You didn't waste my time. I was a willing partner. My point is that if the stuff has no possible use in real life, then it is useless except for exercising your brain.
99% of everyone who learned algebra never uses it again even once in their lifetime. Yet, I would say that, somewhere in the world on a daily basis, every facet of algebra is used to help solve engineering problems. The tool of algebra has practical value and affects our lives in a huge way.
I can't imagine that it ¹matters a whit whether some people think that the set {0,1,2,3,4,5,6,7,8,...} is twice as large as either the set {0,2,4,6,8,...} or the set {0,1,3,5,7,...} or whether they they believe that they are all the same size. Ditto for the concept of one infinite set having the property of being larger than another.
But I haven't been thinking about this for decades like you have, so what I can or can't imagine about it may not be incredibly meaningful. That is why I asked if you could provide an example of its practical value.
¹definition of 'matters a whit' as I used it = can physically affect the lives of people in some way
---Al
