Editing Tetration (section)

== Extensions ==
Tetration can be extended in two different ways; in the equation <math>^na\!</math>, both the base {{mvar|a}} and the height {{mvar|n}} can be generalized using the definition and properties of tetration. Although the base and the height can be extended beyond the non-negative integers to different [[domain of a function|domains]], including <math>{^n 0}</math>, complex functions such as <math>{}^{n}i</math>, and heights of infinite {{mvar|n}}, the more limited properties of tetration reduce the ability to extend tetration.

=== Extension of domain for bases ===
==== Base zero ====
The exponential <math>0^0</math> is not consistently defined. Thus, the tetrations <math>\,{^{n}0}</math> are not clearly defined by the formula given earlier. However, <math>\lim_{x\rightarrow0} {}^{n}x</math> is well defined, and exists:<ref>{{cite web |url=https://math.blogoverflow.com/2015/01/05/climbing-the-ladder-of-hyper-operators-tetration/ |title=Climbing the ladder of hyper operators: tetration |series=Stack Exchange Mathematics Blog |website=math.blogoverflow.com |access-date=2019-07-25}}</ref>
:<math>\lim_{x\rightarrow0} {}^{n}x = \begin{cases}
  1, & n \text{ even} \\
  0, & n \text{ odd}
\end{cases}</math>

Thus we could consistently define <math>{}^{n}0 = \lim_{x\rightarrow 0} {}^{n}x</math>. This is analogous to defining <math>0^0 = 1</math>.

Under this extension, <math>{}^{0}0 = 1</math>, so the rule <math>{^{0}a} = 1</math> from the original definition still holds.

==== Complex bases ====
[[File:Tetration period.png|thumbnail|alt=A colorful graph that shows the period getting much larger|Tetration by period]]
[[File:Tetration escape.png|thumbnail|alt=A colorful graph that shows the escape getting much larger|Tetration by escape]]

Since [[complex number]]s can be raised to powers, tetration can be applied to ''bases'' of the form {{math|''z'' {{=}} ''a'' + ''bi''}} (where {{mvar|a}} and {{mvar|b}} are real). For example, in {{math|{{sup|''n''}}''z''}} with {{math|''z'' {{=}} ''i''}}, tetration is achieved by using the [[principal branch]] of the [[natural logarithm]]; using [[Euler's formula]] we get the relation:

: <math>i^{a+bi} = e^{\frac{1}{2}{\pi i} (a + bi)} = e^{-\frac{1}{2}{\pi b}} \left(\cos{\frac{\pi a}{2}} + i \sin{\frac{\pi a}{2}}\right)</math>

This suggests a recursive definition for {{math|{{sup|''n''+1}}''i'' {{=}} ''a′'' + ''b′i''}} given any {{math|{{sup|''n''}}''i'' {{=}} ''a'' + ''bi''}}:
: <math>\begin{align}
  a' &= e^{-\frac{1}{2}{\pi b}} \cos{\frac{\pi a}{2}} \\[2pt]
  b' &= e^{-\frac{1}{2}{\pi b}} \sin{\frac{\pi a}{2}}
\end{align}</math>

The following approximate values can be derived:
{| class="wikitable"
|+Values of tetration of complex bases
|-
!scope="col"| <math display="inline">{}^{n}i</math>
!scope="col"| Approximate value
|-
!scope="row"| <math display="inline">{}^{1}i = i</math>
| {{math|''i''}}
|-
!scope="row"| <math display="inline">{}^{2}i = i^{\left({}^{1}i\right)}</math>
| {{math|0.2079}}
|-
!scope="row"| <math display="inline">{}^{3}i = i^{\left({}^{2}i\right)}</math>
| {{math|0.9472 + 0.3208''i''}}
|-
!scope="row"| <math display="inline">{}^{4}i = i^{\left({}^{3}i\right)}</math>
| {{math|0.0501 + 0.6021''i''}}
|-
!scope="row"| <math display="inline">{}^{5}i = i^{\left({}^{4}i\right)}</math>
| {{math|0.3872 + 0.0305''i''}}
|-
!scope="row"| <math display="inline">{}^{6}i = i^{\left({}^{5}i\right)}</math>
| {{math|0.7823 + 0.5446''i''}}
|-
!scope="row"| <math display="inline">{}^{7}i = i^{\left({}^{6}i\right)}</math>
| {{math|0.1426 + 0.4005''i''}}
|-
!scope="row"| <math display="inline">{}^{8}i = i^{\left({}^{7}i\right)}</math>
| {{math|0.5198 + 0.1184''i''}}
|-
!scope="row"| <math display="inline">{}^{9}i = i^{\left({}^{8}i\right)}</math>
| {{math|0.5686 + 0.6051''i''}}
|}

