bo198214 Wrote:obviously \( \frac{dx[4]n}{dx}\neq w^n \) and \( \frac{d x[4]n}{dx}\neq \left(w+\frac{\ln(x)}{x}\right)^n \) whatever \( w \) is. What is it btw?
Thanks again. This will help to put it on better footing or dismiss. I was aware that differentiation by x is wrong here since speed is mentioned in relation to x-> infinity. So differentiation, if possible at all , must be either d/d(?) leaving it open, d/d (oo), d/d (i) or something else. Perhaps time calculus as it allows to pick out subsets from real numbers by using graininess function mju(t), so differentiation vs. mixed discrete continuous variable is possible. What if mju(t) would be w(t) of mju(w) ( speculation again
).w=\( \omega \) is Cantors Ordinal number and its relation to speed of growth of functions f(x) as x->oo is mentioned in Conways Book of Numbers page 299.
It mentions Infinitary calculus which uses these notions-i have not been able to find a good reference yet how this infinitary calculus is constructed.
I thought that since cardinals/ordinals have well developed theoretic bacground up to continium hypothesis, applying them to the speed of growth of hyperoperations may simplify proofs of some basic identities and allow classification of hyperoperations as performing certain transformations of number types if speed is fast enough or within some limits.
Idea will be only applicable to certain subsets of reals and other numbers known today , but that will better illuminate differences between these subsets. e.g. why region x<e^(1/e) differs from x>e^(1/e) etc., why negative reals can be turned into complex easily while positive can not and in the region <e^(1/e) even infinite tetration is not fast enough.
For example, i strongly feel (wihout proof) that this will lead to finding of a speed that turns certain numbers ( e.g subset of integers) into surreal numbers, and maybe more.
Ivars
