If an infinite well-ordered set has initial segments of finite cardinality only, is the set isomorphic to $\mathbb N$?

Question

Let $A$ be an infinite well-ordered set. Every initial segment of $A$ is finite. Is $A$ isomorphic to $\mathbb N$?

What's the way to think about it? Should I build an explicit isomorphism? What should I look at?

Answer 1

Yes. This is true. And you should build an explicit isomorphism, because there is really just one isomorphism. Let me give you a hint about that.

HINT: Every initial segment has a unique cardinality.