: if there is infinite numbers between 1 and 2 but none are 3 then between 1 and 3 there are also infinite but less numbers that arnt possible , how can it be

Question

In addition to u/YeetAndMeme ‘s post

In: Other

Answer 1

When we start using infinity in terms of counting, we need to make a few changes to our classical definition. In math, the way we do that is through something called “cardinality.” This is a way to quantify how many things are in a set.

Going back to finite numbers, we do this by seeing if we can find a way to “map” your natural numbers to the set you want to count. For example, take the set {♤, ♡, ◇, ♧}. You can say the spade is the “1st,” the heart is the “2nd,” the diamond is the “3rd,” and the club is the “4th.” So we can associate each object with a number in the set {1, 2, 3, 4}. It should be noted that this association needs to be something you can “undo” in order to make sure it is valid.

When it comes to the infinities between 1 and 2 or 1 and 3, we can make a map that follows these rules, so we can say they have the same amount of stuff. However, a cool thing is that, with infinities, you can often make maps that kick out a finite number of objects and still have one of these undoable maps. But the core here is that these associations is how we say the size of something infinite, which can make our typical notion of counting lead to unintuitive things.