Solving the inverse relation, as in the previous section, yields the expected {{math|{{sup|0}}''i'' {{=}} 1}} and {{math|{{sup|−1}}''i'' {{=}} 0}}, with negative values of {{mvar|n}} giving infinite results on the imaginary axis.{{citation needed|date=January 2025}} Plotted in the [[complex plane]], the entire sequence spirals to the limit {{math|0.4383 + 0.3606''i''}}, which could be interpreted as the value where {{mvar|n}} is infinite.

Such tetration sequences have been studied since the time of Euler, but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the infinitely iterated exponential function. Current research has greatly benefited by the advent of powerful computers with [[fractal]] and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.{{Citation needed|date=July 2019}}

=== Extensions of the domain for different heights ===
==== Infinite heights ====
[[File:Infinite power tower.svg|thumb|alt=A line graph with a rapid curve upward as the base increases|<math>\textstyle \lim_{n\rightarrow \infty} {}^nx</math> of the infinitely iterated exponential converges for the bases <math>\textstyle \left(e^{-1}\right)^e \le x \le e^{\left(e^{-1}\right)}</math>]]
[[File:TetrationConvergence3D.png|thumbnail|alt=A three dimensional Cartesian graph with a point in the center|The function <math>\left| \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} \right|</math> on the complex plane, showing the real-valued infinitely iterated exponential function (black curve)]]

Tetration can be extended to [[Infinity|infinite]] heights; i.e., for certain {{mvar|a}} and {{mvar|n}} values in <math>{}^{n}a</math>, there exists a well defined result for an infinite {{mvar|n}}. This is because for bases within a certain interval, tetration converges to a finite value as the height tends to [[infinity]]. For example, <math>\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}</math> converges to 2, and can therefore be said to be equal to 2. The trend towards 2 can be seen by evaluating a small finite tower:

: <math>\begin{align}
  \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.414}}}}}
    &\approx \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.63}}}} \\
    &\approx \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.76}}} \\
    &\approx \sqrt{2}^{\sqrt{2}^{1.84}} \\
    &\approx \sqrt{2}^{1.89} \\
    &\approx 1.93
\end{align}</math>

In general, the infinitely iterated exponential <math>x^{x^{\cdot^{\cdot^{\cdot}}}}\!\!</math>, defined as the limit of <math>{}^{n}x</math> as {{mvar|n}} goes to infinity, converges for {{math|''e''{{sup|−''e''}} ≤ ''x'' ≤ ''e''{{sup|1/''e''}}}}, roughly the interval from 0.066 to 1.44, a result shown by [[Leonhard Euler]].<ref>Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." ''Acta Acad. Scient. Petropol. 2'', 29–51, 1783. Reprinted in Euler, L. ''Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae''. Leipzig, Germany: Teubner, pp. 350–369, 1921. ([http://math.dartmouth.edu/~euler/docs/originals/E532.pdf facsimile])</ref> The limit, should it exist, is a positive real solution of the equation {{math|1=''y'' = ''x''{{sup|''y''}}}}. Thus, {{math|1=''x'' = ''y''{{sup|1/''y''}}}}. The limit defining the infinite exponential of {{mvar|x}} does not exist when {{math|''x'' > ''e''{{sup|1/''e''}}}} because the maximum of {{math|''y''{{sup|1/''y''}}}} is {{math|''e''{{sup|1/''e''}}}}. The limit also fails to exist when {{math|0 < ''x'' < ''e''{{sup|−''e''}}}}.

This may be extended to complex numbers {{mvar|z}} with the definition:
: <math>{}^{\infty}z = z^{z^{\cdot^{\cdot^{\cdot}}}} = \frac{\mathrm{W}(-\ln{z})}{-\ln{z}} ~,</math>
where {{math|W}} represents [[Lambert's W function]].

