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The irrational arrow notation (無理矢印表記 in Japanese) is a variant of the arrow notation defined by a Japanese Googology Wiki user Jason.[1][2] It is an irrational analogue of another variant called the rational arrow notation defined by Jason.[1]

## Definition

We denote by $$\mathbb{N}_{>0}$$ the set of positive integers, and by $$\mathbb{R}_{>0}$$ the set of positive numbers. We define a map \begin{eqnarray*} \mathbb{N}_{>0}^2 \times \mathbb{R}_{>0}^2 & \to & \mathbb{N}_{>0} \\ ((a,b),(c,\Delta)) & \mapsto & a \uparrow^{c(\Delta)} b \end{eqnarray*} as the unique one satisfying the following relations:

1. If $$b = 1$$ or $$c \leq 1$$, then $$a \uparrow^{c(\Delta)} b := a^b$$.
2. If $$b > 1$$ and $$c > 1$$, then $$a \uparrow^{c(\Delta)} b := a \uparrow^{c'(\Delta')} (a \uparrow^{c(\Delta)} (b-1))$$, where $$c'$$ and $$\Delta'$$ are defined in the following way:

\begin{eqnarray*} c' & := & c \log_{\Delta'+1}(\Delta') \\ \Delta' & := & \left\{ \begin{array}{ll} a \uparrow^{c-1(\Delta)} b & (\Delta \notin (1,a \uparrow^{c-1(\Delta)} b)) \\ \Delta & (\Delta \in (1,a \uparrow^{c-1(\Delta)} b)) \end{array} \right. \end{eqnarray*} Since it is defined on an uncountable set, it is uncomputable by the definition of the computability.

## Analysis

The order type of the structural ordering of the irrational arrow notation is $$\omega 2$$.[3] In order words, there exists a well-founded strict partial ordering $$<$$ on $$\mathbb{N}_{>0}^2 \times \mathbb{R}_{>0}^2$$ such that the two characteristic relations above refer only to the values of the irrational arrow notation whose imput is smaller than $$((a,b),(c,\Delta))$$ with respect to $$<$$, and $$\omega 2$$ is the least ordinal type of such a well-founded strict partial ordering. In particular, the totality of the irrational arrow notation is provable under a weak arithmetic which can prove the well-foundedness of $$\omega 2$$.

On the other hand, the growth rates of the $$1$$-ary function $$x \uparrow^{x(x)} x$$ and $$x \uparrow^{x(0)} x$$ on $$\mathbb{N}$$ are bounded by $$\omega+1$$ in Wainer hierarchy.[4][5]

## Significance

By the argument above, the irrational arrow notation gives a non-trivial example of a large function which can be approximated to an ordinal in FGH with respect to a canonical system of fundamental sequences smaller than the ordinal type of the structural ordering. It is significant because many googologists believe that large functions can be approximated to the ordinal type of the structural ordering in FGH with respect to a canonical system of fundamental sequences. For example, many googologists state that TREE grows as fast as $$f_{\vartheta(\Omega^{\omega} \omega)}(n)$$ as if it had already been verified, but what has been actually verified is that the ordinal type of its structural ordering. As the analysis of the irrational arrow notation implies, such a reasoning is critically incorrect.

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