User:Hyp cos/OCF vs Array Notation p2

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Collapsing higher Mahloness
Now we need another ordinal collapsing function to generate weakly Mahlo cardinals. Similar to "Using weakly Mahlos" section, we need another function after \(\psi\) and \(\chi\), and use a larger cardinal for collapsing - it should be a weakly Mahlo cardinal such that weakly Mahlo cardinals in it are stationary. That leads us to higher Mahloness: an ordinal \(\pi\) is \(\alpha\)-weakly Mahlo if it's an uncountable regular cardinal and the set of \(\gamma\)-weakly Mahlo cardinals in \(\pi\) is stationary for all \(\gamma<\alpha\). So 0-weakly Mahlo cardinals are just uncountable regular cardinals, and 1-weakly Mahlo cardinals are weakly Mahlo cardinals, and the "weakly Mahlo cardinal such that weakly Mahlo cardinals in it are stationary" is a 2-weakly Mahlo cardinal. And Mahloness can extend further. For example, an ordinal \(\pi\) is (1,0)-weakly Mahlo if it's \(\pi\)-weakly Mahlo. Generally, an ordinal \(\pi\) is \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo if it's \((\alpha_1,\alpha_2\cdots,\alpha_i,\gamma,\pi,\underbrace{0,0,\cdots0,0}_{n-i-2})\)-weakly Mahlo for all \(\gamma<\alpha_{i+1}\) and \(0\le i\le n-2\), and the set of \((\alpha_1,\alpha_2,\cdots\alpha_{n-1},\gamma)\)-weakly Mahlo cardinals in \(\pi\) is stationary for all \(\gamma<\alpha_n\).

Let \(M(\alpha_1,\alpha_2\cdots,\alpha_n;0)\) to be the first \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo cardinal, \(M(\alpha_1,\alpha_2\cdots,\alpha_n;\beta+1)\) to be the next \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo cardinal after \(M(\alpha_1,\alpha_2\cdots,\alpha_n;\beta)\), and \(M(\alpha_1,\alpha_2\cdots,\alpha_n;\beta)=\sup\{M(\alpha_1,\alpha_2\cdots,\alpha_n;\gamma)|\gamma<\beta\}\) for limit ordinal \(\beta\). (Notice that there's a semicolon in this notation) So \(M(0;\alpha)=\Omega_{1+\alpha}\), \(M(1;\alpha)=M_{1+\alpha}\), and \(M(2;0)\) is the least "weakly Mahlo cardinal such that weakly Mahlo cardinals in it are stationary".

Let \(A^{\alpha_1,\alpha_2\cdots,\alpha_n}\) be the set of \((\alpha_1,\alpha_2\cdots,\alpha_n)\)-weakly Mahlo cardinals below \(\sup\{M(1,\underbrace{0,0\cdots,0}_{n};0)|n<\omega\}\). So \(M(\alpha_1,\alpha_2\cdots,\alpha_n;\beta)\) is the \(1+\beta\)-th ordinal in the closure of \(A^{\alpha_1,\alpha_2\cdots,\alpha_n}\).

\begin{eqnarray*} C_0(\alpha,\beta) &=& \beta\cup\{0\} \\ C_{n+1}(\alpha,\beta) &=& \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{M(\gamma_1,\gamma_2\cdots,\gamma_k;\delta)|\gamma_1,\gamma_2\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\} \\ &\cup& \{\chi_\pi^{\gamma_1,\gamma_2\cdots,\gamma_k}(\delta)|\pi,\gamma_1,\gamma_2\cdots,\gamma_k,\delta\in C_n(\alpha,\beta)\wedge\delta<\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) \\ \chi_\pi^{\gamma_1,\gamma_2\cdots,\gamma_n}(\alpha) &=& \min(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A^{\gamma_1,\gamma_2\cdots,\gamma_n}\}\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 \(\Omega_1\) (first uncountable cardinal), \(\psi(\alpha)\) is a shorthand for \(\psi_\Omega(\alpha)\), \(\chi_\pi(\alpha)\) is a shorthand for \(\chi_\pi^0(\alpha)\).

The notation gets a bit complex now. We have changes on \(\chi\) and \(\psi\) - a \(\{\pi\}\) is unionized. Now the \(\pi\) isn't restricted to some \(A^{\alpha_1,\alpha_2\cdots,\alpha_n}\), so the \(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A^{\gamma_1,\gamma_2\cdots,\gamma_n}\}\) may be empty for some value such as \(\pi=\Omega\), \(\alpha=0\). So we need to add a harmless \(\{\pi\}\) so that every \(\chi\) results some value. \(\psi\) is similar. And this \(\{\pi\}\) doesn't affect the values if the \(\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\beta\in A^{\gamma_1,\gamma_2\cdots,\gamma_n}\}\) is not empty.