posted on Dec, 27 2012 @ 06:11 PM
Mathematics is all about model building, creating the tools with which we operate on increasingly complex ideas. As such there is no (or just not yet)
absolute "correct" way to explain things. Some models are more useful than others.
For example let's take the infinity thing. Sure, one can say that {2, 4, 6, 8...} has the same size as {1, 2, 3, 4...} because the function F(x) = 2x
produces a one-to-one match. But while any natural number can be used for x not any natural number can be expressed as 2x. In other words, the input
set is larger than the output set. Some people (perhaps most people) would say that it is more intuitive.
Instead of saying that they are of the same size, it is more useful and interesting to say that {1, 2, 3, 4...} is 2 times the size of {2, 4, 6, 8...}
because of the coefficient of 2. As a rule, any function F(x) = Ax for some natural number A produces an output set whose size is 1/A of the input
set, when we are dealing with natural numbers.
edit on 27-12-2012 by Tadeusz because: (no reason given)