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 $f_{\varepsilon_{\omega_1^\text{CK}+1}}(64)$. 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)

(\(\varepsilon_{\omega_1^\text{CK}}=\omega_1^{CK}\)) LittlePeng9 (talk) 07:07, May 4, 2014 (UTC)

Sorry, a minor typo. Ikosarakt1 (talk ^ contribs) 11:50, May 4, 2014 (UTC)

I think, in case of an very fast-growing function, such as Bird's U(n), doubling the FGH growth rate is not enough, I see a very, very naive extension that does that: Define {a,b}₂ = U^b(a), other rules remain the same, and then call U₂(n). Wythagoras (talk) 07:04, May 4, 2014 (UTC)

This trick will work only for functions which are based directly on array notation. We can't extend TREE(n), BB(n) or Rayo(n) in this way. Ikosarakt1 (talk ^ contribs) 11:50, May 4, 2014 (UTC)

Some Examples........ Using BIG FOOT or N........

N + 1

N^N

N!

{LN,N,}N,N

N x N

N!N

EN#N

N^^N

N^^^n

EN

N^N^N

N > N > N (> is the conway chained arrow notation) ^{—Preceding unsigned comment added by Antares.I.G.Harrison (talk • contribs)}

informality

I'm going to explain this edit which removed a bunch of the article. I removed things because they tried to create the impression of formality using terms that are generally informal. I'm really okay with subjective discussion of googological concepts, but not when it tries to disguise itself as math. Specifically i object to these terms:

• if system S, contains concept C, that diagonalizing over S, and using C to extend the diagonalization is a naive extension on S
• doubling the order-type may already be enough to avoid being called a naive extension
• However, as a general rule of thumb, the first non-naive extension is the smallest system which introduces a new concept which does not naturally, or necessarily follow from system S.

The last one I removed because I don't think it's true. Generalizations of googological systems often do "naturally follow," especially when they're natural generalizations. -- vel! 16:58, February 15, 2015 (UTC)

(i wrote this 15 minutes after crashing on the couch for like 12 hours so sorry if this doesnt make sense -- vel! 16:59, February 15, 2015 (UTC))

Yeah, I can agree with this. Wythagoras (talk) 17:17, February 15, 2015 (UTC)

We are now talking things over on IRC about this (sorry wyth -- i should really work on that proxy) -- vel! 17:34, February 15, 2015 (UTC)

Oops, I explain

You might saw in "Wiki activity" that I edited this page then undo it. In fact I thought that the U function that it described was the naive extension, defined by U(a) = something with BEAF. It was a misread, sorry Fluoroantimonic Acid (talk) 17:27, June 30, 2015 (UTC)