Let \(A\) be a subclass of On, and \(n\) a natural number. A limit ordinal \(\alpha\) is said to be \(\Pi_n\)-reflecting on \(A\) (resp. \(\Sigma_n\)-reflecting on \(A\) ) if for any \(\Pi_n\) (resp. \(\Sigma_n\)) formula \(\phi(x)\) and any \(b\in \)\(L\)\(_\alpha\), if \((L_\alpha,\in)\models\phi(b)\) there exists a \(\beta \in A \cap \alpha\) such that \(b\in L_\beta\) and \((L_\beta,\in)\models\phi(b)\).[1] A limit ordinal is said to be \(\Pi_n\)-reflecting ordinal (resp. \(\Sigma_n\)-reflecting ordinal) if it is \(\Pi_n\)-reflecting (resp. \(\Sigma_n\)-reflecting) on \(\textrm{On}\).
Although the short definition above looks like an ill-defined predicate requiring the "quantification of formulae", which is not allowed in the set theory itself because formulae are objects in the metatheory rather than the set theory, it is actually formalisable as a predicate in set theory itself using a non-trivial coding.
Properties
- A limit ordinal \(\alpha\) is \(\Pi_0\)-reflecting on \(A\) if and only if \(\alpha=\sup(A\cap\alpha)\).[2] (this also applies to \(\Pi_1\)-reflecting ordinals by theorems 1.9 (i) and 1.9 (iii) of the same paper)
- An ordinal is \(\Pi_n\)-reflecting if and only if it is \(\Sigma_{n+1}\)-reflecting.[2]
- A limit ordinal is \(\Pi_2\)-reflecting if and only if it is admissible and greater than \(\omega\).[3]
- A limit ordinal is \(\Pi_2\)-reflecting on the class of \(\Pi_2\)-reflecting ordinals if and only if it is recursively Mahlo.[3]
- \(\Pi_3\)-reflecting ordinals are also known as recursively weakly compact ordinals.[4], and in fact the least recursively weakly compact ordinal is much larger than the least recursively Mahlo ordinal[5], similar to how the least weakly compact cardinal is much larger than the least Mahlo cardinal[5]
- \(\Pi_{n+2}\)-reflecting ordinals are considered to be recursive analogues of \(\Pi_n^1\)-indescribable cardinals.[6]
Higher Order Extension
For a natural number \(m\), an ordinal \(\alpha\) is \(\Pi_n^m\)-reflecting on \(A\) if for every \(\Pi_n^m\) sentence[7] \(\phi\), \(L_\alpha\models\phi\rightarrow\exists\beta\in A\cap\alpha(L_\beta\models\phi)\).[7]
A limit ordinal is said to be \(\Pi_n^m\)-reflecting ordinal if it is \(\Pi_n^m\)-reflecting on \(\textrm{On}\).
Note that this definition involves sentences, i.e. formulae without parameters, unlike the previous definition for \(\Pi_n\)-reflection. The notion of a \(\Pi_n^0\)-reflection ordinal is equivalent to that of a \(\Pi_n\)-reflecting ordinal.[citation needed]
Properties
- An ordinal \(\alpha\) is \(\Sigma_1^1\)-reflecting iff it's both \((\alpha^++1)\)-stable and locally countable[8]
References
- ↑ T. Arai, A simplified ordinal analysis of first-order reflection, preprint in arXiv.
- ↑ 2.0 2.1 Richter & Aczel (1973), Inductive Definition and Reflecting Properties of Admissible Ordinals (p.14)
- ↑ 3.0 3.1 T. Arai, Proof theory for theories of ordinals—I: recursively Mahlo ordinals, Annals of Pure and Applied Logic, Volume 122, Issues 1–-3, Pages 1--85, 2003.
- ↑ D. Madore, A Zoo of Ordinals, Ordinal 2.6, preprint.
- ↑ 5.0 5.1 W. Richter, P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.13)
- ↑ M. Rathjen, Proof Theory of Reflection, Annals of Pure and Applied Logic, Volume 68, Pages 181--224, 1994.
- ↑ 7.0 7.1 Richter & Aczel (1973), Inductive Definitions and Reflecting Properties of Admissible Ordinals
- ↑ J. Aguilera, The order of reflection (p.2) (accessed 2021-03-03)
See also
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
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)