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If you haven't done so already, please check out my previous blog where BEAF is formalized and generalized using ordinals. In this blog, I will do the same thing to Cascading-E — define it using ordinals and extend it in parallel to the fast-growing hierarchy. This allows us to extend Saibian's work into \(\Gamma_0\) and far, far beyond.

Sbiis Saibian's Cascading-E notation (E^) extends his Hyper-E notation (xE#) all the way up to epsilon zero. If we were to diagonalize over E^, I expect that we'd get a function comparable to \(f_{\epsilon_0}\) in FGH.

Abstract

xE# and E^ are structurally akin to BEAF. They both have many levels of separators that decompose into smaller levels. However, Bowers' arrays are designed to represent numbers in an array-space, while Saibian treats his arrays more like linear strings. For example, {42,13,37,1(1)42} is the same as {42,13,37(1)42}, but E42#13#37#1##42 is not the same as E42#13#37##42 (even though 1s are default in both). BEAF is dimensional, and E^ is much more linear. For this reason, Bowers' works fit very nicely into the ordinal structure, but Saibian's take a little mangling to operate.

Definition

Define \(E_\gamma(\alpha)\):

\[E_\gamma(\alpha) = \max\{n \in \mathbb{N}_0|\exists \beta_2 < \omega^\gamma, \beta_1: \omega^{\gamma+1} \beta_1 + \omega^\gamma \times n + \beta_2 = \alpha\}\]

Also, define \(Q(\alpha) = \{\gamma|E_\gamma(\alpha) \neq 0\}\), which is the set of all positions with non-zero associated entries.

This is the same as in the BEAF definition — extracting coefficients of \(\omega^\gamma\) in \(\alpha\).

Define \(L(\alpha) = \max(Q(\alpha))\), and \(P(\alpha) = \max\{\gamma < L(\alpha)|\gamma \in Q(\alpha)\}\). In other words, \(L(\alpha)\) is the last non-zero entry and \(P(\alpha)\) is the next-to-last one.

Prime blocks

E^ uses prime blocks, and the prime is the final entry. Prime blocks for zero and successor ordinals are not defined, and other rules kick in for those definition.

Let \(q = \min\{q|\alpha[q] > P(\alpha)\}\).

\[\Pi_p(\alpha) = \{P(\alpha), \alpha[q], \alpha[q + 1], \ldots, \alpha[q + p - 2]\}\]

Example: E100#100#^#^#*#7. Converting this to ordinal yields \(\omega^{\omega^{\omega + 1}}7 + \omega100 + 100\). Since \(P(\alpha) = 1\) and the fundamental sequence of \(\omega^{\omega^\omega\omega}\) (I think) goes \(1\), \(\omega^{\omega^\omega}\), \(\omega^{\omega^\omega2}\), \(\omega^{\omega^\omega\omega3}\), ..., \(q = 1\) and the set is

\[\Pi_p(\omega^{\omega^{\omega + 1}}7 + \omega100 + 100) = \{\omega^{\omega^\omega}, \omega^{\omega^\omega2}, \omega^{\omega^\omega3}, \omega^{\omega^\omega4}, \omega^{\omega^\omega5}, \omega^{\omega^\omega6}\}\]

The Three Rules

Let \(p = E_{L(\alpha)}(\alpha)\).

1. The Base Rule: If \(\alpha < \omega\), \(S_b(\alpha) = b^{\alpha + 1}\).
2. The Catastrophic Rule: If \(L(\alpha) = P(\alpha) + 1\):
\[\alpha' = \sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & \omega^\gamma \times (p - 1) \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\] \[S_b(\alpha) = S_b\left(\sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & 0 \\ \gamma = P(\alpha) : & S_b(\alpha') \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\right)\] 3. The Expansion Rule:
\[S_b(\alpha) = S_b\left(\sum_{\gamma \in Q(\alpha)}\left\{ \begin{array}{rl} \gamma = L(\alpha) : & 0 \\ \gamma \in \Pi_p(L(\alpha)) : & \omega^\gamma \times E_{P(\alpha)}(\alpha) \\ \text{otherwise} : & \omega^\gamma \times E_\gamma(\alpha) \\ \end{array} \right\}\right)\]

That's it! Cascading-E notation is now extensible through all the recursive ordinals.

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