That the union of two disjoint sets, both of which have a cardinality of [math]\aleph_0[/math], has a cardinality of [math]\aleph_0[/math] is logical because we are not dealing with finite sets. The logic of finite sets does not apply to infinite sets. Infinite sets have a logic of their own.
If we have two disjoint sets A = {a1, a2, ...,an, ...} and B = {b1, b2, ...,bn, ...}, both of which can be placed into one-to-one correspondence with [math]\mathbb {N}[/math]:
[math]1 \longleftrightarrow a_1[/math]
[math]2 \longleftrightarrow a_2[/math]
[math]...[/math]
[math]n \longleftrightarrow a_n[/math]
[math]...[/math]
and:
[math]1 \longleftrightarrow b_1[/math]
[math]2 \longleftrightarrow b_2[/math]
[math]...[/math]
[math]n \longleftrightarrow b_n[/math]
[math]...[/math]
then we can interleave these two lists to form a list that is also a one-to-one correspondence with [math]\mathbb {N}[/math]:
[math]1 \longleftrightarrow a_1[/math]
[math]2 \longleftrightarrow b_1[/math]
[math]3 \longleftrightarrow a_2[/math]
[math]4 \longleftrightarrow b_2[/math]
[math]...[/math]
[math]2n-1 \longleftrightarrow a_n[/math]
[math]2n \longleftrightarrow b_n[/math]
[math]...[/math]