List of ordinals

Countable ordinals

In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.

• 0, the smallest ordinal
• 1, the first successor ordinal
• \( \omega \), the first limit ordinal
• \( \omega^{2} \), the second infinite additive principal ordinal
• \( \omega^{3} \)
• \( \omega^{\omega} \)
• \( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \), the PTO of PA and ACA₀^{(sort out page)}
• \( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)^{(decide if own page)}
• \( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \), the second infinite primitive recursively principal ordinal
• \( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \), the Feferman-Schutte ordinal and the PTO of ATR₀
• \( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \), the Ackermann Ordinal^{(decide if keep)}
• \( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \), the SVO (Small Veblen ordinal)
• \( \psi_{0}(\Omega^{\Omega^{\Omega}}) \), the LVO (Large Veblen ordinal)
• \( \psi_{0}(\Omega_{2}) \), the BHO (Bachmann-Howard ordinal)
• \( \psi_{0}(\Omega_{\omega}) \), the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension
• \( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \), the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction
• \( \psi_{0}(\Omega_\Omega) \), sometimes known as Bird's ordinal
• \( \psi_{0}(\Omega_{\Omega_{\dots}}) \), the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion
• \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below)
• \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM
• \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF)
• \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L
• \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second OCF using indescribable cardinals
• PTO of \( \Pi^1_2 \)-comprehension
• PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset
• PTO of \( \text{KP} + "\omega_1 \) exists \( " \)
• PTO of \( \text{ZFC} \)
• \( \omega^{\text{CK}}_{1} \), the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal
• \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension
• The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension^(p.3)
• The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)^(p.3)
• The least recursively hyper-Mahlo ordinal^(p.13)
• The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)
• The least \( (+1) \)-stable ordinal^(p.4)
• The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\)
• The least \( (\cdot 2) \)-stable ordinal
• The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal^(p.4)
• The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals
• The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal^(pp.3,9)
• The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)^(p.20)
• The least \( (^++1) \)-stable ordinal^{(Is this strictly greater than previous entry?)(p.20)}
• The least (next recursively inaccessible ordinal)-stable ordinal
• The least (next recursively Mahlo ordinal)-stable ordinal
• The least (next \( \Pi_n \)-reflecting ordinal)-stable ordnal, for \( 2<n<\omega \)
• The least doubly \( (+1) \)-stable ordinal^(p.4)
• The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal
• The least nonprojectible ordinal^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals^(p.218)
• The least \( \Sigma_2 \)-admissible ordinal^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it^(p.221)
• HIGHER STABILITY STUFF GOES HERE^{(sort out)}
• Some Welch stuff here
• Infinite time Turing machine ordinals
  • \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal
  • \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \)
  • \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal
  • \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability
• The least ordinal in \(E_\eta\), for \(\eta > 0\)
• The least admissible \(\alpha\) so that \(L_\alpha\) satisfies arithmetically quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)
• Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures
• The smallest gap ordinal = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n\)
• Least start of a gap in the constructible universe of length 2
• Least \( \beta \) that starts a gap of length \( \beta \)
• Least \( \beta \) that starts gap of length \( \beta^\beta \)
• Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".^(p.6)
• Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"^(p.6)
• Least height of model of ZFC^(p.6)
• Least stable ordinal^(p.6)(p.9), this is a limit of gap ordinals
• Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)
• The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).^(p.8)

The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is significantly smaller than \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists".

Uncountable ordinals

• \( \Omega \), the smallest uncountable ordinal

Further on, there lie large cardinals, so big that their existence is unprovable, but which are useful if they do exist:

• \( I \), the smallest inaccessible cardinal
• \( M \), the smallest Mahlo cardinal
• \( K \), the smallest weakly compact cardinal