Building a scale by diagonalization and transfinite induction

Question

I'm asking a question following the reading of Set Theory from Jech (page 126).

For the record, a $\kappa$-sequence of functions $\langle f_\alpha : \alpha < \kappa\rangle$ is called a $\kappa$-scale if $f_\beta < f_\alpha$ whenever $\beta < \alpha$, and if for every $g\colon \omega \rightarrow \omega$ there exists an $\alpha$ such that $g < f_\alpha$.

Suppose that $\aleph_1 = 2^{\aleph_0}$. A scale $\langle f_\alpha : \alpha < \omega_1 \rangle$ is constructed by transfinite induction to $\omega_1$:

Let $\{g_\alpha : \alpha < \omega_1\}$ enumerate all functions from $\omega$ to $\omega$. At stage $\alpha$, we construct, by diagonalization, a function $f_\alpha$ such that for all $\beta < \alpha$, $f_\alpha > f_\beta$ and $f_\alpha > g_\beta$. Then $\langle f_\alpha : \alpha < \omega_1 \rangle$ is an $\omega_1$-scale.

I don't see how to manage the transfinite induction in order to avoid for limit ordinals that $f_\alpha$ becomes infinite.

Thanks for your support!

Answer 1

The inequality relation between functions $\omega\to\omega$ that is written $f<g$ in the question must mean that $f$ is eventually (not everywhere) below $g$, i.e., that $f(n)<g(n)$ holds for all sufficiently large $n$. (If it meant "for all $n$, then the induction would indeed break down at the first limit ordinal.) Now to construct an upper bound for countably many functions $f_\alpha$, first re-enumerate them in an $\omega$-sequence, say as $g_n$, and then use the function $k\mapsto 1+\max\{g_n(k):n<k\}$.

By the way, the usual notation for the "eventually $<$" relation is not $<$ but $<^*$.