User:TGReddy/TGR's Playground/tOCF Analysis


 * \( S \) is any string

Mahlos

 * \( \alpha \vartriangleleft_{0} A \) is true iff for all functions \( f : \alpha \rightarrow \alpha \), there exists \( \kappa \in A \) that is closed under \( f \)
 * \( [0]M = \{ \alpha : \alpha \vartriangleleft Ord \} \)
 * \( [S,\alpha + 1]\Xi_{0} = \{ \beta : \beta \vartriangleleft [\alpha]\Xi_{0} \} \)
 * \( [S,\alpha]\Xi_{0} = \bigcap_{\beta<\alpha} [\beta]\Xi_{0} \) iff \( \alpha \in Lim \)
 * \( [1,0,S]\Xi_{0} = \{ \alpha : \alpha = \bigcap_{\beta<\alpha} [\beta,S]\Xi_{0} \} \)
 * \( A(\alpha) = \text{enum}(A) \)
 * \( r(A,[\alpha]\Xi_{0}(S,\beta + 1,0)) = \{ \gamma : \gamma = [\alpha]\Xi_{0}(S,\beta,\gamma) \land \gamma \in A \} \)

Shorthand

 * \( r([\alpha]\Xi_{0},[\alpha]\Xi_{0}(S))(0) = [\alpha]\Xi_{0}(S) \)
 * \( [0]\Xi_{0}(S) = \Omega(S) \)
 * \( [1]\Xi_{0}(S) = M(S) \)
 * \( \Omega(\alpha) = \Omega_{\alpha} \)
 * \( M(\alpha) = M_{\alpha} \)
 * \( \Omega_{0} = \Omega \)
 * \( \Omega(1,0) = I \)
 * \( M_{0} = M \)
 * \( [1]\Xi_{0}(1,0) = N \) (in UNOCF)
 * \( [2]\Xi_{0}(0) = N \) (intended)

tOCF Definition

 * \( C^{0}(\alpha,\nu) = \nu \cup \{ 0,G \} \)
 * \( C^{n+1}(\alpha,\nu) = \{ \beta + \gamma, \psi_{\theta}(\eta) : \beta,\gamma,\theta,\eta \in C^{n}(\alpha,\nu) \land \eta < \alpha \land \theta < \nu \} \)
 * \( C(\alpha,\nu) = \bigcup_{n \in \mathbb{N}} C^{n}(\alpha,\nu) \)
 * \( \psi_{\nu}(\alpha) = min\{ \beta : \beta \notin C(\alpha,\nu) \cap \nu \} \)