Re:How can some infinities be bigger than others?
by tsimionescu on Wednesday April 19, 2023 @09:10PM
(#35635219)
> but why does the presence of a mapping imply anything about the "size" of an infinity?
It is used like this because it corresponds to an intuitive property of size. If I say that set X is larger than set Y, it comes naturally to assume that, if I were to lay out their elements one by one in pairs, at some point I would run out of elements from Y but still have more elements in X.
For example, even without knowing how many fingers I have, I can check whether there are more pebbles on a beach than fingers on my hand by putting a pebble on every finger. If there are no more pebbles and I have free fingers, the size of the set of fingers was actually larger than the set of pebbles.
And while of course I would never finish if I started doing this with the naturals and the rationals, I can still prove that it can be done if given infinite time; but that, given infinite time, when comparing the naturals to the reals in the same way, we would run out of naturals and still have more reals left.
It is used like this because it corresponds to an intuitive property of size. If I say that set X is larger than set Y, it comes naturally to assume that, if I were to lay out their elements one by one in pairs, at some point I would run out of elements from Y but still have more elements in X.
For example, even without knowing how many fingers I have, I can check whether there are more pebbles on a beach than fingers on my hand by putting a pebble on every finger. If there are no more pebbles and I have free fingers, the size of the set of fingers was actually larger than the set of pebbles.
And while of course I would never finish if I started doing this with the naturals and the rationals, I can still prove that it can be done if given infinite time; but that, given infinite time, when comparing the naturals to the reals in the same way, we would run out of naturals and still have more reals left.