User:Vel!/BEAF


 * You are 100% free to edit this page and help improve definitions and such.

It's finally here! This is hopefully the One True Formalization of BEAF: unambiguous, clean, and consistent with Bowers' writings. Here, BEAF serves as both a large number function and a powerful ordinal notation.

Current status: The definition of BEAF is complete. So far we have proven things up to around \(\vartheta(\Omega)\).

Definition
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 \(S = \Pi(\pi) \backslash \{\pi, \kappa\}\). Define BEAF \(v : \text{set of all arrays} \mapsto \Omega\) as follows:


 * 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 \(A'(1) = n + 1\).
 * \(v(A) = v(A')\).

The traditional function restricts the codomain of arrays to \(\omega \backslash \{0\}\).

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 \(\lambda[n] = \omega^{\alpha_1} + \omega^{\alpha_2} + \ldots + \omega^{\alpha_{k - 1}} + \omega^{\alpha_k}[n]\).
 * \(\omega^{\alpha + 1}[n] = \omega^\alpha n\)
 * \(\omega^\alpha[n] = \omega^{\alpha[n]},\,\,\,\alpha \in \text{Lim}\)
 * All further fundseqs are defined in the \(v\) function above.

Constructibility
An ordinal is constructible using BEAF iff it can be expressed using finite applications of 0, 1, \(\omega\), addition, multiplication, and BEAF expressions, where each \((x, y) \in A\) has either \(y = 1\) or previously constructed \(x\) and \(y\). This requires us to use only ordinals definable using "pure" BEAF.

Conjecture: Totality
Conjecture. The \(v\) function is total. That is, it is defined for all possible arrays and cannot create an infinite loop.

Incomplete proof. We assign an ordinal to each array, according to the following definition:

\[A = \sum_{(\alpha, \beta) \in A} \Omega^\alpha \cdot \left\{ \begin{array}{ll} \beta < \omega & : \beta - 1 \\ \beta \geq \omega & : \beta \end{array} \right.\]

Conjecture: Normality
Conjecture. If one of an array's entries is set to a free variable \(x\), the function \(x \mapsto v(A)\) is either normal or homogeneous.

Epsilon-zero
Theorem. \(〈\omega, \omega, 2〉 = \varepsilon_0\).

Proof. By 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*}

We can also see that this gives us the fundseq \(\varepsilon_0[1] = \omega\) and \(\varepsilon_0[n + 1] = \omega^{\varepsilon_0[n]}\). (BEAF fundseqs start at 1.)

Epsilon-numbers
Theorem. \(〈\omega,\omega(1+\alpha), 2〉 = \varepsilon_{\alpha}\). (For example, \(\omega \uparrow\uparrow (\omega 2) = \varepsilon_1\) and \(\omega \uparrow\uparrow \omega^2 = \varepsilon_{\omega}\).)

Proof: By transfinite induction. The base case is \(\{\omega, \omega, 2\} = \varepsilon_0\), as proven before. For the inductive step, we apply the Limit Rule, and then the Infinite Prime Rule. \begin{eqnarray*} 〈\omega, \omega(1+\alpha + 1), 2〉 &=& \sup\{〈\omega, \omega(1+\alpha), 2〉, 〈\omega, \omega(1+\alpha)+1, 2〉, 〈\omega, \omega(1+\alpha)+2, 2〉, \ldots\} \\ &=& \sup\{〈〈\omega, \omega(1+\alpha), 2〉, 1, 2〉, 〈〈\omega, \omega(1+\alpha), 2〉, 2, 2〉, 〈〈\omega, \omega(1+\alpha), 2〉, 3, 2〉, \ldots\} \\ &=& \sup\{\varepsilon_{\alpha}, \varepsilon_{\alpha}^{\varepsilon_{\alpha}},\varepsilon_{\alpha}^{\varepsilon_{\alpha}^{\varepsilon_{\alpha}}}, \ldots\} \\ &=& \varepsilon_{\alpha+1}. \\ \end{eqnarray*}

Finally, we have the "limit case" for limit ordinals \(\alpha\), which is easily verified by the Limit Rule and the fact that \(\alpha \mapsto \omega(1 + \alpha)\) is a normal function.

