Ideal OCF \(\kappa\) is hereditarily \(\Pi_n\)-reflecting if and only if it is \(\Pi_n\)-reflecting or \(\Pi_m\)-reflecting on the hereditarily \(\Pi_n\)-reflecting ordinals. \(C_\kappa(\alpha)\) is the closure of \(\kappa\cup{\text{least }\Pi_n\text{-reflecting}\mid n>1}\) under \(\beta,\gamma\mapsto\beta+\gamma\) and \(\beta,\gamma,\delta\mapsto\chi_\beta(\gamma,\delta)\) provided \(\gamma<\alpha\). Suppose \(\kappa\) is \(\Pi_{n+1}\)-reflecting (\(n>0\)) but not \(\Pi_{n+2}\)-reflecting. Then set \(\mathcal A_\kappa^\alpha\) to be the set of \(\pi<\kappa\) such that: If

There’s a 2009 article by Arai on interated recursive Mahloness. The main idea of the paper is that stronger systems of reflection can be analysed in terms of iterated $\Pi_2$-reflection. In section 5, Arai writes the following: A $\Pi_3$-reflecting ordinal $K$ is understood to be …

I’m back. Surprise! I’m making a simplified collapsing function reaching $\Pi_\omega$-ref. $K$ denotes the least $+1$-stable, First, let’s set up our reflection configurations. We define them by stating that for ordinal $\kappa$, integer $n>0$, ordinal $\alpha$, and reflection configurations $\mathbb{X},\ldots$, $(\kappa,n,\alpha,\mathbb{X},\ldots)$ is a reflection configuration, provided $\kappa$ is $\Pi_{n+1}$-reflecting on $\mathcal A\mathbb X\cap\ldots$. The variables

The post is meant to extend https://googology.fandom.com/wiki/User_blog:Hyp_cos/Attempt_of_OCF_up_to_Stability. Firstly, I’ll verify that it’s worth extending. I compared it with the other OCFs (Stegert, Duchhardt). It is virtually identical to Stegert, if we associate the second reflection instance element be the position in the hierarchy. The only worrying difference is that Stegert’s hierarchies require their elements to