As the limit {{math|1=''y'' = {{sup|∞}}''x''}} (if existent on the positive real line, i.e. for {{math|''e''{{sup|−''e''}} ≤ ''x'' ≤ ''e''{{sup|1/''e''}}}}) must satisfy {{math|1=''x''{{sup|''y''}} = ''y''}} we see that {{math|1=''x'' ↦ ''y'' = {{sup|∞}}''x''}} is (the lower branch of) the inverse function of {{math|1=''y'' ↦ ''x'' = ''y''{{sup|1/''y''}}}}.

==== Negative heights ====
We can use the recursive rule for tetration,
: <math>{^{k+1}a} = a^{\left({^{k}a}\right)},</math>

to prove <math>{}^{-1}a</math>:
: <math>^{k}a = \log_a \left(^{k+1}a\right);</math>

Substituting −1 for {{mvar|k}} gives
: <math>{}^{-1}a = \log_{a} \left({}^0 a\right) = \log_a 1 = 0</math>.<ref name="tetration extensions">{{cite web |url=http://www.mpmueller.net/reihenalgebra.pdf |last=Müller |first=M. |title=Reihenalgebra: What comes beyond exponentiation? |access-date=12 December 2018 |archive-date=2013-12-02  |archive-url=https://web.archive.org/web/20131202235053/http://www.mpmueller.net/reihenalgebra.pdf |url-status=dead }}</ref>

Smaller negative values cannot be well defined in this way. Substituting −2 for {{mvar|k}} in the same equation gives
: <math>{}^{-2}a = \log_{a} \left( {}^{-1}a \right) = \log_a 0 = -\infty</math>

which is not well defined. They can, however, sometimes be considered sets.<ref name="tetration extensions" />

For <math>n = 1</math>, any definition of <math>\,\! {^{-1}1}</math> is consistent with the rule because
: <math>{^{0}1} = 1 = 1^n</math> for any <math>\,\! n = {^{-1}1}</math>.

==== Linear approximation for real heights ====
[[File:Real-tetration.png|thumbnail|alt=A line graph with a figure drawn on it similar to an S-curve with values in the third quadrant going downward rapidly and values in the first quadrant going upward rapidly|<math>\,{}^{x}e</math> using linear approximation]]
A [[linear approximation]] (solution to the continuity requirement, approximation to the differentiability requirement) is given by:
: <math>{}^{x}a \approx \begin{cases}
  \log_a\left(^{x+1}a\right) &  x \le -1 \\
                       1 + x & -1 < x \le 0 \\
    a^{\left(^{x-1}a\right)} &  0 < x
\end{cases}</math>

hence:
{| class="wikitable"
|+Linear approximation values
!scope="col"| Approximation
!scope="col"| Domain
|-
!scope="row"| <math display="inline">{}^x a \approx x + 1</math>
| for {{math|−1 < ''x'' < 0}}
|-
!scope="row"| <math display="inline">{}^x a \approx a^x</math>
| for {{math|0 < ''x'' < 1}}
|-
!scope="row"| <math display="inline">{}^x a \approx a^{a^{(x-1)}}</math>
| for {{math|1 < ''x'' < 2}}
|}

and so on. However, it is only piecewise differentiable; at integer values of {{mvar|x}}, the derivative is multiplied by <math>\ln{a}</math>. It is continuously differentiable for <math>x > -2</math> if and only if <math>a = e</math>. For example, using these methods <math>{}^\frac{\pi}{2}e \approx 5.868...</math> and <math>{}^{-4.3}0.5 \approx 4.03335...</math>

A main theorem in Hooshmand's paper<ref name="uxp" /> states: Let <math>0 < a \neq 1</math>. If <math>f:(-2, +\infty)\rightarrow \mathbb{R}</math> is continuous and satisfies the conditions:

* <math>f(x) = a^{f(x-1)} \;\; \text{for all} \;\; x > -1, \; f(0) = 1,</math>
* <math>f</math> is differentiable on {{open-open|−1, 0}},
* <math>f^\prime</math> is a nondecreasing or nonincreasing function on {{open-open|−1, 0}},
* <math>f^\prime \left(0^+\right) = (\ln a) f^\prime \left(0^-\right) \text{ or } f^\prime \left(-1^+\right) = f^\prime \left(0^-\right).</math>

then <math>f</math> is uniquely determined through the equation
: <math>f(x) = \exp^{[x]}_a \left(a^{(x)}\right) = \exp^{[x+1]}_a((x)) \quad \text{for all} \; \; x > -2,</math>

where <math>(x) = x - [x]</math> denotes the fractional part of {{mvar|x}} and <math>\exp^{[x]}_a</math> is the <math>[x]</math>-[[iterated function]] of the function <math>\exp_a</math>.

