User blog:Vel!/Ordinal BEAF

Breaking news: I think I just formalized BEAF.

Prime blocks
Define \(\Pi_p(n)\) 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 \(E_\gamma(\alpha)\) to be the coefficient of \(\omega^\gamma\) in \(\alpha\). Formally:

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

e.g. \(E_1(\omega^2 \times 4 + \omega \times 5 + 6) = 5\), since \(5\) is the coefficient of \(\omega^1\) (which can be obtained by setting \(\beta_1 = 4\) and \(\beta_2 = 6\)). Note that \(E_\gamma(\alpha)\) always exists since \(\beta_1 = \alpha\) and \(n = \beta_2 = 0\) is a solution.

Thanks to Deedlit for correcting some of this definition.

Pilots and copilots
Define \(P(\alpha)\) (pilot) like so:

\[P(\alpha) = \min\{\gamma > 1|E_\gamma(\alpha) \neq 0\}\]

This is the first non-zero term after the prime in \(\alpha\), reading terms from smallest to largest. \(P(\alpha)\) does not exist iff \(\alpha < \omega^2\).

Define \(CP(\alpha) = \{\gamma|\gamma + 1 = P(\alpha)\}\). This is the set of copilots, which has one member iff the pilot is a successor ordinal, and no members otherwise.

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

The Three Rules
Let \(b = E_0(\alpha)\) and \(p = E_1(\alpha)\), and let \(\mu > \alpha\).

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

Some notes about the last one:
 * In the definition of \(N(\alpha)\), the conditions are evaluated from top to bottom.
 * \(E_1(\alpha) - 1\) and \(E_{P(\alpha)}(\alpha) - 1\) are well-defined. In both situations, the left-hand term is greater than one. (If not, the first one would cause the prime rule to kick in, and the second one would cause the base rule to kick in).