List of countable ordinals

This is a list of some countable ordinals in increasing order.
 * 0, the least ordinal
 * 1, the least successor ordinal
 * \(\omega\), 1st transfinite ordinal
 * \(\omega+1\), 1st transfinite successor ordinal
 * \(\omega2\), 2nd transfinite limit ordinal
 * \(\omega2+1\), 1st successor ordinal after the 2nd transfinite limit ordinal
 * \(\omega^2\), \(\omega\)th transfinite limit ordinal
 * \(\omega^2+\omega\), 1st limit ordinal after the \(\omega\)th transfinite limit ordinal
 * \(\omega^3\), of
 * \(\omega^\omega\), PTO of, RCA0 and WKL0 , least limit of smaller additive principal numbers below
 * \(\varepsilon_0=\psi_0(\Omega)=\psi_0(\psi_1(0))\) with respect to Buchholz's ordinal collapsing function, and \(\varphi(1,0)\) in Veblen's function, PTO of PA and ACA0
 * \(\omega^{\varepsilon_0+1}=\varepsilon_0\times\omega\), least additive principal above an epsilon number
 * \(\varepsilon_1=\varphi(1,1)\) in Veblen's function.
 * \(\varepsilon_{\varepsilon_0}=\varphi(1,\varphi(1,0))\) with respect to Veblen's function, PTO of ACA
 * \(\zeta_0=\psi_0(\Omega^2)=\psi_0(\psi_1(\psi_1(0)))\) with respect to Buchholz's ordinal collapsing function, and \(\varphi(2,0)\) in Veblen's function also called Cantor's ordinal
 * \(\eta_0\)\(=\varphi(3,0)=\psi_0(\psi_1(\psi_1(0)+\psi_1(0)))\) with respect to Veblen's function and Buchholz's ordinal collapsing function
 * \(\varphi(\omega,0)=\psi_0(\psi_1(\psi_1(1)))\) with respect to Buchholz's ordinal collapsing function, the closure ordinal of ordinal functions.
 * Feferman-Schütte ordinal \(\varphi(1,0,0)=\Gamma_0=\psi_0(\psi_1(\psi_1(\psi_1(0))))\) with respect to Buchholz's ordinal collapsing 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 ordinal collapsing function
 * Small Veblen ordinal, \(\begin{pmatrix}1 \\ \omega\end{pmatrix}\) with respect to Schutte Klammersymbolen, \(\psi_0(\psi_1(\psi_1(\psi_1(\psi_0(0)))))\) with respect to Buchholz's ordinal collapsing 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 ordinal collapsing function
 * Bachmann-Howard ordinal \(\psi_0(\psi_2(0))\) with respect to Buchholz's ordinal collapsing function, PTO of KP set theory + axiom of infinity
 * \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ordinal collapsing function, PTO of \(\Pi_1^1\textrm{-CA}_0\)
 * Takeuti-Feferman-Buchholz ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) with respect to Buchholz's ordinal collapsing function, PTO of \(\Pi_1^1-\textrm{CA}+\textrm{BI}\)
 * \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) with respect to Extended Buchholz's ordinal collapsing function, the countable limit of Extended Buchholz's ordinal collapsing function
 * \(\psi_{\Omega}(\varepsilon_{\text{I}+1})\) with respect to Buchholz's ordinal collapsing function based on the least weakly inaccessible cardinal, PTO of KPi
 * \(\psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1 }} (0)}(0))\) with respect to Rathjen's ordinal collapsing function based on the least weakly Mahlo cardinal, PTO of KPM
 * \(\Psi^0_\Omega(\varepsilon_{K+1})\) with respect to Rathjen's ordinal collapsing function based on a weakly compact cardinal, greater than or equal to the PTO of KP + \(\Pi_3\) reflection
 * \(\psi_\Omega(\varepsilon_{\mathbb{K}+1})\) with respect to Arai's ordinal collapsing function using the least \(\Pi_N\)-reflecting ordinal \(\mathbb{K}\) for \(N\ge 3\), PTO of KP with a \(\Pi_N\)-reflecting universe under ZF + V = L.
 * The limit \(\Psi_{\mathbb{X }} ^{\varepsilon_{\Upsilon+1 }} \) of Jan-Carl Stegert's second ordinal collapsing function using indescribable cardinals. This is greater than or equal to the ordinal \(\vert\textsf{Stability}\vert_{\Sigma_1^{\omega_1^{CK }}} \), related to a theory called Stability.
 * PTO of Z2
 * Limit of Taranovsky's C function (conjectured to be larger than 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
 * (Weakenings of) stable 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 \(\Sigma\)
 * Gap ordinals
 * Stable ordinals, ordinals \(\alpha\) where \(L_\alpha\) is a \(\Sigma_1\)-elementary-substructure of \(L\)