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A gap ordinal is an ordinal \(\alpha\) where \((L_{\alpha+1}-L_\alpha)\cap\mathcal P\omega=\varnothing\), where \(\mathcal P\omega\) denotes the powerset of \(\omega\).[1] 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.[2]. In fact, for \(\beta\) starting a gap, \(L_\beta\cap\mathcal P\omega\) is closed under definitions from analysis[3]. 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 \(\omega_1^L\).[1]

The first gap ordinal is sometimes called the "ordinal of ramified analysis" and written as \(\beta_0\).[4] 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 \(L_{\beta^+}\vDash\beta\textrm{ is uncountable}\)[5]. 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 recursion[6].

Extensions

There are several directions of extensions of the notion of a gap ordinal.

Based on Definability

Putnam's result on the unboundedness under \(\omega_1^L\) is extended to a generalised notion. Given a \(\Sigma_1\)-definable(in what structure?) function \(\Phi \colon \textrm{On}^m \to \textrm{On}\) with 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 \((\beta_1,\ldots,\beta_{m-1}) \in (\omega_1^L)^{m-1}\).[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 \(\mathcal{A}_3\).[1]

Relative sizes

The least third-order gap ordinal is greater than the least starting points of many of the gaps based on definability. 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, \(\beta=\textrm{min}\{\xi:L_\xi\vDash\textrm{ZFC}-\textrm{Powerset+}``\omega_1\textrm{ exists}\!"\}\)[7], \(L_{\gamma^+}\vDash``\omega_1\textrm{ exists}\!"\)[8], \(L_{\gamma^+}\not\vDash\textrm{ZFC}-\textrm{Powerset}\)[9], and \(\beta\neq\gamma\).

Sources

  1. 1.0 1.1 1.2 1.3 Marek & Srebrny, Gaps in the Constructible Universe, Annals of Mathematical Logic 6 (1974), pp. 359--394.
  2. 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.
  3. M. Lucian, Gap-minimal systems of notations and the constructible hierarchy (p.23)
  4. D. Madore, A Zoo of Ordinals (2017) (p.6)
  5. T. Arai, A sneak preview for proof theory of ordinals (p.17) (accessed 2021-03-07)
  6. The least α for which E(α) is inadmissible (p.149) (accessed 2021-03-07)
  7. D. Madore, A Zoo of Ordinals (p.6). Accessed 2021-05-05
  8. [Arai, A sneak preview of proof theory of ordinals (1997) (p.17)]
  9. γ+ isn't a limit of admissible ordinals, so γ+ isn't Σ2-admissible[citation needed], so γ+ isn't a gap ordinal (cf. [MarekSrebrny73 (p.368)]

See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation
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)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert'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: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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