zlacker

Countable infinity does not work like that: two countable infinities are not more than one countable infinity. I think it falls into the "not even wrong" category of statements.

The Wikipedia article is fairly useful: https://en.wikipedia.org/wiki/Countable_set

>>usgrou+(OP)
Yes, if you put two (or three, or countably many) countable sets together, you obtain a set that is also countable. The problem is, we want to explicitly describe a bijection between the combined set and the natural numbers, so that each element is visited at some time. Constructing such a bijection between the natural numbers and a countably-infinite tree is perfectly possible, but it's less trivial than just DFS or BFS.

If we're throwing around Wikipedia articles, I'd suggest a look at https://en.wikipedia.org/wiki/Order_type. Even if your set is countable, it's possible to iterate through its elements so that some are never reached, not after any length of time.

For instance, suppose I say, "I'm going to search through all positive odd numbers in order, then I'm going to search through all positive even numbers in order." (This has order type ω⋅2.) Then I'll never ever reach the number 2, since I'll be counting through odd numbers forever.

That's why it's important to order the elements in your search strategy so that each one is reached in a finite time. (This corresponds to having order type ω, the order type of the natural numbers.)