In addition to u/YeetAndMeme ‘s post
In: Other
Infinite does not mean all inclusive. It means impossible to measure. So even though 3 does not fall in between 1 and 2 there are still so many numbers that we cannot physically measure it
Infinite simply means “not finite”. It’s a property, not a specific number, the same way that even and odd are properties. Lots of things can fail to be finite, while also not being equal to each other, just as lots of numbers can fail to be even but they aren’t equal to one another.
In terms of the sizes of sets, we say that two sets have equal size if there is a scheme for transforming the elements of one set into the elements of the other set without any overlaps, and vice versa. For example, {1,2,3} and {4,5,6} are the same size because we can take each element of the first set, add 3 to it, and get the elements of the second set. Likewise, we can subtract 3 from each element of the second set to get the elements of the first set. Because we have schemes for both directions and neither scheme transforms two different elements into the same element, we say that the sets have the same size. There can be many different schemes for the given sets, but as long as there’s at least one for each direction, the sizes are considered equal.
Now, to transform the elements of the interval [1,2] into the elements of [1,3], you can simply take each element, double it, then subtract 1. Likewise, you can go from [1,3] to [1,2] by taking each element, adding 1 to it, and then halving the sum. We have schemes for both directions and neither cause overlaps, so the sizes are equal.
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.
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