You need to be much more specific on your definition of infinity. Say it is the set of all integers (we usually call this objects the integers). What does it me

You need to be much more specific on your definition of infinity. Say it is the set of all integers (we usually call this objects the integers). What does it mean to multiply it by a real number. One interpretation would mean to add it to itself (we are only multiplying by integers, so this is a simple definition). If we don't allow repeat elements, then this is trivially a NO-OP. If we do allow repeat elements, than 2infinity has 'twice' as many elements as infinity. However, we can still draw a 1:1 corraspondence between these two sets, so they are therefore the same size. IE. 1_0<>1,1_1<>2,2_0<>3,2_1<>4... Using this, we simplify to infinity/infinity=2/3. I cannot think of any good way to define infinity/infinity, but 2/3 does not seem like a good answer. >We know that the outer circle circumference is larger than the inner circle circumference. Who is talking about circumference, we are talking about the number of points. Consider that a point has a length of 0, and a circle has an infinite number of points, Using that, we get that the length of both circles is 0infinity, which is undefined. What Cantor is saying is that the infinity is the same object for both circles. >How do you transition from that - to counting infinitely small dots on those circles and saying they have the same number of those dots? You don't, the circumference has nothing to do with the number of points.