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 \(p = E_1(A)\).
- The Base Rule: If \(A\) has no pilot, \(N(A) = b^p\).
- The Prime Rule: If \(p = 1\), \(N(A) = b\).
- The Catastrophic Rule: Otherwise:
\[A' := \bigcup_{\gamma \in Q(A)}\left\{ \begin{array}{rl} \gamma = 1 : & p \\ \text{otherwise} : & E_\gamma(A) \\ \end{array} \right\}\] \[N(\alpha) = N\left(\bigcup_{\gamma \in Q(A)}\left\{ \begin{array}{rl} \gamma = P(A) : & E_{P(A)}(A) \\ \gamma = CP(A) : & N(A') \\ \gamma \in \Pi_p(P(A)) : & b \\ \text{otherwise} : & E_\gamma(A) \\ \end{array} \right\}\right)\]