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
• \( \omega^{2} \)
• \( \omega^{3} \)
• \( \omega^{\omega} \)
• \( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)^{(sort out page)}
• \( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)^{(decide if own page)}
• \( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)
• \( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \), 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)
• \( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \), the TFBO (Takeuti-Feferman-Buchholz ordinal)
• \( \psi_{0}(\Omega_\Omega) \), sometimes known as Bird's ordinal
• \( \psi_{0}(\Omega_{\Omega_{\dots}}) \), the EBO (Extended Buchholz ordinal)
• \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi (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
• \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second OCF using indescribable cardinals
• PTO of \( \text{Z}_{2} \)
• PTO of \( \text{ZFC} \)
• \( \omega^{\text{CK}}_{1} \)
• RECURSIVELY LARGE ORDINALS GO HERE^{(sort out)}
• The least recursively inaccessible ordinal^(p.3)
• The least recursively Mahlo ordinal^(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 \( (^+) \)-stable ordinal = 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 = 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 doubly \( (+1) \)-stable ordinal^(p.4)
• The least nonprojectible ordinal^(p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below^(p.218)
• The least \( \Sigma_2 \)-admissible ordinal^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below^(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
  • \( \gamma \), the supremum of all clockable ordinals
  • \( \zeta \), the supremum of all eventually writable ordinals
  • \( \Sigma \), the supremum of all accidentally writable ordinals
• Welch's \(E_0\)-ordinals
• The smallest gap ordinal
• 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)

Uncountable ordinals

• \( \Omega \), the smallest uncountable ordinal
• \( I \), the smallest inaccessible cardinal
• \( M \), the smallest mahlo cardinal
• \( K \), the smallest weakly compact cardinal