Journal of Symbolic Logic

Bounds for the Closure Ordinals of Essentially Monotonic Increasing Functions

Andreas Weiermann

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Abstract

Let $\Omega := \aleph_1$. For any $\alpha < \varepsilon_{\Omega + 1} := \min \{\xi > \Omega:\xi = \omega^\xi\}$ let $E_\Omega (\alpha)$ be the finite set of $\varepsilon$-numbers below $\Omega$ which are needed for the unique representation of $\alpha$ in Cantor-normal form using 0, $\Omega, +$, and $\omega$. Let $\alpha^\ast := \max (E_\Omega(\alpha) \cup \{0\})$. A function $f: \varepsilon_{\Omega + 1} \rightarrow \Omega$ is called essentially increasing, if for any $\alpha < \varepsilon_{\Omega + 1}; f(\alpha) \geq \alpha^\ast: f$ is called essentially monotonic, if for any $\alpha,\beta < \varepsilon_{\Omega + 1}$; $\alpha \leq \beta \wedge \alpha^\ast \leq \beta^\ast \Rightarrow f(\alpha) \leq f(\beta).$ Let $\mathrm{Cl}_f(0)$ be the least set of ordinals which contains 0 as an element and which satisfies the following two conditions: (a) $\alpha,\beta \epsilon \mathrm{Cl}_f(0) \Rightarrow \omega^\alpha + \beta \epsilon \mathrm{Cl}_f(0)$, (b) $E_\Omega\alpha \subseteq \mathrm{Cl}_f(0) \Rightarrow f(\alpha) \epsilon \mathrm{Cl}_f(0)$. Let $\vartheta_{\varepsilon_{\Omega + 1}}$ be the Howard-Bachmann ordinal, which is, for example, defined in [3]. The following theorem is shown: If $f:\varepsilon_{\Omega + 1} \rightarrow \Omega$ is essentially monotonic and essentially increasing, then the order type of $\mathrm{Cl}_f(0)$ is less than or equal to $\vartheta\varepsilon_{\Omega + 1}$.

Article information

Source
J. Symbolic Logic, Volume 58, Issue 2 (1993), 664-671.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744254

Mathematical Reviews number (MathSciNet)
MR1233931

Zentralblatt MATH identifier
0787.03045

JSTOR
links.jstor.org

Citation

Weiermann, Andreas. Bounds for the Closure Ordinals of Essentially Monotonic Increasing Functions. J. Symbolic Logic 58 (1993), no. 2, 664--671. https://projecteuclid.org/euclid.jsl/1183744254


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