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 → Π_{3}-rfl.) because (iii) condition from Devlin is Π_{3}, 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 Π_{n+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 Σ_{1}-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

- If an ordinal \(\alpha\) is (+1)-stable, then \(\alpha\) is also admissible
- W. Richter and P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.44)

- An ordinal \(\alpha\) is admissible iff there is no \(\Sigma_1\) projection with parameters in \(L_\alpha\) that maps some \(\gamma\in\alpha\) cofinally into \(\alpha\)
- K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.38) (Note that there is a typo in theorem 40)

- If an ordinal \(\alpha\) is \(\Sigma_0\)-admissible, then it's also \(\Sigma_1\)-admissible
- K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.17)

- If a cardinal \(\kappa\) is subtle, then it's also stationary on the set of \(\kappa\)-shrewd cardinals below
- Mentioned on the Shrewd cardinal Wikipedia article and Cantor's attic ("subtle" section), possibly a result of M. Rathjen, The Higher Infinite in Proof Theory (p.21)

- If a (multiplicative principal) ordinal \(\beta\) starts a gap of length \(\beta^+\) (where \(^+\) denotes "next admissible" operation), then \(L_{\beta^+}\vDash``\textrm{there is no injective map }f:\omega\rightarrow\beta\!"\)
- [Arai, A sneak preview of proof theory of ordinals (1997) (p.17)], then apply [Arai, A sneak preview of proof theory of ordinals (1997) (p.20)]

- The property of being \(\Pi_n\)-reflecting is \(\Pi_{n+1}\)-definable (in terms of satisfaction)
- T. Arai, Iterating the recursively Mahlo operations (2010) (p.1)

- For a set of formulae \(\Gamma\) let \(\prec_\Gamma\) denote the \(\Gamma\)-elementary-substructure relation. For \(n>1\), \(\Sigma_n\)-admissibility and the property of "\(\alpha\) being \(\Pi_{n+1}\)-reflecting on \(\{\alpha'\in\alpha:L_{\alpha'}\prec_{\Sigma_{n-1}}L_\alpha\}\)" are equivalent. (At first glance in this paper, this result seems extendable to Kranakis's \(S_\alpha^n\). However a reason why it's not is explained here. Also note \((L_\alpha\vDash\Sigma_n\textrm{-coll.})\not\rightarrow(L_\alpha\vDash\Sigma_n\textrm{-sep.}\cup\Sigma_n\textrm{-coll.})\))
- E. Kranakis, Reflection and partition properties of admissible ordinals (1982) (p.221)

- The map \((\lambda\alpha.L_\alpha)\upharpoonright\omega_1\) is \(\Sigma_1\).
- Greenberg, Knight, Computable Structure Theory using Admissible Recursion Theory on ω
_{1}(p.4)

- Greenberg, Knight, Computable Structure Theory using Admissible Recursion Theory on ω

## 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