Editing Knuth's up-arrow notation (section)

==Generalizations==
Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an ''n''-arrow operator  <math>\uparrow^n</math> is useful (and also for descriptions with a variable number of arrows), or equivalently, [[hyper operator]]s.

Some numbers are so large that even that notation is not sufficient. The [[Conway chained arrow notation]] can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful.

:<math>
  \begin{matrix}
   a\uparrow^n b & = & a [n+2] b & = & a\to b\to n \\
   \text{(Knuth)} & & \text{(hyperoperation)} & & \text{(Conway)}
  \end{matrix}
 </math>
: <math>6\uparrow\uparrow4 = \underbrace{6^{6^{.^{.^{.^{6}}}}}}_4</math>, Since <math>6\uparrow\uparrow4 = 6^{6^{6^{6}}} = 6^{6^{46,656}}</math>, Thus the result comes out with <math>\underbrace{6^{6^{.^{.^{.^{6}}}}}}_4</math>

: <math>10\uparrow(3\times10\uparrow(3\times10\uparrow15)+3) = \underbrace{100000\ldots000}_{ \underbrace{300000\ldots003}_{\underbrace{300000\ldots000}_{15} }}</math> or <math>10^{3\times10^{3\times10^{15}}+3}</math>

Even faster-growing functions can be categorized using an [[ordinal number|ordinal]] analysis called the [[fast-growing hierarchy]]. The fast-growing hierarchy uses successive function iteration and diagonalization to systematically create faster-growing functions from some base function <math>f(x)</math>. For the standard fast-growing hierarchy using <math>f_0(x) = x+1</math>, <math>f_2(x)</math> already exhibits exponential growth, <math>f_3(x)</math> is comparable to tetrational growth and is upper-bounded by a function involving the first four hyperoperators;. Then, <math>f_\omega(x)</math> is comparable to the [[Ackermann function]], <math>f_{\omega + 1}(x)</math> is already beyond the reach of indexed arrows but can be used to approximate [[Graham's number]], and <math>f_{\omega^2}(x)</math> is comparable to arbitrarily-long Conway chained arrow notation.

These functions are all computable. Even faster computable functions, such as the [[Goodstein sequence]] and the [[TREE(3)|TREE sequence]] require the usage of large ordinals, may occur in certain combinatorical and proof-theoretic contexts. There exist functions which grow uncomputably fast, such as the [[Busy Beaver]], whose very nature will be completely out of reach from any up-arrow, or even any ordinal-based analysis.