User blog comment:Edwin Shade/A Question Concerning Cardinal Infinities/@comment-24920136-20171025011457

The meaningful distinction is being able to map the elements from one set to the elements of the other, such that to every element from set A corresponds an element from set B.

You can do such a map with the set of natural numbers  and any other countable set. For example the set of values of 2n+1   starting with n=1.

1 2 3 ...

3 5 7 ...

Cantor's diagonal argument shows that you CAN'T create such a map from the integers to the reals.

I'd recommend this video series on cardinalities aleph_0 and aleph_1