The proof is that the second through fourth conditions trivially imply that  {{mvar|f}}  is a linear function on {{closed-closed|−1, 0}}.

The linear approximation to natural tetration function <math>{}^xe</math> is continuously differentiable, but its second derivative does not exist at integer values of its argument. Hooshmand derived another uniqueness theorem for it which states:

If <math> f: (-2, +\infty)\rightarrow \mathbb{R}</math> is a [[continuous function]] that satisfies:

* <math>f(x) = e^{f(x-1)} \;\; \text{for all} \;\; x > -1, \; f(0) = 1,</math>
* <math>f</math> is convex on {{open-open|−1, 0}},
* <math>f^\prime \left(0^-\right) \leq f^\prime \left(0^+\right).</math>

then <math>f = \text{uxp}</math>. [Here <math>f = \text{uxp}</math> is Hooshmand's name for the linear approximation to the natural tetration function.]

The proof is much the same as before; the recursion equation ensures that <math>f^\prime (-1^+) = f^\prime (0^+),</math> and then the convexity condition implies that <math>f</math> is linear on {{open-open|−1, 0}}.

Therefore, the linear approximation to natural tetration is the only solution of the equation <math>f(x) = e^{f(x-1)} \;\; (x > -1)</math> and <math>f(0) = 1</math> which is [[convex function|convex]] on {{open-open|−1, +∞}}. All other sufficiently-differentiable solutions must have an [[inflection point]] on the interval {{open-open|−1, 0}}.

==== Higher order approximations for real heights ====
[[File:Approximations of 0.5 tetratrated to the x.png|thumb|alt=A pair of line graphs, with one drawn in blue looking similar to a sine wave that has a decreasing amplitude as the values along the x-axis increase and the second is a red line that directly connects points along these curves with line segments|A comparison of the linear and quadratic approximations (in red and blue respectively) of the function <math>^{x}0.5</math>, from {{math|1=''x'' = −2}} to {{math|1=''x'' = 2}}]]
Beyond linear approximations, a [[quadratic approximation]] (to the differentiability requirement) is given by:
: <math>{}^{x}a \approx \begin{cases}
  \log_a\left({}^{x+1}a\right) & x \le -1 \\
  1 + \frac{2\ln(a)}{1 \;+\; \ln(a)}x - \frac{1 \;-\; \ln(a)}{1 \;+\; \ln(a)}x^2 & -1 < x \le 0 \\
  a^{\left({}^{x-1}a\right)} & x >0
\end{cases}</math>

which is differentiable for all <math>x > 0</math>, but not twice differentiable. For example, <math>{}^\frac{1}{2}2 \approx 1.45933...</math> If <math>a = e</math> this is the same as the linear approximation.<ref name="uwu" />

Because of the way it is calculated, this function does not "cancel out", contrary to exponents, where <math>\left(a^\frac{1}{n}\right)^n = a</math>. Namely,
: <math>
  {}^n\left({}^\frac{1}{n} a\right)
  = \underbrace{
      \left({}^\frac{1}{n}a\right)^{
        \left({}^\frac{1}{n}a\right)^{
          \cdot^{\cdot^{\cdot^{\cdot^{
            \left({}^\frac{1}{n}a\right)
          }}}}
        }
      }
    }_n
  \neq a
</math>.

Just as there is a quadratic approximation, cubic approximations and methods for generalizing to approximations of degree {{mvar|n}} also exist, although they are much more unwieldy.<ref name="uwu" /><ref name=SolveAnalyt>Andrew Robbins. [https://web.archive.org/web/20090201164821/http://tetration.itgo.com/paper.html Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm]. The extensions are found in part two of the paper, "Beginning of Results".</ref>

=== Complex heights ===
[[File:Tetration analytic extension.svg|thumb|alt=A complex graph showing mushrooming values along the x-axis|Drawing of the analytic extension <math>f = F(x+{\rm i}y)</math> of tetration to the complex plane. Levels <math>|f| = 1, e^{\pm 1}, e^{\pm 2}, \ldots</math> and levels <math>\arg(f) = 0, \pm 1, \pm 2, \ldots</math> are shown with thick curves.]]

