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Temporary notes here

We set \(\psi(\delta)\equiv\exists(\gamma\in\textrm{Ord})(L_\gamma\prec_{\Sigma_2}L_\delta)\), and \(\psi\equiv\exists(\delta\in\textrm{Ord})\psi(\delta)\), and let \(\psi\langle x\rangle\) denote \(\textrm{sup}\{\delta:\psi^x(\delta)\}\). Set \(\chi\equiv\exists\eta(``\eta\textrm{ is a limit ordinal}\! "\land\psi_x\in\eta\land\psi^{L_\eta})\). Now we need Pi_1^1-reflection with Σ_1-stb. witness

!(Nonprojectibility → Π3-rfl.) because (iii) condition from Devlin is Π3, so by contradiction least isn't minimal

The reason Σ_n-admissibility stronger than Π_(n+1)-reflection is because there are two reflections, the domain and range, which are both Πn+1-defined, then apply Barwise's theorem 7.8??

The universal quantification in Π_2-reflection represents totality of the α-recursive function (in the witness). Note that recursively Mahlo is stronger than this, but such α-recursive functions must still be total. This might not be too much of a problem, total functions can be reverse-engineered by a Σ1-cof argument?

I put some things on User blog:C7X/Drafts

I'm able to delete blog posts if you want one of yours to be deleted (IDK if there's a setting that lets everyone delete their own blog posts)

List of some stability-related ordinals and properties: User:C7X/Stability

Triple xi \(\xi^{\!\!\!\!\!\!\phantom{!}\phantom{!}\phantom{!}^\zeta}\)

Inline/small tree \(\substack{\circ&-&\circ&-&\circ \\ \vert&&\vert&& \\ \circ&&\circ&& \\ \vert&&&& \\ \circ&&&&}\) text

Useful Facts and Citations

Conventions

I use these conventions on most of my things (unless otherwise specified)

  • For a class \(A\), \(\mathcal P(A)\) denotes the class of subsets of \(A\)
  • \(0\in\mathbb{N}\)
  • \(-\) is relative complement between sets
  • \(\subset\) is strict subset
  • \(\subseteq\) is subset or equal
  • \(|x|\) is cardinality of \(x\)
  • For a cardinal \(\kappa\), a set \(x\) has "\(\kappa\)-many elements" if \(|x|=\kappa\)
  • Let \(S\) be a set of Booleans. \(\bigwedge S\) and \(\bigvee S\) are conjunction and disjunction of all (possibly vacuous) members of the set (can be thought of as \(\forall(t\in S)(t)\) and \(\exists(t\in S)(t)\), however formulae are objects in the metatheory), for example, \(\bigvee\{\textrm{False,True,False}\}=\textrm{True}\))
  • "Kripke-Platek set theory" includes the axiom of infinity
  • I often write in terms of terminology defined here
  • Quantification:
    • For example, \(\exists(\beta>\alpha)(\cdots)\) means \(\exists(\beta)(\alpha<\beta\land\cdots)\) (if \(\alpha\) and \(\beta\) are ordinals, \(\alpha<\beta\) denotes \(\alpha\in\beta\))

I usually use ZFC+Generalized Continuum Hypothesis, often unless otherwise specified

Types of OCFs

If a function is called some variant of the symbol \(\psi\), it probably refers to something that acts similarly to Buchholz psi (it may or may not use regular subscripts, what's important is that \(\psi(\psi(\Omega)+1)\le\psi(\psi(\Omega))\))

If a function is called some variant of the symbol \(\vartheta\), it probably refers to something that acts similarly to Weiermann vartheta (it satisfies an equality similar to \(\vartheta(\varepsilon_0+1)=\omega^{\varepsilon_0+1}\))

If a function is called some variant of the symbol \(\theta\), it probably refers to something that acts similarly to Feferman theta

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