Talk:Naive extension

I see at least 2 levels of naive extensions:

1) Suppose we have the number $$\Sigma(64)$$. The most naive thing is ignore the fact that it is 64-th member of some sequence and just take $$A = \Sigma(64)$$. Then we want to beat the number A and define A+1, "two times A", "thousand times A", "A times A", "1 followed by A zeroes" or "A factorial". This is the "layman" level - for the people who aren't used to think about numbers as the members of sequences.

2) If we have $$\Sigma(64)$$, why not have $$\Sigma(65), \Sigma(66), \Sigma(\Sigma(64)), \Sigma^3(64), \Sigma^64(64)$$? It seems to be better and more logical strategy, as there is non-trivial gulf between $$\Sigma(65)$$ and $$\Sigma(64)$$. However, when we extend it in this way, we completely ignore the definition of $$\Sigma(n)$$ itself and thus we get functions not much powerful in terms of FGH.

The non-naive extension for $$\Sigma(64)$$ which pretends to dominate it in large number race, must be independent from it and $$\Sigma(n)$$ in general. We can do it in the following way: define $$\omega_1^\text{CK}[n]$$ and prove that $$f_{\omega_1^\text{CK}}(n) \approx \Sigma(n)$$. Then define the number $$f_{\varepsilon_{\omega_1^\text{CK}}}(64)$$. Only for now we can say that guy who defined this number dominated the guy who defined $$\Sigma(64)$$. Ikosarakt1 (talk ^ contribs) 06:29, May 4, 2014 (UTC)