User blog comment:B1mb0w/Strong D Function/@comment-1605058-20150702183610/@comment-10262436-20150809022331

OK. Now this is interesting.

It is easy to define D(2,n) = n.3+8 = g(n). These are elementary functions.

Then because D(3,0) = g(g(3)). We then define D(3,n) = A, where A is a term built from the variable n, constant 3, and previously given functions g and D (i.e F). This is easy to do for n=1 because D(3,1) = g(3).3+8 and with "simple recursion", it can be done for any n. For simplicity, lets call all the constructions in A, h(n).

At this point we have D(4,0) = h(h(4)). Now it gets confusing. If we want to define D(4,n) = B, where A is a term built from the variable n, constant 4, and previously given functions g, h and D.

Then how do we do this ? We may only use previously given functions of D, which are, D(2,n), D(3,n) and D(4,0).

D(4,n) = simple recursion of h(h(h(...h(4)) with n nested functions. Do we now define a new function j(n) to represent this function ?

If we do, then the rest of the constructions follow OK because they are all simple recursion nested functions.

But what does this prove. The sequence of g,h,j functions simply have the same behaviour as the D function so we have just shifted the proof problem from D to g,h,j....

Is this correct ?

Because if we do not progressively define the functions h,j using simple recursion, then it seems to me, that it is not possible to define D(m,n) = A using constructions of variables, constants, previously defined D functions and only g, using simple recursion (you will need far more powerful recursion to do it).

Is this correct ?