Are Some Infinities Bigger than Other Infinities?

No John Green, they are not.

(Below is an attempt at an answer solely informed by intuition. After a formal research and study of the question and its background, I have written a second article addressing the same issue (read it here). Both articles reach the same conclusion, i.e., that some infinities are not bigger than other infinities)

The statement “there are infinite numbers between 1 & 2 and there are infinite numbers between 1& 3. So, the later infinity is bigger than the first infinity” is invalid on many levels.

1. The range [1, 2] is by definition a finite range, comprised of finite units, same for [1, 3].

2. Infinity entails having no upper bound, but when you talk about infinite number of values between two bounds, you are contradicting this definition.

3. This is what there is: an infinite level of precision or granularity. But at any one level of granularity there are only a finite number of values between two bounds. But the number of sets of different granularities is infinite. In other words, the number line is infinitely dense but once you have chosen a specific zoom level (precision), then any range at that level contains only a finite number of items.

4. You can’t say there are infinite values between 1 & 2 like 1, 1.5, 1.51, 1.511, 1.512…. because this is not a valid range with valid regularity. An infinite set is one that has no upper bound when we continue to add x unit of the same level of granularity to the previous value.

5. The longer and more explicit name for a range is “a range of units at the same level of granularity”. You can talk about
[1, 2] => where the counting coefficient (let’s call it x) is 1.
[1, 1.5, 2] => x = 0.5
[1, 1.25, 1.5, 1.75, 2] => x=0.25
….
but at any one time, you are talking about a range at one of these levels of granularities, which are finite sets by definition.

6. The depth of the infinite sets of ranges with different precisions is equal for the range [1, 2] and [1, 3]. At any given precision, the finite sets at the different ranges are not equal, obviously. But both ranges can be described by an equal, infinite, number of sets of different granularities. Hence, some infinities are not greater than other infinities, at least not by this reasoning.