I'm reading a book on stochastic games which apparently needs ordinals somewhere. This is the first time I meet a concept, and although the definition is clear to me, the lack of practice makes it harder to think of them. That's why I need some help with examples.
As far as I understood, the ordinals are exactly equivalence classes w.r.t. order isomorphism of well-ordered sets. In the Wikipedia article on the topic there is a remark, saying that formally such classes are too big to be sets in ZF system of axioms, so one shall rather talk about order types - OK for me as well, so far it is all quite natural.
I can also imagine sets which have order types of the ordinal $\xi = 5$ or $7$, or any other natural number - this would be just any well-ordered set with $5$ or $7$ elements, right? It's easy for me to think of $\omega$ as well: one of the examples that has such order type is the set of all natural numbers.
What is harder for me, is to find an example of a set which has order types of $\omega+1$, $2\cdot \omega$, $\omega^2$ or $\omega^\omega$. Even more, for the latter I used to think that this is an unordered set of all infinite sequences of natural numbers. However, still it reads that $\omega^\omega$ is a countable ordinal.
It would be nice if the example for $\omega+1$ would be something more intuitive than $\omega+1$ itself (which perhaps is not even a set, right?), such as $\Bbb N$ for $\omega$.