When it's impossible to change or eliminate something, it's ineradicable. Unfortunately, your dad's ineradicable dislike of furry animals means you'll have to wait until you have a place of your own to get that dog you dream of.
The adjective ineradicable comes from eradicate, which means "totally destroy," and the prefix in-, "opposite of." So when something's ineradicable, whether it's a quality like hostility or a situation like inequality, there's just no way to destroy it. Though negative things are most often described as ineradicable, you can also praise your friend's ineradicable optimism or admire your teacher's ineradicable enthusiasm.