User:Hyp cos/OCF vs Array Notation p3

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In November 2018, Dmytro Taranovsky updated the analysis of TON (Degrees of Reflection). In Degrees of Reflection, \(C(\Omega,0)\) is first \(\Pi_2\)-reflecting ordinal (i.e. admissible ordinal); \(C(\Omega^\Omega,0)\) is first \(\Pi_3\)-reflecting ordinal; \(C(\Omega^{\Omega^\Omega},0)\) is first \(\Pi_4\)-reflecting ordinal; \(C(\Omega^{\Omega^{\Omega^\Omega}},0)\) is first \(\Pi_5\)-reflecting ordinal, etc. Addition on n-th level of \(\Omega\)-power tower corresponds to consecutive \(\Pi_n\)-reflection, so for example From 1-stable to next-admissible-stable, they are also similar to Guess 3. For example, The OCFs in this part aim to follow those correspondences.
 * \(C(\Omega^\Omega+\Omega^2+\Omega+1,0)\) is first \(\Pi_1\)-reflecting on \(\Pi_2\)-reflectings that are also \(\Pi_1\)-reflecting on ordinals both \(\Pi_2\)-reflecting on \(\Pi_2\)-reflectings and \(\Pi_1\)-reflecting on \(\Pi_3\)-reflectings (i.e. limit of admissible limits of recursively Mahlo limits of \(\Pi_3\)-reflectings)
 * \(C(\Omega^{\Omega^\Omega+\Omega+1},0)\) is first \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings that are also \(\Pi_2\)-reflecting on \(\Pi_4\)-reflectings
 * \(C(\Omega^{\Omega^{\Omega^\Omega+\Omega+1}},0)\) is first \(\Pi_3\)-reflecting on \(\Pi_4\)-reflectings that are also \(\Pi_3\)-reflecting on \(\Pi_5\)-reflectings
 * \(C(\Omega^{\varepsilon_{\Omega+1}+\Omega^\Omega+1},0)\) is first \(\Pi_2\)-reflecting on \(\Pi_4\)-reflectings that are also (for all finite \(n\)) \(\Pi_2\)-reflecting on \(\Pi_n\)-reflectings
 * \(C(\varepsilon_{\Omega2},0)\) is first 1-stable
 * \(C(\varphi(2,\Omega2),0)\) is first \(\omega\)-stable
 * \(C(\varphi(3,\Omega2),0)\) is first \(\omega^2\)-stable
 * \(C(\varphi(C(\varphi(\Omega,1),0),\Omega),0)\) is first \(\alpha\) that is \(\alpha\)-stable
 * \(C(\varphi(\Omega,1),0)\) is first \(\alpha\) that is \(\beta\)-stable where \(\beta\) is next admissible ordinal after \(\alpha\)

Unlike previous parts, symbols \(\Omega\), \(I\), \(M\) and \(K\) will denote large countable ordinals instead of cardinals. In this part, \(X=X_1\) is first X-ordinal, \(X_{\alpha+1}\) is next X-ordinal after \(X_\alpha\), and \(X_\alpha=\sup\{X_\beta|\beta<\alpha\}\) for limit \(\alpha\), where \(X=\Omega\) means "admissible", \(X=I\) means "recursively inaccessible", \(X=M\) means "recursively Mahlo", and \(X=K\) means "\(\Pi_3\)-reflecting".

Ordinal \(\alpha\) is \(\Pi_n\)-reflecting on class \(A\) if for all \(\Pi_n\) formula \(\phi\), \(L_\alpha\models\phi\rightarrow\exists\beta\in A\cap\alpha(L_\beta\models\phi)\). An ordinal is called \(\Pi_n\)-reflecting if it is \(\Pi_n\)-reflecting on \(\text{Ord}\).

Due to the following properties, the \(\Pi_2\)-reflection will cover the relative largeness of admissibles (or uncountable regular cardinals) and recursively Mahloness (or Mahloness). So \(\Pi_2\)-reflectings on \(\Pi_2\)-reflectings will be recursively Mahlos, and \(\Pi_2\)-reflectings on \(\Pi_2\)-reflectings on \(\Pi_2\)-reflectings will be recursively 2-Mahlos, and so on. Recursively inaccessible ordinals can also be named as \(\Pi_2\)-reflectings that are \(\Pi_1\)-reflecting on \(\Pi_2\)-reflectings.
 * 1) \(\alpha\) is \(\Pi_1\)-reflecting on class \(A\) iff \(\alpha=\sup(A\cap\alpha)\), i.e. \(\alpha\) is a limit point of \(A\)
 * 2) \(\alpha\) is \(\Pi_2\)-reflecting iff \(\alpha>\omega\) and is admissible
 * 3) \(\alpha\) is \(\Pi_2\)-reflecting on class \(A\) iff \(\alpha\) is recursively Mahlo on \(A\)

\(\Pi_2\)-reflection over \(\Pi_3\)-reflectings
After "\(\Pi_3\)-reflecting limit of \(\Pi_3\)-reflecting limits of ... \(\Pi_3\)-reflecting limit of \(\Pi_3\)-reflectings" (or "weakly compact limit of weakly compact limits of ... weakly compact limits of weakly compacts"), we need another collapsing function to diagonalize over them, which needs a larger ordinal to collapse. An ordinal \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings (or a weakly Mahlo cardinals in which weakly compact cardinals are stationary) suits. Further, we can have more of such ordinals, and higher tier limits of such ordinals e.g. ordinals both \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings and limit of \(\Pi_2\)-reflectings on \(\Pi_3\)-reflectings. Finally, there are ordinals \(\Pi_2\)-reflecting on \(\Pi_2\)-reflectings on \(\Pi_3\)-reflectings (level-2 \(\Pi_2\)-reflectings on \(\Pi_3\)-reflectings), and higher levels of iterated \(\Pi_2\)-reflection.

The resulting OCF will contain a \(B^\alpha\) class of level-\(\alpha\) \(\Pi_2\)-reflectings on \(\Pi_3\)-reflectings (where \(B^0\) are \(\Pi_3\)-reflectings), and \(\chi_\pi^\alpha(\beta)\) take ordinals in them. Another \(A_\pi(\alpha)\) work similar to "Using weakly compacts" but start with \(A_\pi(0)=\pi\), and \(\Psi_\pi(\alpha,\beta)\) take ordinals in them.

But we don't stop here. We can also introduce a \(\Pi_3\)-reflecting ordinal that is \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings (or a weakly compact cardinal in which weakly compact cardinals are stationary), for a collapsing function diagonalizing over the \(\alpha\) in \(B^\alpha\), making the \(B\)-series similar to the \(A\)-series introduced in "Using a weakly compact". Rename \(B(\alpha)\) to \(A^1(\alpha)\), and now we have 2 similar series. We can have 3 series as we introduce a \(\Pi_3\)-reflecting ordinal that is \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings that are also \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings, 4 series as we introduce a \(\Pi_3\)-reflecting ordinal that is \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings that are also \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings that are also \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings, and and higher levels of iterated \(\Pi_2\)-reflection over \(\Pi_3\)-reflectings.

So here is the OCF. Let \(X^0=\text{Ord}\), \(X^\alpha\) be the class of \(\Pi_3\)-reflecting ordinals that are (for all \(\beta<\alpha\)) \(\Pi_2\)-reflecting on \(X^\beta\).