I guess it is a matter of taste. But I do think that countability is a natural concept since it essentially corresponds to counting. Countability in a finite setting then gives rise to countability in an infinite setting.
@blackiris wrote a nice post about the basic concept of infinity. If you have free time check it out :) https://steemit.com/mathematics/@blackiris/lamentations-in-a-sanatorium
Well, the title of the post sure does sound promising! :D I will make sure to give it a read.
Thanks for your input! :)