Basically, if two quantities are the same, you can pair them off. It’s possible to prove you cannot pair off all real numbers with all integers. (It works for integers and all rational numbers, though)
How many infinities you accept as meaningful is a matter of preference, really. You don’t even have to accept basic infinity or normal really big numbers as real, if you don’t want to. Accepting “all of them” tends to lead to contradictions; not accepting, like, 3 is just weird and obtuse.
Basically, if two quantities are the same, you can pair them off. It’s possible to prove you cannot pair off all real numbers with all integers. (It works for integers and all rational numbers, though)
How many infinities you accept as meaningful is a matter of preference, really. You don’t even have to accept basic infinity or normal really big numbers as real, if you don’t want to. Accepting “all of them” tends to lead to contradictions; not accepting, like, 3 is just weird and obtuse.