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 WKL0M. Rathjen, Proof theory: From arithmetic to set theory. Retrieved 2021-05-22., 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.D. Madore, A Zoo of Ordinals. Retrieved 2021-02-25.
* \(\varepsilon_{\varepsilon_0}=\varphi(1,\varphi(1,0))\) 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.H. Levitz, Transfinite Ordinals and Their Notations: For The Unitiated (p.8). Retrieved 2021-05-11
* 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 ATR0D. Madore, A Zoo of Ordinals (2017, #1.16). Retrieved 2020-11-19.
* 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 \(\Pi_1^1\textrm{-CA}_0\)W. Buchholz, A new system of proof-theoretic ordinal functions (1984). Retrieved 2021-06-19.
* Takeuti-Feferman-Buchholz ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) with respect to Buchholz's ordinal collapsing function, PTO of \(\Pi_1^1-\textrm{CA}+\textrm{BI}\)Buchholz, Feferman, Pohlers, Sieg, Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies (1981)
* \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) with respect to Extended Buchholz's ordinal collapsing function, the countable limit of Extended Buchholz's ordinal collapsing function
* \(\psi_{\Omega}(\varepsilon_{I+1})\) with respect to Buchholz's ordinal collapsing function based on the least weakly inaccessible cardinal, PTO of KPiD. Madore, A Zoo of Ordinals (2017) (p.2, entry 1.22)
* \(\psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) with respect to Rathjen's ordinal collapsing function based on the least weakly Mahlo cardinal, PTO of KPMRathjen, Proof-theoretic analysis of KPM (p.27)
* \(\Psi^0_\Omega(\varepsilon_{K+1})\) with respect to Rathjen's ordinal collapsing function based on a weakly compact cardinal, greater than or equal to the PTO of KP + \(\Pi_3\) reflectionRathjen, Proof theory of reflection (p.47)
* \(\psi_\Omega(\varepsilon_{\mathbb{K}+1})\) 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 universeArai, A simplified analysis of first-order reflection (p.2) 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.Jan-Carl Stegert, Ordinal Proof Theory of Kripke-Platek Set Theory Augmented by Strong Reflection Principles (2010) (p.141)
* PTO of Z2
* Limit of Taranovsky's C function (conjectured to be larger than PTO of Z2D. Taranovsky, Ordinal Notation.)
* 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 \(L\)
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