Strictly increasing function from $\alpha$ to $\aleph_0$

Question

Given that $\alpha < \aleph_1$, does there exists a function $f: \alpha \rightarrow \aleph_0$ that is a strictly increasing function?

This seems like it would be true since $|\alpha| = \aleph_0$, but I am not sure how to show it.

Answer 1

First of all, $\aleph_0$ denotes the cardinal, whereas $\omega$ denotes the ordinal.

But to your question, no. If $f\colon\alpha\to\beta$ is strictly increasing then $\alpha\leq\beta$.