Editing Tetration (section)

== Terminology ==
There are many terms for tetration, each of which has some logic behind it, but some have not become commonly used for one reason or another. Here is a comparison of each term with its rationale and counter-rationale.

* The term ''tetration'', introduced by Goodstein in his 1947 paper ''Transfinite Ordinals in Recursive Number Theory''<ref>{{cite journal |author=R. L. Goodstein |title=Transfinite ordinals in recursive number theory |jstor=2266486 |journal=Journal of Symbolic Logic |volume=12 |year=1947 |issue=4 |doi=10.2307/2266486 |pages=123–129|s2cid=1318943 }}</ref> (generalizing the recursive base-representation used in [[Goodstein's theorem]] to use higher operations), has gained dominance. It was also popularized in [[Rudy Rucker]]'s ''[[Infinity and the Mind]]''.
* The term ''superexponentiation'' was published by Bromer in his paper ''Superexponentiation'' in 1987.<ref>{{cite journal |author=N. Bromer |title=Superexponentiation |journal=Mathematics Magazine |volume=60 |issue=3 |year=1987 |pages=169–174 |jstor=2689566|doi=10.1080/0025570X.1987.11977296 }}</ref> It was used earlier by Ed Nelson in his book Predicative Arithmetic, [[Princeton University Press]], 1986.
* The term ''hyperpower''<ref>{{cite journal |author=J. F. MacDonnell |title=Somecritical points of the hyperpower function <math>x^{x^{\dots}}</math> |journal=International Journal of Mathematical Education |year=1989 |volume=20 |issue=2 |pages=297–305 |mr=994348 |url=http://www.faculty.fairfield.edu/jmac/ther/tower.htm |doi=10.1080/0020739890200210|url-access=subscription }}</ref> is a natural combination of ''hyper'' and ''power'', which aptly describes tetration. The problem lies in the meaning of ''hyper'' with respect to the [[hyperoperation]] sequence. When considering hyperoperations, the term ''hyper'' refers to all ranks, and the term ''super'' refers to rank 4, or tetration. So under these considerations ''hyperpower'' is misleading, since it is only referring to tetration.
* The term ''power tower''<ref>{{MathWorld |urlname=PowerTower |title=Power Tower}}</ref> is occasionally used, in the form "the power tower of order {{mvar|n}}" for <math>{\ \atop {\ }} {{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}} \atop n}</math>. Exponentiation is easily misconstrued: note that the operation of raising to a power is right-associative (see [[#Direction of evaluation|below]]). Tetration is iterated ''exponentiation'' (call this [[Operator associativity#rightAssociative|right-associative]] operation ^), starting from the top right side of the expression with an instance a^a (call this value c). Exponentiating the next leftward a (call this the 'next base' b), is to work leftward after obtaining the new value b^c. Working to the left, use the next a to the left, as the base b, and evaluate the new b^c. 'Descend down the tower' in turn, with the new value for c on the next downward step.

Owing in part to some shared terminology and similar [[Mathematical notation|notational symbolism]], tetration is often confused with closely related functions and expressions. Here are a few related terms:

{|class="wikitable"
|+Terms related to tetration
!scope="col"| Terminology
!scope="col"| Form
|-
!scope="row"|Tetration
|<math>a^{a^{\cdot^{\cdot^{a^a}}}}</math>
|-
!scope="row"|Iterated exponentials
|<math>a^{a^{\cdot^{\cdot^{a^x}}}}</math>
|-
!scope="row"|Nested exponentials (also towers)
|<math>a_1^{a_2^{\cdot^{\cdot^{a_n}}}}</math>
|-
!scope="row"|Infinite exponentials (also towers)
|<math>a_1^{a_2^{a_3^{\cdot^{\cdot^\cdot}}}}</math>
|}

In the first two expressions {{mvar|a}} is the ''base'', and the number of times {{mvar|a}} appears is the ''height'' (add one for {{mvar|x}}). In the third expression, {{mvar|n}} is the ''height'', but each of the bases is different.

Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean [[iterated function|iterated]] [[power (mathematics)|powers]] or iterated [[exponential function|exponentials]].