A question related to cofinality and cofinal subclass

Question

I am confusing about the definition of cofinal subclass and cofinality.To be more precise, I don't understand why there exists some regular ordinals β such that cf(β)=β. In my understanding way, if I choose the cofinal subclass of β just β for any ordinal β(As β⊆β), then I have cf(β)=1 for any ordinal. Where is the problem?

Answer 1

The simplest example is $\omega$. As $\omega \not \lt \omega$, you can't choose that. If you choose any finite ordinal it is not cofinal with $\omega$, so you have to choose an infinite subset. Therefore $cf(\omega)=\omega$. Your approach works for any successor ordinal, but not limit ones.