Simplified Reflection OCF

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 of a reflection configuration are all ordinals appearing in the reflection configuration not as a second component, and the inductive variables are the ordinals appearing as a third component.

With reflection configurations, we can then define collapsing hierarchies $\mathcal A$ associated to a reflection configuration $\mathbb X=(\kappa,n,\alpha,\mathbb Y,\ldots)$:

  • If $\alpha=0$, $\mathcal A\mathbb X=\kappa\cap\mathcal A\mathbb Y\cap\ldots$.
  • Otherwise, $\mathcal A\mathbb X$ is the set of $\pi\in\mathcal A(\kappa,n,0,\mathbb Y,\ldots)$ such that:
    • $C(\alpha,\pi)\cap\kappa=\pi$,
    • For every $\beta\in C(\alpha,\pi)\cap\alpha$, $\pi$ is $\Pi_n$-reflecting on $\mathcal A(\kappa,n,\beta,\mathbb Y,\ldots)$.

Collapsing functions are defined by $\psi=\min\mathcal A$.

Then $C(\alpha,\kappa)$ is the closure of $\kappa\cup\{0,K\}$ under:

  • $\beta+\gamma$,
  • $\omega^\beta$,
  • $\psi\mathbb X$, if all variables of $\mathbb X$ are previously constructed and all inductive variables are less than $\alpha$.

Then $\psi(\omega_1^{CK},1,\varepsilon_{K+1})$ is most likely the proof-theoretic ordinal of $\Pi_\omega$-ref.

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