## Journal of Symbolic Logic

### Bounds for the Closure Ordinals of Essentially Monotonic Increasing Functions

Andreas Weiermann

#### 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

https://projecteuclid.org/euclid.jsl/1183744254

Mathematical Reviews number (MathSciNet)
MR1233931

Zentralblatt MATH identifier
0787.03045

JSTOR