In 2017, it was proved<ref name="PAU10">{{cite journal
  |author-first1=W.
  |author-last1=Paulsen
  |author-first2=S.
  |author-last2=Cowgill
  |title=Solving <math>F(z+1) = b^{F(z)}</math> in the complex plane
  |journal=Advances in Computational Mathematics
  |volume=43
  |pages=1–22
  |date=March 2017
  |doi=10.1007/s10444-017-9524-1
  |url=http://myweb.astate.edu/wpaulsen/tetration2.pdf
  |s2cid=9402035
}}</ref> that there exists a unique function <math>F</math> satisfying
<math>F(z + 1) = \exp\bigl(F(z)\bigr)</math> (equivalently <math>F(z+1) = b^{F(z)}</math> when <math>b=e</math>), with the auxiliary conditions
<math>F(0) = 1</math>, and
<math>F(z) \to \xi_{\pm}</math> (the attracting/repelling fixed points of the logarithm, roughly <math>0.318 \pm 1.337\,\mathrm{i}</math>) as <math>z \to \pm i\infty</math>.  Moreover, <math>F</math> is holomorphic on all of <math>\mathbb{C}</math> except for the cut along the real axis at <math>z \le -2</math>.  This construction was first conjectured by Kouznetsov (2009)<ref name="MOC09">{{cite journal
  |author-first=D.
  |author-last=Kouznetsov
  |title=Solution of <math>F(z + 1) = \exp(F(z))</math> in complex <math>z</math>-plane
  |journal=Mathematics of Computation
  |volume=78
  |issue=267
  |pages=1647–1670
  |date=July 2009
  |doi=10.1090/S0025-5718-09-02188-7
  |url=http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/S0025-5718-09-02188-7.pdf
}}</ref> and rigorously carried out by Kneser in 1950.<ref name="hellmuth50">{{cite journal
  |author-first=H.
  |author-last=Kneser
  |title=Reelle analytische Lösungen der Gleichung <math>\varphi(\varphi(x)) = e^x</math> und verwandter Funktionalgleichungen
  |journal=Journal für die reine und angewandte Mathematik
  |volume=187
  |pages=56–67
  |date=1950
  |language=de
}}</ref>  Paulsen &amp; Cowgill’s proof extends Kneser’s original construction to any base <math>b>e^{1/e}\approx1.445</math>, and subsequent work showed how to allow <math>b \in \mathbb{C}</math> with <math>|b|>e^{1/e}</math>.<ref name="PAU18">{{cite journal
  |author-first=W.
  |author-last=Paulsen
  |title=Tetration for complex bases
  |journal=Advances in Computational Mathematics
  |volume=45
  |pages=243–267
  |date=June 2018
  |doi=10.1007/s10444-018-9615-7
}}</ref>

In May 2025, Vey gave a unified, holomorphic extension for arbitrary complex bases <math>b\in \mathbb{C}\setminus\{0,1\}</math> and complex heights <math>z\in\mathbb{C}</math> by means of Schröder’s equation.  In particular, one constructs a linearizing coordinate near the attracting (or repelling) fixed point of the map <math>f(w)=b^w</math>, and then patches together two analytic expansions (one around each fixed point) to produce a single function <math>F_{b}(z)</math> that satisfies
<math>F_{b}(z+1)=b^{\,F_{b}(z)}</math> and <math>F_{b}(0)=1</math> on all of <math>\mathbb{C}</math>.  The key step is to define
<math>\displaystyle
  \Phi_{b}(w)=\lim_{n\to\infty}\;s^{n}\Bigl(f^{\circ n}(w)-\alpha\Bigr),
</math>
where <math>\alpha</math> is a fixed point of <math>f(w)=b^w</math>, <math>s = f'(\alpha)</math>, and <math>f^{\circ n}</math> denotes <math>n</math>-fold iteration.  One then solves Schröder’s functional equation
<math>\Phi_{b}\bigl(b^{\,w}\bigr)\;=\;s\;\Phi_{b}(w)</math>
locally (for <math>w</math> near <math>\alpha</math>), extends both branches holomorphically, and glues them so that there is no monodromy except the known cut-lines.  Vey also provides explicit series for the coefficients <math>a_{n}^{(b)}</math> in the local Schröder expansion:
<math>\Phi_{b}(w)
  = \sum_{n=0}^{\infty} a_{n}^{(b)}\,(w-\alpha)^{n},
</math>
and gives rigorous bounds proving factorial convergence of <math>a_{n}^{(b)}</math>.<ref name="VEY25">{{cite web 
  |author-first=Vincent
  |author-last=Vey
  |title=Holomorphic Extension of Tetration to Complex Bases and Heights via Schröder’s Equation
  |date=May 2025
  |url=http://dx.doi.org/10.13140/RG.2.2.10348.48008
}}</ref>

