Reflecting ordinal

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 \((L_\beta,\in)\models\phi(b)\).T. Arai, A simplified ordinal analysis of first-order reflection, preprint in arXiv. 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 \(\alpha=\sup(A\cap\alpha)\).Richter & Aczel (1973), Inductive Definition and Reflecting Properties of Admissible Ordinals (p.14) (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.
# A limit ordinal is \(\Pi_2\)-reflecting if and only if it is admissible and greater than \(\omega\).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.
# A limit ordinal is \(\Pi_2\)-reflecting on the class of \(\Pi_2\)-reflecting ordinals if and only if it is recursively Mahlo.
# \(\Pi_3\)-reflecting ordinals are also known as recursively weakly compact ordinals.D. Madore, A Zoo of Ordinals, Ordinal 2.6, preprint., and in fact the least recursively weakly compact ordinal is much larger than the least recursively Mahlo ordinalW. Richter, P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.13), similar to how the least weakly compact cardinal is much larger than the least Mahlo cardinal
# \(\Pi_{n+2}\)-reflecting ordinals are considered to be recursive analogues of \(\Pi_n^1\)-indescribable cardinals.M. Rathjen, Proof Theory of Reflection, Annals of Pure and Applied Logic, Volume 68, Pages 181--224, 1994.
== Higher Order Extension ==
For a natural number \(m\), an ordinal \(\alpha\) is \(\Pi_n^m\)-reflecting on \(A\) if for every \(\Pi_n^m\) sentenceRichter & Aczel (1973), Inductive Definitions and Reflecting Properties of Admissible Ordinals \(\phi\), \(L_\alpha\models\phi\rightarrow\exists\beta\in A\cap\alpha(L_\beta\models\phi)\).
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 \(\Pi_n^0\)-reflecting ordinal is equivalent to that of a \(\Pi_n\)-reflecting ordinal.
===Properties===
#If an ordinal \(\alpha\) is both \((\alpha^++1)\)-stable and locally countable, then it is \(\Sigma_1^1\)-reflecting.J. Aguilera, The order of reflection (p.2) (accessed 2021-03-03)
== References ==
== See also ==
ja:反映順序数
Category:Transfinite ordinals
Category:Set theory