Irrational arrow notation

Irrational arrow notation (無理矢印表記 in Japanese) is a variant of the arrow notation defined by the Japanese Googology Wiki user Jason.

= Definition =

For a natural number \(n\), we denote by \(\mathbb{N}_{>n}\) the set of natural numbers greater than \(n\). We define a map \begin{eqnarray*} \mathbb{N}_{>0}^2 \times \mathbb{R}^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: \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 uncomputable set, it is uncomputable. On the other hand, its restriction to the subset of tuples whose entries are computable numbers is computable. Indeed, the two relations characterising the irrational arrow notation give recursive definition of its restriction.
 * 1) If \(b = 1\) or \(c \leq 1\), then \(a \uparrow^{c(\Delta)} b := a^b\).
 * 2) If \(b > 0\) 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:

= Analysis =

The ordinal type of the structural ordering of the irrational arrow notation is \(\omega 2\), and hence its totality 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.

= Significance =

By the argument above, irrational arrow notation is a non-trivial example of a computable 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 fas 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 irrational arrow notation implies, such a reasoning is critically incorrect.

= Sources =