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

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

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