User blog:Hyp cos/General fundamental sequences for OCFs

In definitions of some ordinal collapsing functions (OCFs), there is always a series of \(C_n(\text{some arguments})\) sets. \(C_0(\text{something})\) may contains some argument in the "something", and may contains a "large" ordinal for collapsing; \(C_{n+1}(\text{something})\) is usually obtained from \(C_n(\text{something})\) by applying some operations with some limitations (avoiding "loop" definition); and \(C(\text{something})\) is the union of all \(C_n(\text{something})\). By this structure, we can define general fundamental sequences for OCFs.

An example
Here this "general" definition works on the notation using a weakly compact cardinal. The definition of the ordinal notation is:
 * Let \(K\) denote the weakly compact cardinal, \(\Omega_0=0\) and \(\Omega_\alpha\) is the \(\alpha\)-th uncountable cardinal. Then,

\begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0,K\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi(\xi,\gamma)|\pi,\xi,\gamma\in C_n(\alpha,\beta)\wedge\xi<\alpha\wedge\gamma<\alpha\} \\ &\cup& \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \\ C(\alpha,\beta) &=& \bigcup_{n<\omega} C_n(\alpha,\beta) \\ A(\alpha) &=& \{\beta<K|C(\alpha,\beta)\cap K\subseteq\beta\wedge\beta\text{ is uncountable regular} \\ & & \wedge(\forall\xi\in C(\alpha,\beta)\cap\alpha)A(\xi)\text{ is stationary in }\beta\} \\ \chi_\pi(\xi,\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A(\xi)\}\cup\{\pi\}) \\ \psi_\pi(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}\cup\{\pi\}) \end{eqnarray*}
 * And \(\Omega\) is a shorthand for \(\Omega_1\), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\).

Fundamental sequences are defined as follows: \begin{eqnarray*} C_0 &=& \{0,K\} \\ C_{n+1} &=& \{\alpha+\beta|\alpha,\beta\in C_n\} \\ &\cup& \{\Omega_\alpha|\alpha\in C_n\} \\ &\cup& \{\chi_\pi(\xi,\alpha)|\pi,\xi,\alpha\in C_n\} \\ &\cup& \{\psi_\pi(\alpha)|\pi,\alpha\in C_n\} \\ L(\alpha) &=& \min\{n<\omega|\alpha\in C_n\} \\ \alpha[n] &=& \max\{\beta<\alpha|L(\beta)\le L(\alpha)+n\} \end{eqnarray*} This definition is very similar to the one for Taranovsky's ordinal notation. Every ordinal below \(\psi(\Omega_{\Omega_{\cdots_{\Omega_{K+1}}}})\) can be expressed using addition, \(\Omega_\alpha\), \(\chi\) and \(\psi\), so these must be some n such that \(C_n\) contain the ordinal. So \(L(\alpha)\) indicates the complexity of \(\alpha\), and \(\alpha[n]\) is a "largest but below \(\alpha\)" ordinal under limited complexity.

Disadvantage
But this kind of FS doesn't work so "regular" as these fundamental sequences - the problem mainly happen on the addition. e.g. \(\omega[0]=0\), \(\omega[1]=1\), \(\omega[2]=2\), \(\omega[3]=4\), \(\omega[4]=8\), \(\omega[5]=16\), etc. At those points, \(f_\alpha(n)\) will be comparable to \(f_\alpha(f_2(n))\) in those using "regular" FS; and \(H_\alpha(n)\) will be comparable to \(H_{\alpha+\omega^2}(n)\) in those using "regular" FS.

This problem applies on most ordinals, except those obtained from some kind of fixed points, e.g. \(\psi(\Omega)=\varepsilon_0\), \(\psi(\Omega2)\), \(\psi(\Omega^2)\), \(\psi(\Omega^\Omega)\), \(\psi(\Omega_2)\), \(\psi(\Omega_\Omega)\), \(\psi(\psi_I(1))\), \(\psi(\psi_K(1))\), \(\psi(K)\), etc.