For each cardinal number $u$, there exists a smallest ordinal number $\alpha$ such that card$\alpha$ =$u$ .

Question

For each cardinal number $u$, there exists a smallest ordinal number $\alpha$ such that $card$$\alpha$ =$u$. ,

I'm having trouble proving this theorem. How can I show this?

Answer 1

Whenever you see "smallest ordinal number such that...", it ought to remind you of one of the most important properties of the ordinals: they are well-ordered. All you need to show is that there is at least one such ordinal, and then the existence of a smallest one is guaranteed.

Note that when you prove existence like this you won't always know which ordinal you get (by which I mean that it won't always be easy to prove any other properties such an ordinal has), and that's all right. You were tasked with proving that it exists, so that's all you need to do.