Veblen function
Conjecture. \(\{\omega,\omega(1+\alpha), \beta + 1\} = \varphi_{\beta}(\alpha)\) for \(\beta \geq 1\).

Incomplete proof. The definition of \(\varphi_{\gamma + 1}\) is that it enumerates the fixed points of \(\varphi_\gamma\), so we want to prove that \(\alpha \mapsto 〈\omega, \omega(1 + \alpha), \beta + 1〉\) enumerates the fixed points of \(\alpha \mapsto 〈\omega, \omega(1 + \alpha), \beta〉\).

The base case is \(〈\omega, \omega(1+\alpha), 2〉 = \varepsilon_\alpha = \varphi_1(\alpha)\) for \(\beta = 1\) as proven before.

Theta function
Theorem. \(\{\omega, \omega, \beta + 1\} = \theta(\beta)\).

Proof. Suppose we know \(\{\omega, \omega, \beta + 1\} = \theta(\beta)\) for a given \(\beta\). We can prove that \(〈\omega,\omega(1+\alpha), \beta〉 = \theta(\beta,\alpha)\).

With the knowledge \(〈\omega,\omega, \beta+1〉 = \theta(\beta)\), we can prove \(〈\omega,\omega(1+\alpha), \beta+1〉 = \theta(\beta, \alpha)\)

\begin{eqnarray*} 〈\omega, \omega(1+\alpha + 1), \beta〉 &=& \sup\{〈\omega, \omega(1+\alpha), \beta〉, 〈\omega, \omega(1+\alpha)+1, \beta〉, 〈\omega, \omega(1+\alpha)+2, \beta〉, \ldots\} \\ &=& \sup\{〈〈\omega, \omega(1+\alpha), \beta〉, 1, \beta〉, 〈〈\omega, \omega(1+\alpha), \beta〉, 2, \beta〉, 〈〈\omega, \omega(1+\alpha), \beta〉, 3, \beta〉, \ldots\} \\ &=& \sup\{\theta(\beta,\alpha), \theta(\beta-1,\theta(\beta,\alpha)),\theta(\beta-1,\theta(\beta-1,\theta(\beta,\alpha))), \ldots\} \\ &=& \theta(\beta,\alpha+1) \\ \end{eqnarray*}

Proof. With the knowledge \(〈\omega,\omega(1+\alpha), \beta+1〉 = \theta(\beta, \alpha)\), we can prove \(〈\omega,\omega, \beta+2〉 = \theta(\beta+1)\)

\begin{eqnarray*} \{\omega, \omega, \beta+2\} &=& \sup\{〈\omega, 1, \beta+2〉, 〈\omega, 2, \beta+2〉, 〈\omega, 3, \beta+2〉, \ldots\} \\ &=& \sup\{〈\omega, \omega, \beta+1〉, 〈\omega, 〈\omega, \omega, \beta+1〉, \beta+1〉, 〈\omega, 〈\omega, 〈\omega, \omega, \beta+1〉, \beta+1〉, \beta+1〉, \ldots\} \\ &=& \sup\{\theta(\beta) ,\theta(\beta,\theta(\beta)) ,\theta(\beta,\theta(\beta,\theta(\beta))), \ldots\} \\ &=& \theta(\beta+1) \\ \end{eqnarray*}

With the base cases lemma 1 and lemma 2 we have proven lemma 3.

Comments

 * The fourth equality needs to be explained. It's best if we avoid arrow notation. FB100Z &bull; talk &bull; contribs 20:41, January 11, 2014 (UTC)
 * It is all fixed, with this complete new proof! Wythagoras (talk) 07:35, January 12, 2014 (UTC)
 * Still seems like there are some missing pieces. I'll patch what I can. FB100Z &bull; talk &bull; contribs 21:56, January 12, 2014 (UTC)

Lemma 4: Gamma-zero
Lemma 4. \(\{\omega, \omega, 1, 2\} = \vartheta(\Omega)\).

