Fun with infinity

And to Kajiti, the infinity for even numbers is the same exact infinity for natural numbers. There are exactly as many even numbers as there are natural numbers. The only set of numbers that are more infinite are the set of real numbers.

in short the degrees of infinity for the different sets of numbers is as follows: Natural = Even = Rational < Real

What's the line of logic behind that? Also, just so I have clarity, what's the difference between natural and real numbers? Natural numbers are integers, whereas real numbers are any non-imaginative number, including decimals? (bleh I'm BSing, I don't really know the difference)

And Devil's Sunrise, what I mean by different degrees of infinity can be demonstrated by limit problems.

So say you have f(x) = x^2, and g(x) = x^3. Say we try to measure both f(x) and g(x) as x goes to infinity. They would both be infinite right? Say we define h(x) = f(x) / g(x). We try to measure h(x) as x goes to infinity. Does h(x) go to infinity, or does it go to zero? Since f(x) / g(x) = x^(-1), we can establish that we can pick apart different types of infinity.

infinity - infinity = undefined, since infinity "isn't an exact finite quantity".

Code:

Assume ∞ - ∞ = 0
∞ - ∞ + 1 = 1
Since ∞ + 1 = ∞,
∞ - ∞ = 1
1 = 0

Whoops!