User:Vel!/BEAF


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An array is a function \(A : \Omega \mapsto \Omega \backslash \{0\}\), where the number of outputs greater than 1 is finite. Let \(b = A(0)\), \(p = A(1)\), \(\pi = \min\{\alpha > 1: A(\alpha) > 1\}\), \(\kappa = A(A(\min\{\alpha > 1: A(\alpha) > 1\})-1)\), and finally \(\lambda = \min\{\alpha > 0: A(\alpha) \in \text{Lim}\}\).

Define the prime block \(\Pi(\alpha)\):


 * \(\Pi(0) = \{\}\)
 * \(\Pi(\alpha + 1) = \{\alpha\} \cup \Pi(\alpha)\)
 * \(\Pi(\alpha) = \Pi(\alpha[p])\) if \(\alpha\) is a limit ordinal

Define the passengers as \(S = \Pi(\pi) \backslash \{\pi, \kappa\}\).


 * 1) Base rule. If \(\pi\) does not exist, \(v(A) = b^p\).
 * 2) Prime rule. If \(p = 1\), \(v(A) = p\).
 * 3) Limit rule. If \(\lambda\) exists:
 * 4) * Define \(A_n'\) as \(A\) with \(A_n'(\lambda) := A(\lambda)[n]\).
 * 5) * \(v(A)[n] = v(A_n')\) and \(v(A) = \sup\{n < \omega : v(A)[n]\}\).
 * 6) Catastrophic rule. If \(p < \omega\), define \(A'\) as \(A\) with the following modifications:
 * 7) * Define \(B\) as \(A\), but with \(B(1) := p - 1\).
 * 8) * If \(\kappa\) exists, \(A'(\kappa) := v(B)\).
 * 9) * \(A'(\pi) := A(\pi) - 1\).
 * 10) * \(A'(\sigma) := b\) for \(\sigma \in S\).
 * 11) * \(v(A) = v(A')\).
 * Infinite Catastrophic Rule. If \(p > \omega\):
 * Let \(\alpha + n = p\), where \(\alpha\) is a limit ordinal and \(n < \omega\).
 * Define \(B\) as \(A\), but with \(B(1) = \alpha\).
 * Define \(A'\) as \(A\), but with \(A'(0) = v(B)\) and \(A'(1) = n + 1\).
 * \(v(A) = v(A')\).

Fundamental sequences

 * If \(\lambda = \omega^{\alpha_1} + \omega^{\alpha_2} + \ldots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k}\) for \(\alpha_1 \geq \alpha_2 \geq \ldots \geq \alpha_{k - 1} \geq \alpha_k\), then \(\lambda[n] = \omega^{\alpha_1} + \omega^{\alpha_2} + \ldots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k}[n]\).
 * \(\omega^{\alpha + 1}[n] = \omega^\alpha n\)
 * \(\omega^\alpha[n] = \omega^{\alpha[n]},\,\,\,\alpha \in \text{Lim}\)

Specific ordinals
Lemma 1. \(\{\omega, \omega, 2\} = \varepsilon_0\).

Proof. Applying the Limit Rule and then the Prime Rule:

\begin{eqnarray*} \{\omega, \omega, 2\} &=& \sup\{\{\omega, 1, 2\}, \{\omega, 2, 2\}, \{\omega, 3, 2\}, \ldots\} \\ &=& \sup\{\omega, \omega^\omega, \omega^{\omega^\omega}, \ldots\} \\ &=& \varepsilon_0. \\ \end{eqnarray*}

Lemma 2. \(\{\omega,\omega(1+\alpha)+\omega, 2\} = \varepsilon_{\alpha+1}\) (note that the \(\omega\) is before the \(\alpha\), therefore the \(\omega\) doesn't matter if \(\alpha \geq \omega^\omega\))

'''Proof. '''Proof by induction.

Base case: \(\{\omega, \omega, 2\} = \varepsilon_0\), by lemma 1.

