Think of it this way: The set {even numbers} is enumerable and the set {odd numbers} is enumerable. And the union of the sets, that is to say, the positive integers are enumerable. But the set {{evens}, {odds}} is not enumerable. It is of magnitude aleph_1.
You miss the ball here: the set {{evens},{odds}} is enumerable, it has cardinal number aleph_0 and has Omega+Omega as ordinal number. For the set {{evens},{odds}} stands in 1-1 correspondence with {natural numbers}.
"Fat" Cantor Sets: do these paradoxical objects bother anyon
in Metaphysics and Epistemology
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You miss the ball here: the set {{evens},{odds}} is enumerable, it has cardinal number aleph_0 and has Omega+Omega as ordinal number. For the set {{evens},{odds}} stands in 1-1 correspondence with {natural numbers}.