List of ordinals

Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the recursive ordinals have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the natural numbers disappear and the primary objects being studied include elementary embeddings, cofinality, cardinality and abstract reflection or partition properties.

As such, apeirology is linked to:

• Set theory (which includes study of large cardinals)
• α-recursion theory (the study of generalising recursion on the natural numbers to on L_α for admissible ordinals α)
• β-recursion theory (the generalisation of α-recursion theory to non-admissible α)
• Proof theory and ordinal analysis (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded)
• Googology (which translates recursive ordinal notations into systems for constructing large finite numbers).

Below we list some milestone ordinals.

Countable ordinals

In this list we assume there is a transitive model of ZFC. The \(\psi\) is Extended Buchholz unless specified.

• 0, the smallest ordinal
• 1, the first successor ordinal
• \( \omega \), the first limit ordinal
• \( \omega^{2} \), the second infinite additive principal ordinal
• \( \omega^{3} \)
• \( \omega^{\omega} \)
• \( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \), the PTO of PA and ACA₀^{(sort out page)}
• \( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)^{(decide if own page)}
• \( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \), the second infinite primitive recursively principal ordinal
• \( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \), the Feferman-Schutte ordinal and the PTO of ATR₀. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).
• \( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \), the Ackermann Ordinal^{(decide if keep)}
• \( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \), the SVO (Small Veblen ordinal)
• \( \psi_{0}(\Omega^{\Omega^{\Omega}}) \), the LVO (Large Veblen ordinal)
• \( \psi_{0}(\Omega_{2}) \), the BHO (Bachmann-Howard ordinal)
• \( \psi_{0}(\Omega_{\omega}) \), the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension
• \( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \), the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction
• \( \psi_{0}(\Omega_\Omega) \), sometimes known as Bird's ordinal
• \( \psi_{0}(\Omega_{\Omega_{\dots}}) \), the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion
• \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below)
• \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM
• \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF)
• \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L
• \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second OCF using indescribable cardinals
• PTO of \( \Pi^1_2 \)-comprehension
• PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset
• PTO of \( \text{KP} + "\omega_1 \) exists \( " \)
• PTO of \( \text{ZFC} \)
• \( \omega^{\text{CK}}_{1} \), the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal
• \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)^p.24
• The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-comprehension^(p.3)
• The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)^(p.3)
• The least recursively hyper-Mahlo ordinal^(p.13)
• The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)
• The least \( (+1) \)-stable ordinal^(p.4)
• The least \( (+\alpha) \)-stable ordinal, for small \(\alpha\)
• The least \( (\cdot 2) \)-stable ordinal
• The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal^(p.4)
• The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals
• The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal^(pp.3,9)
• The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)^(p.20)
• The least \( (^++1) \)-stable ordinal^{Each class of \((\sigma^1_1)^n\)-rfl. ordinals is nonempty below this ordinal (p.20)}
• The least (next recursively inaccessible ordinal)-stable ordinal
• The least (next recursively Mahlo ordinal)-stable ordinal
• The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \)
• The least doubly \( (+1) \)-stable ordinal^(p.4)
• The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)^p.24
• The least nonprojectible ordinal^(p.5) = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals^(p.218)
• The least \( \Sigma_2 \)-admissible ordinal^(pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it^(p.221)
• HIGHER STABILITY STUFF GOES HERE^{(sort out)}
• Some Welch stuff here
• Infinite time Turing machine ordinals
  • \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal
  • \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \)
  • \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal
  • \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability
• The least ordinal in \(E_1\), in unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\)
• The least ordinal in \(E_\eta\), for \(\eta > 1\)
• The least admissible \(\alpha\) so that \(L_\alpha\) satisfies \(\mathsf{AQI}\), arithmetical quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)Template:Verification failed
• Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures
• The smallest gap ordinal = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)
• Least start of a gap in the constructible universe of length 2
• Least \( \beta \) that starts a gap of length \( \beta \)
• Least \( \beta \) that starts gap of length \( \beta^\beta \)
• Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a model of KP+"\( \omega_1 \) exists".^(p.6)
• Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"^(p.6)
• Least height of model of ZFC^(p.6)
• Least stable ordinal^(p.6)(p.9), this is a limit of gap ordinals, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)^p.23
• Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)
• The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).^(p.8)

The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is significantly smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. ^{Corollary 21}

Uncountable ordinals

• \( \Omega \), the smallest uncountable ordinal

Further on, there lie large cardinals, so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist:

• \( I \), the smallest inaccessible cardinal
• \( M \), the smallest Mahlo cardinal
• \( K \), the smallest weakly compact cardinal