: 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

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.