11,582
pages

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

## 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