This is a list of some countable ordinals in increasing order.
- 0, the least ordinal
- 1, the least successor ordinal
- 2, the second 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
- \(\omega^3\), PTO of EFA
- \(\omega^\omega\), PTO of PRA, RCA0 and WKL0[1], 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.[2]
- \(\varepsilon_{\varepsilon_0}=\varphi(1,\varphi(1,0))\) 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 primitive recursive ordinal functions.[3]
- 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[4]
- 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 \(\Pi_1^1\textrm{-CA}_0\)[5]
- Takeuti-Feferman-Buchholz ordinal \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) with respect to Buchholz's ordinal collapsing function, PTO of \(\Pi_1^1-\textrm{CA}+\textrm{BI}\)[6]
- \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) 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 KPi[7]
- \(\psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0))\) with respect to Rathjen's ordinal collapsing function based on the least weakly Mahlo cardinal, PTO of KPM[8]
- \(\Psi^0_\Omega(\varepsilon_{K+1})\) 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[9]
- \(\psi_\Omega(\varepsilon_{\mathbb{K}+1})\) 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[10] 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.[11]
- PTO of Z2
- Limit of Taranovsky's C function (conjectured to be larger than PTO of Z2[12])
- 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 \(L\)
Sources[]
- ↑ M. Rathjen, Proof theory: From arithmetic to set theory. Retrieved 2021-05-22.
- ↑ D. Madore, A Zoo of Ordinals. Retrieved 2021-02-25.
- ↑ H. Levitz, Transfinite Ordinals and Their Notations: For The Unitiated (p.8). Retrieved 2021-05-11
- ↑ D. Madore, A Zoo of Ordinals (2017, #1.16). Retrieved 2020-11-19.
- ↑ W. Buchholz, A new system of proof-theoretic ordinal functions (1984). Retrieved 2021-06-19.
- ↑ Buchholz, Feferman, Pohlers, Sieg, Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies (1981)
- ↑ D. Madore, A Zoo of Ordinals (2017) (p.2, entry 1.22)
- ↑ Rathjen, Proof-theoretic analysis of KPM (p.27)
- ↑ Rathjen, Proof theory of reflection (p.47)
- ↑ Arai, A simplified analysis of first-order reflection (p.2)
- ↑ Jan-Carl Stegert, Ordinal Proof Theory of Kripke-Platek Set Theory Augmented by Strong Reflection Principles (2010) (p.141)
- ↑ D. Taranovsky, Ordinal Notation.
Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)