List of countable ordinals


 * \(\omega\)
 * \(\omega+1\), 1st transfinite successor ordinal
 * \(\omega2\), 2nd transfinite limit ordinal
 * \(\omega^3\), 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 \(\varphi(1,0,0)=\Gamma_0=\psi_0(\psi_1(\psi_1(\psi_1(0))))\) with respect to Buchholz's function, PTO of ATR0
 * Ackermann ordinal \(\varphi(1,0,0,0)=\psi_0(\psi_1(\psi_1(\psi_1(0)+\psi_1(0))))\) with respect to Buchholz's function
 * Small Veblen ordinal \(\psi_0(\psi_1(\psi_1(\psi_1(\psi_0(0)))))\) with respect to Buchholz's function, PTO of \(\text{ACA}_0+\Pi_2^1-\text{BI}\)
 * Large Veblen ordinal \(\psi_0(\psi_1(\psi_1(\psi_1(\psi_1(0)))))\) with respect to Buchholz's function
 * Bachmann-Howard ordinal \(\psi_0(\psi_2(0))\) with respect to Buchholz's function, PTO of KP set theory + axiom of infinity
 * \(\psi_0(\Omega_\omega)\) with respect to Buchholz's function, PTO of \(\Pi_1^1-\text{CA}_0\)
 * Takeuti-Feferman-Buchholz ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) with respect to Buchholz's function, PTO of
 * \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) with respect to Extended Buchholz's function, the countable limit of Extended Buchholz's function
 * \(\psi_{\Omega_1}(\varepsilon_{\text{I}+1})\) with respect to Buchholz's ordinal collapsing function based on the least weakly inaccessible cardinal, PTO of KP + "the universe is recursively inaccessible" (KPi)
 * \(\psi_{\Omega_1}(\varepsilon_{\text{M}+1})\) with respect to Rathjen's ordinal collapsing function based on the least weakly Mahlo cardinal, PTO of KP + "the universe is recursively Mahlo" (KPM)
 * \(\Psi_{\Omega_1}(0,\varepsilon_{\text{K}+1})\) with respect to Rathjen's ordinal collapsing function based on reflection, PTO of KP + \(\Pi_3\) reflection
 * Limit of Taranovsky's C function (conjecturally larger than PTO of Z2 )
 * 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
 * Reflecting ordinals
 * Stable ordinals
 * Gap 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 \(\Sigma\)