Using Kneser’s (and Vey’s) tetration, example values include
<math>{}^{\tfrac{\pi}{2}}e \approx 5.82366\ldots</math>,
<math>{}^{\tfrac{1}{2}}2 \approx 1.45878\ldots</math>, and
<math>{}^{\tfrac{1}{2}}e \approx 1.64635\ldots</math>.

The requirement that tetration be holomorphic on all of <math>\mathbb{C}</math> (except for the known cuts) is essential for uniqueness.  If one relaxes holomorphicity, there are infinitely many real‐analytic “solutions” obtained by pre‐ or post‐composing with almost‐periodic perturbations.  For example, for any fast‐decaying real sequences <math>\{\alpha_{n}\}</math> and <math>\{\beta_{n}\}</math>, one can set
<math>
  S(z)
  =
  F_{b}\Bigl(\,
    z
    +\sum_{n=1}^{\infty}\sin(2\pi n\,z)\,\alpha_{n}
    +\sum_{n=1}^{\infty}\bigl[1 - \cos(2\pi n\,z)\bigr]\,\beta_{n}
  \Bigr),
</math>
which still satisfies <math>S(z+1)=b^{S(z)}</math> and <math>S(0)=1</math>, but has additional singularities creeping in from the imaginary direction.

<syntaxhighlight lang="wikitext">
<!-- Example of “calling” Vey’s solution in pseudocode (series form) -->
function ComplexTetration(b, z):
    # 1) Find attracting fixed point alpha of w ↦ b^w
    α ← the unique solution of α = b^α near the real line
    # 2) Compute multiplier s = b^α · ln(b)
    s ← b**α * log(b)
    # 3) Solve Schröder’s equation coefficients {a_n} around α:
    #    Φ_b(w) = ∑_{n=0}^∞ a_n · (w − α)^n,   Φ_b(b^w) = s · Φ_b(w)
    {a_n} ← SolveLinearSystemSchroeder(b, α, s)
    # 4) Define inverse φ_b⁻¹ via the local power series around 0
    φ_inv(u) = α + ∑_{n=1}^∞ c_n · u^n   # (coefficients c_n from series inversion)
    # 5) Put F_b(z) = φ_b⁻¹(s^(-z) · Φ_b(1))
    return φ_inv( s^(−z) * ∑_{n=0}^∞ a_n · (1 − α)^n )
</syntaxhighlight>
<!-- End of code example -->
<!-- Note: Vey’s actual paper shows how to bound |a_n| ≲ C·n!·R⁻ⁿ so that the
     series converges for all w in a suitable neighborhood of α; see Vey 2025. -->
=== Ordinal tetration ===
Tetration can be defined for [[ordinal numbers]] via [[transfinite induction]].  For all {{math|''&alpha;''}} and all {{math|''&beta;'' > 0}}:
<math display=block>{}^0\alpha = 1</math>
<math display=block>{}^\beta\alpha = \sup(\{\alpha^{{}^\gamma\alpha} : \gamma < \beta\})\,.</math>

== Non-elementary recursiveness ==
Tetration (restricted to <math>\mathbb{N}^2</math>) is not an [[ELEMENTARY|elementary recursive function]]. One can prove by induction that for every elementary recursive function {{mvar|f}}, there is a constant {{mvar|c}} such that
: <math>f(x) \leq \underbrace{2^{2^{\cdot^{\cdot^{x}}}}}_c.</math>
We denote the right hand side by <math>g(c, x)</math>. Suppose on the contrary that tetration is elementary recursive. <math>g(x, x)+1</math> is also elementary recursive. By the above inequality, there is a constant {{mvar|c}} such that <math>g(x,x) +1 \leq g(c, x)</math>. By letting <math>x=c</math>, we have that <math>g(c,c) + 1 \leq g(c, c)</math>, a contradiction.
<!-- who originally posted this proof? -->