How many numbers are there between 1 and 4 inclusive? That is, what is the number of distinct numbers found within the closed set [1,4]?
How many numbers are there between -2 and 7? That is, what is the number distinct numbers found within the closed set [-2,7]?
Wouldn't the answer to the latter be an infinity equal to exactly three times the former? If not, then why? Similarly, wouldn't the answer to the number of numbers between two real numbers exclusive (an open set, (n1,n2)) be equal to exactly 2 fewer than inclusive closed cardinal set [n1,n2] (which is exactly one more than either (n1,n2] or [n1,n2))?
Extended to all real numbers, wouldn't the number of different numbers between (-∞,-∞) be greater than those restricted to domain of, say, (-4,-1)? If so; then by what degree, respectively notated how exactly? If not; then why not, and what would the most correct notation of the infinitude be?
I have read about different types of numbers, bijections, and a little of set theories, but none has proven to me why this would not be an intuitively clear example of varying scale of infinitude, in fact to the contrary. If I am mistaken, please explain to me why. If my intuition is correct, please substantiate.
When comparing the cardinality of infinite sets, we do it by bijections. Any two sets that can be put in bijection with each other are the same size. The simplest example is the naturals and the even naturals. Your argument would claim there are twice as many naturals as even naturals, but there are the same number of each. We call that $\aleph_0$, the only countable infinity.
All intervals of reals have the same cardinality as each other and as all the reals. You can make a simple linear bijection between any two intervals. One or two endpoints left over is no problem-we can hide those. We can also make a bijection between any interval and the reals.
One's intuition needs a lot of updating when it comes to infinite sets.