User:Ubersketch/KOCF

\(C_0 (\alpha, \beta) = \beta \cup \lbrace 0, K \rbrace\)

\(C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, \Xi(\eta, \gamma), \Psi_\pi(\epsilon, \eta)\)

\(| \gamma, \delta, \epsilon, \eta, \pi \in C_n (\alpha, \beta); \epsilon \le \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace\)

\(C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta)\)

\(M(0) = \lbrace \beta < K : C(0, \beta) \cap K = \beta \rbrace\)

\(For \(\alpha > 0\),\)

\(M(\alpha) = \lbrace \beta < K : C(\alpha, \beta) \cap K = \beta \wedge \beta \text { is regular } \wedge (\forall \gamma \in C(\alpha, \beta) \cap \alpha )\)

\((M(\gamma) \text { is stationary in } \beta ) \rbrace\)

\(\Xi(\alpha, \beta) = the \(\beta\)th ordinal in \(M(\alpha)\)

\(\Psi_\pi (\alpha, \beta) = \min (\lbrace \gamma : \gamma \in M(\alpha) \cap \pi \wedge C( \beta, \gamma) \cap \pi \subseteq \gamma \wedge \pi \in C( \beta, \gamma) \rbrace \cup \lbrace \pi \rbrace)\)

\(L(\alpha) &=& \min\{n<\omega|\alpha\in C_n\}\)

\(\alpha[n] &=& \max\{\beta<\alpha|L(\beta)\le L(\alpha)+n\}\)

KOCF by Deedlit11.