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

How to make light theme for the logo: Paste `.wds-community-header__wordmark{filter:invert(100%);}`

into your common.css

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 \(\alpha\) (i.e. there is no \(\Sigma_1\)-definable subset of \(\alpha\) (with parameters in \(L_\alpha\)) that is unbounded in \(\alpha\) and has order type \(<\alpha\)??)
- 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, 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)]

## 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 \(A\)
- \(0\in\mathbb{N}\)
- \(\varphi\) is Veblen varphi
- Function \(\psi\)=Buchholz psi (\(\psi\) is \(\psi_0\))
- Function \(\theta\)=Feferman theta
- Function \(\vartheta\)=Weiermann vartheta
- \(-\) 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
- Quantification:
- For example, \(\exists(\beta>\alpha)(\cdots)\) means \(\exists(\beta)(\alpha\in\beta\land\cdots)\)

I use ZFC+Generalized Continuum Hypothesis+\(\exists\text{A proper class of inaccessibles}\)

### 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 \(\psi(\varepsilon_0+1)\neq\omega^{\varepsilon_0+1}\))

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