The cardinal set difference one is not well defined. If I remove the evens from the integers, I have infinitely many left over. If I remove the integers from the integers, I have nothing.
The limits are also not well defined with addition and subtraction, as you can add a function that goes to infinity with one that goes to negative infinity and get all sorts of stuff. Hyperreals are what you get when you take the limits and try to make them well defined under that stuff.
The cardinal set difference one is not well defined. If I remove the evens from the integers, I have infinitely many left over. If I remove the integers from the integers, I have nothing.
The limits are also not well defined with addition and subtraction, as you can add a function that goes to infinity with one that goes to negative infinity and get all sorts of stuff. Hyperreals are what you get when you take the limits and try to make them well defined under that stuff.