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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

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

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

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

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

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

...

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

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

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

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

...

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

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

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

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

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