User blog comment:B1mb0w/Mapping D(l,0,1) to epsilon nought/@comment-1605058-20150702203134

With this blog post I think I see where your mistake is. I'm not sure how to explain it best, but I'll try.

Note that getting from \(D(2,0,n)\) to \(D(2,0,n+1\) we just apply to the former number function \(F:n\mapsto D(1,n,n)\) (by definition of D function). Now let's see what you are doing at different steps.

When going from \(D(2,0,1)\) to \(D(2,0,2\) you go from bound \(f_{\omega+2}(3)\) to bound \(f_{\omega+2}^2(3)\), suggesting that application of \(F\) corresponds to application of \(f_{\omega+2}\). But later, say when going from \(D(2,0,5)\) to \(D(2,0,6\), you go from bound \(f_{\omega.2+1}(3)\) to \(f_{\omega.2+1}^2(3)\), making one think that application of \(F\) will be at least \(f_{\omega.2+1}\). This is precisely where your mistake is - at each step you do the same thing to the left hand side, namely apply \(F\) to \(D(2,0,n)\), and you think that at each step you are doing the same thing to your bound, but that's not true. Going from \(f_{\omega+2}(3)\) to \(f_{\omega+2}^2(3)\) is entirely different thing than from \(f_{\omega.2+1}(3)\) to \(f_{\omega.2+1}^2(3)\).

Hopefully this clarifies some things.