What is the difference between a set that is countable and infinite and one that is countably infinite? I am rereading my analysis notes and I came on this remark in the section on countability:
We have proved that Q is countable, and certainly Q is not ﬁnite, because N ⊆ Q. We
have not proved that Q is countably inﬁnite.
The way this is phrased suggests a difference between something being countably infinite and something being countable and infinite. Is there a difference?
Thanks.
 A: There is no difference. Finite sets are by definition countable, so countably infinite is used to clarify that the set in question, whilst countable, is not finite.
A: This is speculation, but perhaps your text means to say that, even if we have a surjection $\mathbb{N}\to S$, and we know that $S$ is not in bijection with any finite set, we still need to prove that there is a bijection $S\to\mathbb{N}$.  This is true, but it's good form to prove it.
A: If it may help, I will follow the exposition in Patrick Suppes, Axiomatic set theory (1960 - Dover reprint).
Suppes starts with Tarski's definition of finite set [page 100] :

Definition 5. [A set] $A$ is finite if and only if every non-empty family of subsets of $A$ has a minimal element.

An alternative definition is due to Dedekind [page 107] :

Definition 6. A set is Dedekind finite if and only if it is not equipollent to any of its proper subsets.

Then we have :

Theorem 46. If a set is finite then it is Dedekind finite.

Page 150 :

Definition 23. A set is infinite if and only if it is not finite.
Theorem 38. if $A$ is infinite and $A ≈ B$ then $B$ is infinite.
Theorem 39. If $A \subseteq B$ and $A$ is infinite then $B$ is infinite.
Theorem 40. A set $A$ is infinite if and only if for every natural number $n$ there is a subset of $A$ equipollent to $n$.
Theorem 41. The set $\omega$ of natural numbers is infinite.

Page 151 :

Definition 24. A set is denumerable [also : countable infinite] if and only if it is equipollent to the set $\omega$ of all natural numbers.

Page 152 :

As an immediate consequence of Theorems 38 and 41 we have:
Theorem 43. Every denumerable set is infinite.
Definition 25. A set is Dedekind infinite if and only if it is not Dedekind finite.
Theorem 44. A set is Dedekind infinite if and only if it has a proper subset equipollent to it.


Comment
We have to note the interplay between definition and theorems : stating the basic definition of denumerable set, Suppes is able to prove that a denumerable set is infinite.