Inductive step: \begin{eqnarray*} \{\omega, \omega(1+\alpha)+\omega, 2\} &=& \sup\{\{\omega, \omega(1+\alpha), 2\}, \{\omega, \omega(1+\alpha)+1, 2\}, \{\omega, \omega(1+\alpha)+2, 2\}, \ldots\} \\ &=& \sup\{\omega\uparrow\uparrow\omega(1+\alpha), \omega\uparrow\uparrow(\omega(1+\alpha)+1),\omega\uparrow\uparrow(\omega(1+\alpha)+2), \ldots\} \\ &=& \sup\{\omega\uparrow\uparrow\omega(1+\alpha), (\omega\uparrow\uparrow(1+\alpha)\omega)^{(\omega\uparrow\uparrow(1+\alpha)\omega)},(\omega\uparrow\uparrow\omega(1+\alpha))^{(\omega\uparrow\uparrow(1+\alpha)\omega)^{(\omega\uparrow\uparrow(1+\alpha)\omega)}}, \ldots\} \\ &=& \sup\{\varepsilon_{\alpha}, \varepsilon_{\alpha}^{\varepsilon_{\alpha}},\varepsilon_{\alpha}^{\varepsilon_{\alpha}^{\varepsilon_{\alpha}}}, \ldots\} \\ &=& \varepsilon_{\alpha+1}. \\ \end{eqnarray*}

Lemma 3. \(\{\omega, \omega(1+\alpha)+\omega, \beta + 1\} = \theta(\beta,\alpha+1)\)

'''Proof. '''Proof by induction.

Base case: This is lemma 2. Inductive step: \begin{eqnarray*} \{\omega, \omega(1+\alpha)+\omega, \beta + 1\} &=& \sup\{\{\omega, \omega(1+\alpha), 3\}, \{\omega, \omega(1+\alpha)+1, \beta + 1\}, \{\omega, \omega(1+\alpha)+2, \beta + 1\}, \ldots\} \\ &=& \sup\{\omega\uparrow^{\beta + 1}\omega(1+\alpha), \omega\uparrow^{\beta + 1}(\omega(1+\alpha)+1),\omega\uparrow^{\beta + 1}(\omega(1+\alpha)+2), \ldots\} \\ &=& \sup\{\omega\uparrow^{\beta + 1}\omega(1+\alpha), (\omega\uparrow^{\beta + 1}(1+\alpha)\omega)\uparrow^{\beta}(\omega\uparrow^{\beta + 1}(1+\alpha)\omega),(\omega\uparrow^{\beta + 1}\omega(1+\alpha))\uparrow^{\beta}(\omega\uparrow^{\beta + 1}(1+\alpha)\omega)\uparrow^{\beta}(\omega\uparrow^{\beta + 1}(1+\alpha)\omega), \ldots\} \\ &=& \sup\{\theta(\beta,\alpha), \theta(\beta,\theta(\beta,\alpha)),\theta(\beta,\theta(\beta,\theta(\beta,\alpha))), \ldots\} \\ &=& \theta(\beta+1). \\ \end{eqnarray*}

Conjecture. \(\{\omega, \omega, 1, 2\} = \Gamma_0\).

Conjecture. \(\{\omega, \omega, 1, 1, 2\} = \vartheta(\Omega^2)\).

Conjecture. \(\{\omega, \omega (1) 2\} = \vartheta(\Omega^\omega)\).

Conjecture. \(\{\omega, \omega, 2 (1) 2\} = \vartheta(\Omega^\Omega)\).

Conjecture. \(\{\omega, \omega (0, 1) 2\} = \vartheta(\varepsilon_{\Omega + 1})\).

Conjecture. \(\{\omega, \omega / 2\} = \vartheta(\Omega_\omega)\).

Conjecture. \(\{\omega, \omega // 2\} = \vartheta(\Omega_{\Omega_\omega})\).

Single-argument form
Define \(\&(\alpha) = \{(0, \omega), (1, \omega), (\alpha, 2)\}\).

Conjectures:


 * \(\&(2) = \varepsilon_0\)
 * \(\&(3) = \Gamma_0\)
 * \(\&(4) = \vartheta(\Omega^2)\)
 * \(\&(\omega) = \vartheta(\Omega^\omega)\)
 * \(\&(\varepsilon_0) = \vartheta(\varepsilon_{\Omega + 1})\)
 * \(\&(\vartheta(\Omega_\omega)) = \vartheta(\Omega_\omega)\)