Gap ordinal

A gap ordinal is an ordinal \(\alpha\) where \((L_{\alpha+1}-L_\alpha)\cap\mathcal P\omega=\varnothing\), where \(\mathcal P\omega\) denotes the of \(\omega\).Marek & Srebrny, Gaps in the Constructible Universe, Annals of Mathematical Logic 6 (1974), pp. 359--394. Leeds and Putnam proved that for a gap ordinal \(\alpha\), \(L_\alpha\cap\mathcal P\omega\) is a β-model of second-order arithmetic \(\mathcal{A}_2\) using the recursion-theoretic result that \(L_\alpha\cap\mathcal P\omega\) is closed under hyperjump.S. Leeds and H. Putnam, An intrinsic characterization of the hierarchy of Constructible sets of integers, Logic Colloquium '69 (1971), Proceedings of the Summer school and Colloquium in Mathematical Logic, Manchester, August 1969 / Ed. by R.O. Gandy and C.M.E. [sic] Yates.. In fact, for \(\beta\) starting a gap, \(L_\beta\cap\mathcal P\omega\) is closed under definitions from analysisM. Lucian, Gap-minimal systems of notations and the constructible hierarchy (p.23). Currently, gap ordinals are not significantly useful in googology due to it being too high up the hierarchy of countable ordinals, but it is likely that in the future, it can be useful in analysis of TON, or other similarly strong concepts.
== Existence ==
The existence of a gap ordinal immediately follows when we work in \(\textrm{ZFC}\) augumented by \(V = L\) because of the constructibility of \(\mathcal P\omega\) itself under the assumption, but a stronger result by Putnam given as the corollary of the classification of elementary substructures of \(L_{\omega_1^L}\) by Devlin is known: The set of gap ordinals is unbounded in \(\omega_1^L\).
The first gap ordinal is sometimes called the "ordinal of ramified analysis" and written as \(\beta_0\).D. Madore, A Zoo of Ordinals (2017) (p.6) By the unboundedness result, \(\beta_0\) is smaller than \(\omega_1^L\).
==Relation to cardinals==
Arai has shown that ordinals \(\beta\) starting gaps of length \(\beta^+\) satisfy \(L_{\beta^+}\vDash\beta\textrm{ is uncountable}\)T. Arai, A sneak preview for proof theory of ordinals (p.17) (accessed 2021-03-07). In fact, it's even possible to find non-admissible \(\theta\) such that \(L_\beta\vDash\exists\theta\exists p(p=\aleph_1^{L_\theta}\land\theta\textrm{ is not admissible})\), and even stronger results than this, which involve primitive recursionThe least α for which E(α) is inadmissible (p.149) (accessed 2021-03-07).
== Extensions ==
There are several directions of extensions of the notion of a gap ordinal.
=== Based on Σ1-definable functions ===
Putnam's result on the unboundedness under \(\omega_1^L\) is extended to a generalised notion. Let \(m\) be a positive integer, and call a function \(\Phi \colon \textrm{On}^m \to \textrm{On}\) "\(\Sigma_1\)-definable in \(V\)" if there exists a \(\Sigma_1\) formula \(\chi\) with \(m+1\) free variables such that \(\Phi(\alpha,\beta_1,\ldots,\beta_{m-1})=\gamma\) iff \(\chi(\alpha,\beta_1,\ldots,\beta_{m-1},\gamma)\). Given a function \(\Phi \colon \textrm{On}^m \to \textrm{On}\) that's \(\Sigma_1\)-definable in \(V\) with arity of a positive integer \(m\), the set \(\{\alpha \in \omega_1^L \mid (L_{\Phi(\alpha,\beta_1,\ldots,\beta_{m-1})}-L_{\alpha})\cap\mathcal P\omega=\varnothing\}\) is unbounded in \(\omega_1^L\) for any \((\beta_1,\ldots,\beta_{m-1}) \in (\omega_1^L)^{m-1}\).
=== Based on Higher Order ===
The notion of a gap ordinal is extended to the higher order setting by replacing \(\mathcal P\omega\) by \(\mathcal P^m\omega\) for a natural number \(m\), where the superscript denotes iteration. Namely, an ordinal \(\alpha\) is said to be a gap of \(n\)-th order for a natural number \(n\) if \((L_{\alpha+1}-L_{\alpha})\cap\mathcal P^m\omega=\varnothing\) for any natural number \(m < n\). For example, every ordinal is a gap of \(0\)-th order, an infinite ordinal is precisely a gap of first order, a gap ordinal is precisely a gap of second order, and the notion of a gap of third order is characterised by a property using the notion of a \(\beta\)-model of third order arithmetic \(\mathcal{A}_3\).
===Relative sizes===
The least third-order gap ordinal is greater than the least starting points of many of the gaps based on \(\Sigma_1\)-definable functions. For example, the least \(\beta\) starting a third-order gap is greater than the least \(\gamma\) such that \((L_{\gamma^+}-L_\gamma)\cap\mathcal P\omega=\varnothing\). As evidence, \(\beta=\textrm{min}\{\xi:L_\xi\vDash\textrm{ZFC}-\textrm{Powerset+}``\omega_1\textrm{ exists}\!"\}\)D. Madore, A Zoo of Ordinals (p.6). Accessed 2021-05-05, \(L_{\gamma^+}\vDash``\omega_1\textrm{ exists}\!"\)[Arai, A sneak preview of proof theory of ordinals (1997) (p.17)], \(L_{\gamma^+}\not\vDash\textrm{ZFC}-\textrm{Powerset}\)\(\gamma^+\) isn't a limit of admissible ordinals, so \(\gamma^+\) isn't \(\Sigma_2\)-admissible, so \(\gamma^+\) isn't a gap ordinal (cf. [MarekSrebrny73 (p.368)], and \(\beta\neq\gamma\).
== Sources ==
== See also ==
Category:Transfinite ordinals