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


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:

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

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