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Here is a simpler alteration to ordinal BEAF. It uses integers indexed by ordinals instead of ordinal polynomials.

Example: $$\{(0, 10), (1, 100), (\omega^\omega, 2)\}$$, which I believe would be {10, 100 (0, 1) 2}.

Only a finite number of non-1 entries are allowed.

Prime blocks

Define $$\Pi_p(\alpha)$$ like so:

• $$\Pi_p(0) = \emptyset$$
• $$\Pi_p(\alpha + 1) = \{\alpha\} \cup \Pi_p(\alpha)$$
• $$\Pi_p(\alpha) = \Pi_p(\alpha[p])$$ if $$\alpha$$ is a limit ordinal

Entries

Define $$E_\gamma(A)$$ to be the entry in position $$\gamma$$ in $$A$$. That is, $$E_\gamma(A) = \{n|(\gamma, n) \in A\}$$.

Also, define $$Q(A) = \max\{\gamma|E_\gamma(A) \neq 1\} + 1$$, which is the set of all entries in the array up to the last non-1 entry.

Pilots and copilots

Define $$P(A)$$ (pilot) like so:

$P(A) = \min\{\gamma > 1|E_\gamma(A) > 1\}$

This is the first non-one term after the prime in $$A$$, reading terms from smallest to largest. $$P(\alpha)$$ may not exist.

Define $$CP(A) = P(A) - 1$$. This only exists if $$P(A)$$ is a successor ordinal.

The airplane is $$\Pi_p(P(A))$$, and the passengers are $$\Pi_p(P(A)) \backslash \{P(A), CP(A)\}$$.

The Three Rules

Let $$b = E_0(A)$$ and $$p = E_1(A)$$.

1. The Base Rule: If $$A$$ has no pilot, $$N(A) = b^p$$.
2. The Prime Rule: If $$p = 1$$, $$N(A) = b$$.
3. The Catastrophic Rule: Otherwise:

$A' := \bigcup_{\gamma \in Q(A)}\left\{ \begin{array}{rl} \gamma = 1 : & p \\ \text{otherwise} : & E_\gamma(A) \\ \end{array} \right\}$ $N(\alpha) = N\left(\bigcup_{\gamma \in Q(A)}\left\{ \begin{array}{rl} \gamma = P(A) : & E_{P(A)}(A) \\ \gamma = CP(A) : & N(A') \\ \gamma \in \Pi_p(P(A)) : & b \\ \text{otherwise} : & E_\gamma(A) \\ \end{array} \right\}\right)$