Proof. Repetition of Lemma 3.

\begin{eqnarray*} \{\omega, \omega, 1,2\} &=& \sup\{〈\omega, \omega, \omega〉, 〈\omega, \omega, 〈\omega, \omega, \omega〉〉, 〈\omega, \omega, 〈\omega, \omega, 〈\omega, \omega, \omega〉〉〉, \ldots\} \\ &=& \sup\{\vartheta(\omega), \vartheta(\vartheta(\omega)),\vartheta(\vartheta(\vartheta(\omega))),\ldots\} \\ &=& \vartheta(\Omega). \\ \end{eqnarray*}

Comments

 * How come \(〈\omega, \omega, 〈\omega, \omega, \omega〉〉 = \vartheta(\vartheta(\omega))\)? Where's the base case and inductive step? FB100Z &bull; talk &bull; contribs 20:39, January 11, 2014 (UTC)
 * It is not a proof by induction, and \(〈\omega, \omega, 〈\omega, \omega, \omega〉〉 = 〈\omega, \omega, \vartheta(\omega)〉,\vartheta(\vartheta(\omega))\) is simple by the now properly proven lemma 3. Wythagoras (talk) 07:37, January 12, 2014 (UTC)
 * But it's a different theta function as before, so it's kind of confusing. FB100Z &bull; talk &bull; contribs 21:56, January 12, 2014 (UTC)

Lemma 5: Ackermann ordinal
Lemma 5. \(\{\omega, \omega, 1, 1, 2\} = \vartheta(\Omega^2)\).

Proof. Proof using lemma 4 on fourth entry.

\begin{eqnarray*} \{\omega, \omega, 1,1,2\} &=& \sup\{〈\omega, \omega, \omega, \omega〉, 〈\omega, \omega, \omega, 〈\omega, \omega, \omega, \omega〉〉, 〈\omega, \omega, \omega, 〈\omega, \omega, \omega, 〈\omega, \omega, \omega, \omega〉〉〉, \ldots\} \\ &=& \sup\{\vartheta(\Omega \omega + \omega), \vartheta(\Omega \vartheta(\Omega \omega + \omega) + \omega),\vartheta(\Omega \vartheta(\Omega \vartheta(\Omega \omega + \omega) + \omega) + \omega),\ldots\} \\ &=& \vartheta(\Omega^2). \\ \end{eqnarray*}


 * Again, a more detailed explanation of the inductive proof is needed. FB100Z &bull; talk &bull; contribs 20:40, January 11, 2014 (UTC)

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

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

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

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

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

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

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

Conjecture. \(\{\omega, \omega, 1, 2\}\&\omega = \vartheta(\Omega_2)\).

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

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

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

Conjecture. \(\{L,X\}_{\omega, \omega} = \psi(\psi_I(0))\).

Conjecture. \(\{L,X+1\}_{\omega, \omega} = \psi(\psi_I(\Omega_\omega))\).

Conjecture. \(\{L,X*2\}_{\omega, \omega} = \psi(\psi_I(\psi_I(0)))\).

Conjecture. \(\{L,X^2\}_{\omega, \omega} = \psi(I)\).

Conjecture. \(\{L,X^X\}_{\omega, \omega} = \psi(I^\omega)\).

Conjecture. \(\{L,L\}_{\omega, \omega} = \psi(\Omega_{I+\omega})\).

Conjecture. \(\{L,X,2\}_{\omega, \omega} = \psi(\psi_{I_\omega}(0))\).

Single-argument form
Define \(\&(\alpha) = \{(0, \omega), (1, \omega), (\alpha, 2)\}\). It basically expresses the entire power of BEAF.

Conjectures:


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