I put most things on User blog:C7X/Drafts List of some stable ordinals (ordered by increasing size of first ordinal satisfying the condition): 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 \(\alpha\) is (+1)-stable, then \(\alpha\) is also admissible
- W. Richter and P. Aczel, "Inductive Definitions and Reflecting Properties of Admissible Ordinals" (1973) (p.44)

## Conventions

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

- \(\varphi\) is Veblen varphi
- \(\psi\)=Buchholz psi (\(\psi\) is \(\psi_0\))
- \(\theta\)=Feferman theta
- \(\vartheta\)=Weiermann vartheta
- \(-\) is relative complement between sets
- \(\subset\) is strict subset
- \(\subseteq\) is subset or equal
- \(f^n\) is iteration of the function \(f\)
- \(|x|\) is cardinality of \(x\)
- \(\land S\) and \(\lor S\) are conjunction and disjunction of all (possibly vacuous) members of the set (can be defined as \(\forall(t\in S)(t)\) and \(\exists(t\in S)(t)\)), where S is a set of Booleans (for example, \(\lor\{\textrm{False,True,False}\}=\textrm{True}\))

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 uses 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