List of countable ordinals


 * \(\omega\)
 * \(\omega^3\), of
 * \(\omega^\omega\), PTO of, RCA0undefinedand WKL0
 * \(\varepsilon_0=\omega\uparrow^2\omega=\vartheta(\Omega)\), PTO of PA and ACA0
 * \(\varepsilon_{\varepsilon_0}\), PTO of ACA
 * Cantor's ordinal \(\zeta_0\)
 * Feferman-Schütte ordinal \(\Gamma_0=\vartheta(\Omega^2)\), PTO of ATR0
 * Ackermann ordinal \(\vartheta(\Omega^3)\)
 * Small Veblen ordinal \(\vartheta(\Omega^\omega)\), PTO of \(\text{ACA}_0+\Pi_2^1-\text{BI}\)
 * Large Veblen ordinal \(\vartheta(\Omega^\Omega)\)
 * Bachmann-Howard ordinal \(\vartheta(\varepsilon_{\Omega + 1})\), PTO of KP set theory + axiom of infinity
 * \(\psi(\Omega_\omega)\), PTO of \(\Pi_1^1-\text{CA}_0\)
 * Takeuti-Feferman-Buchholz ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\), PTO of
 * Ψ(ψᵢ(0)), PTO of \(\Pi_1^1-\text{TR}_0\)
 * \(\psi_{\Omega_1}(\varepsilon_{\text{I}+1})\), PTO of KP + "there exists a recursively inaccessible ordinal" (KPI)
 * \(\psi_{\Omega_1}(\varepsilon_{\text{M}+1})\), PTO of KP + "there exists a recursively Mahlo ordinal" (KPM)
 * \(\Psi_{\Omega_1}(0,\varepsilon_{\text{K}+1})\), PTO of KP + \(\Pi_3\) reflection
 * Limit of Taranovsky's C function
 * PTO of Z2
 * PTO of ZFC
 * Omega one chess \(\omega_1^{\mathfrak{Ch}}\)
 * Omega one of 3D chess \(\omega_1^{\mathfrak{Ch}_3}\)
 * Admissible ordinals, ordinals \(\alpha\) such that \(L_\alpha\) is a model of KP
 * Church-Kleene ordinal \(\omega_1^{\text{CK}}\), first nonrecursive ordinal and first admissible ordinal>\(\omega\)
 * Recursively inaccessible ordinals
 * Recursively Mahlo ordinals
 * Infinite time Turing machine ordinals
 * Supremum of all writable ordinals \(\lambda\)
 * Supremum of all clockable ordinals \(\gamma\)
 * Supremum of all eventually writable ordinals \(\zeta\)
 * Supremum of all accidentally writable ordinals \(\Sigma\)