Yes, sometimes. Cantor proved this. Basically, each set has something called cardinality, and two sets with the same cardinality can be mapped to each other, element for element. For instance, there is no way to take the integers and reals and pair them up, element by element. There is, however, a way to pair up the integers and rationals (fractions). Therefore, integers and rationals are the same size, but the set of reals is bigger than both.

When dealing with infinite numbers, things can get confusing. For example, there are exactly the same number of even integers are there are integers. If the sets were finite, you'd get twice as many integers as even ones, but because they are infinite, there are exactly as many.

As another example, suppose you take the set containing all the integers. I take all the elements of your set and put them in mine, and then sprinkle a couple of fractions in there, and maybe a car or a boat. My set is still the same size as yours.

There's a neat and related thing called the continuum hypothesis. I think that it would be very interesting to do math assuming that it isn't true, but I haven't seen much in that vein. Most people like it a lot.

There's also something called the Lebesgue measure that can be used to measure the "size" of infinite sets. Some sets, however, are non-measurable, and you get cool things like the Banach-Tarski paradox.

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"It's turtles all the way down."