In my textbook Introduction to Set Theory, the authors define singular cardinal as follows:
I propose an equivalent definition as follows:
An infinite cardinal $\kappa$ is called singular if there exists an increasing sequence of ordinals $\langle \alpha_\nu \mid \nu<\vartheta \rangle$ such that $\vartheta\neq\kappa$ is a limit ordinal and $\kappa=\lim_{\nu\to\vartheta}\alpha_\nu$.
Next I prove that my proposed definition implies one by the authors:
- $\alpha_\nu<\kappa$ for all $\nu<\vartheta$
If not, there exists $\nu'<\vartheta$ such that $\alpha_{\nu'}\ge\kappa$. Since $\vartheta$ is a limit ordinal, $\nu'+1<\vartheta$. Since $\langle \alpha_\nu \mid \nu<\vartheta \rangle$ is increasing, $\alpha_{\nu'+1}>\alpha_{\nu'}\ge\kappa$. Then $\kappa=\lim_{\nu\to\vartheta}\alpha_\nu \ge \alpha_{\nu'+1}>\kappa$, which is a contradiction.
- $\vartheta<\kappa$
If not, $\kappa < \vartheta$ and thus $\kappa+1 < \vartheta$. Since $\langle \alpha_\nu \mid \nu<\vartheta \rangle$ is increasing, $\alpha_\nu \ge \nu$ for all $\nu<\vartheta$. It follows that $\alpha_{\kappa+1}>\alpha_\kappa\ge \kappa$. Then $\kappa=\lim_{\nu\to\vartheta}\alpha_\nu \ge \alpha_{\kappa+1}>\kappa$, which is a contradiction.
Could you please verify my attempt? Thank you so much!