Definition 1.8.2.19.  Let κ be a cardinal number. We define:
κ is countable  :⇔  κ ≤ ℵ0  ⇔  ∃ M ⊆ ℕ : κ = |M|
κ is uncountable  :⇔  κ is not countable
Equivalence.  No proof.
Remarks.

See also the definition of a $../../Sets/countable set, which references this definition as one alternative.