(f_{\omega+1}^m(m))$$

$$= f_{\omega+1}^{m.3-3}(f_{\omega+2}(m)) >> f_{\omega+1}^m(f_{\omega+2}(m))$$

Next calculation - general formula for $$D(2,m,n)$$

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

$$>> f_{\omega+1}(f_{\omega+1}^m(f_{\omega+2}(m))) = f_{\omega+1}^{m+1}(f_{\omega+2}(m))$$

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

Next calculation - general formula for $$D(2,n,n)$$

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

Fifth proof: $$D(l,0,n) >> f_{\phi}^2(n)$$

assume

$$D(l-1,0,n) >> f_{\phi-1}^2(n))$$

$$D(l-1,m,n) >> f_{\phi-1}^{m+n}(f_{\phi}(m))$$

then

$$D(l,0,0) = D(l-1,D(l,l-1,l-1),D(l,l-1,l-1)) >> f_{\phi-1}^{D(l,l-1,l-1))+D(l,l-1,l-1))}(f_{\phi}(D(l,l-1,l-1))))$$

$$>> f_{\phi}(D(l,l-1,l-1)) >> f_{\phi}(D(l-1,D(l,l-1,l-2),D(l,l-1,l-2)))$$

$$>> f_{\phi}(f_{\phi-1}^{D(l,l-1,l-2)+D(l,l-1,l-2)}(f_{\phi}(D(l,l-1,l-2)))))$$

$$>> f_{\phi}(f_{\phi-1}(f_{\phi}(D(l,l-1,l-2)))) >> f_{\phi}(f_{\phi}(D(l,l-1,l-2))) >> f_{\phi}^2(D(l,l-1,l-2))$$

$$>> f_{\phi}^2(3)$$

$$D(l,0,3) >> D(l,0,0) >> f_{\phi}^2(3))$$

assume

$$D(l,0,n-1) >> f_{\phi}^2(n-1)$$

then

$$D(l,0,n) = D(l-1,D(l,0,n-1),D(l,0,n-1)) >> f_{\phi-1}^{D(l,0,n-1))+D(l,0,n-1))}(f_{\phi}(D(l,0,n-1))))$$

$$>> f_{\phi}(D(l,0,n-1)) >> f_{\phi}(f_{\phi}^2(n-1))$$

$$= f_{\phi}^2(f_{\phi}(n-1)) >> f_{\phi}^2(n)$$

Sixth proof: $$D(l,m,n) >> f_{\phi}^{m+n}(f_{\phi+1}(m))$$

assume

$$D(l-1,m,n) >> f_{\phi-1}^{m+n}(f_{\phi}(m))$$

$$D(l,0,n) >> f_{\phi}^2(n)$$

then

$$D(l,3,0) = D(l-1,D(l,2,3),D(l,2,3)) >> f_{\phi-1}^{D(l,2,3).2}(f_{\phi}(D(l,2,3)))$$

$$>> f_{\phi}(D((l,2,3)) >> f_{\phi}(D(l-1,D(l,2,2),D(l,2,2)))$$

$$>> f_{\phi}(f_{\phi-1}^{D(l,2,2)+D(l,2,2)}(f_{\phi}(D(l,2,2))))$$

$$>> f_{\phi}(f_{\phi}(D(l,2,2))) >> f_{\phi}^2(D(l-1,D(l,2,1),D(l,2,1))$$

$$>> f_{\phi}^2(D(l,2,1)) >> f_{\phi}^2(f_{\phi}(D(l-1,D(l,2,0),D(l,2,0)))$$

$$>> f_{\phi}^3(D(l,2,0)) >> f_{\phi}^3(f_{\phi}(D(l-1,D(l,1,2),D(l,1,2)))$$

$$>> f_{\phi}^4(D(l,1,2)) >> f_{\phi}^4(D(l,0,3)) >> f_{\phi}^4(f_{\phi}^2(3)) = f_{\phi}^3(f_{\phi}^3(3))$$

$$= f_{\phi}^3(f_{\phi+1}(3))$$

assume

$$D(l,m-1,0) >> f_{\phi}^m(f_{\phi+1}(m-1))$$

then

$$D(l,m,n) = D(l-1,D(l,m,n-1),D(l,m,n-1)) >> f_{\phi-1}^{D(l,m,n-1)+D(l,m,n-1)}(f_{\phi}(D(l,m,n-1)))$$

$$>> f_{\phi}(D((l,m,n-1)) >> f_{\phi}(D(l-1,D(l,m,n-2),D(l,m,n-2)))$$

$$>> f_{\phi}(f_{\phi-1}^{D(l,m,n-2)+D(l,m,n-2)}(f_{\phi}(D(l,m,n-2))))$$

$$>> f_{\phi}(f_{\phi}(D(l,m,n-2))) >> f_{\phi}^2(D(l-1,D(l,m,n-3),D(l,m,n-3))$$

$$>> f_{\phi}^2(D(l,m,n-3)) >> f_{\phi}^2(f_{\phi}(D(l-1,D(l,m,n-4),D(l,m,n-4)))$$

...

$$>> f_{\phi}^{n-1}(D(l,m,n-n)) >> f_{\phi}^{n-1}(D(l,m,0) >> f_{\phi}^{n-1}(f_{\phi}(D(l-1,D(l,m-1,m),D(l,m-1,m)))$$

$$>> f_{\phi}^n(D((l,m-1,m)) >> f_{\phi}^n(D(l-1,D(l,m-1,m-1),D(l,m-1,m-1)))$$

$$>> f_{\phi}^{n+1}(D((l,m-1,m-1)) >> f_{\phi}^{n+1}(D(l-1,D(l,m-1,m-2),D(l,m-1,m-2)))$$

$$>> f_{\phi}^{n+2}(D((l,m-1,m-2)) >> f_{\phi}^{n+1}(D(l-1,D(l,m-1,m-3),D(l,m-1,m-3)))$$

...

$$>> f_{\phi}^{n+m}(D((l,m-1,m-m)) >> f_{\phi}^{n+m}(D(l-1,m-1,0)) >> f_{\phi}^{n+m}(f_{\phi}^m(f_{\phi+1}(m-1)))$$

$$= f_{\phi}^{n+m.2}(f_{\phi+1}(m-1)) = f_{\phi}^{n+m.2}(f_{\phi}^{m-1}(m-1))$$

$$= f_{\phi}^{n+m.3-2}(f_{\phi}(m-1)) >> f_{\phi}^{n+m.3-2}(m) = f_{\phi}^{n+m.2-2}(f_{\phi}^{m}(m))$$

$$= f_{\phi}^{n+m.2-2}(f_{\phi+1}(m))$$