Fourth Hyperoperation is Tetration Operation

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Theorem

The $4$th hyperoperation is the tetration operation restricted to the positive integers:

$\forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = x \uparrow \uparrow y$

where $\uparrow \uparrow$ denotes tetration:

$x \uparrow \uparrow n := \begin{cases} 1 & : n = 0 \\ x \uparrow \left({x \uparrow \uparrow \left({n - 1}\right)}\right) & : n > 0 \\ \end{cases}$


In this context, $x \uparrow y$ is the Knuth notation for powers:

$x \uparrow y := x^y$


Proof

By definition of the hyperoperation sequence:

$\forall n, x, y \in \Z_{\ge 0}: H_n \left({x, y}\right) = \begin{cases} y + 1 & : n = 0 \\ x & : n = 1, y = 0 \\ 0 & : n = 2, y = 0 \\ 1 & : n > 2, y = 0 \\ H_{n - 1} \left({x, H_n \left({x, y - 1}\right)}\right) & : n > 0, y > 0 \end{cases}$


Thus the $4$th hyperoperation is defined as:

$\forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = \begin{cases} 1 & : y = 0 \\ H_3 \left({x, H_4 \left({x, y - 1}\right)}\right) & : y > 0 \end{cases}$


From Third Hyperoperation is Integer Power Operation:

$(1): \quad \forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = \begin{cases} 1 & : y = 0 \\ x \uparrow H_3 \left({x, y - 1}\right) & : y > 0 \end{cases}$


The proof proceeds by induction.

For all $y \in \Z_{\ge 0}$, let $P \left({y}\right)$ be the proposition:

$\forall x \in \Z_{\ge 0}: H_4 \left({x, y}\right) = x \uparrow \uparrow y$


Basis for the Induction

$P \left({0}\right)$ is the case:

\(\ds H_4 \left({x, 0}\right)\) \(=\) \(\ds 1\) from $(1)$
\(\ds \) \(=\) \(\ds x \uparrow \uparrow 0\) Definition of Tetration


Thus $P \left({0}\right)$ is seen to hold.


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 0$, then it logically follows that $P \left({k + 1}\right)$ is true.


So this is the induction hypothesis:

$\forall x \in \Z_{\ge 0}: H_4 \left({x, k}\right) = x \uparrow \uparrow k$


from which it is to be shown that:

$\forall x \in \Z_{\ge 0}: H_4 \left({x, k + 1}\right) = x \uparrow \uparrow \left({k + 1}\right)$


Induction Step

This is the induction step:


\(\ds H_4 \left({x, k + 1}\right)\) \(=\) \(\ds x \uparrow H_4 \left({x, \left({k + 1}\right) - 1}\right)\) from $(1)$
\(\ds \) \(=\) \(\ds x \uparrow H_4 \left({x, k}\right)\)
\(\ds \) \(=\) \(\ds x \uparrow \left({x \uparrow \uparrow k}\right)\) Induction Hypothesis
\(\ds \) \(=\) \(\ds x \uparrow \uparrow \left({k + 1}\right)\) Definition of Tetration

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall x, y \in \Z_{\ge 0}: H_3 \left({x, y}\right) = x \uparrow \uparrow y$

$\blacksquare$