Comments by "eggynack" (@eggynack) on "TED-Ed" channel.

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  40. Yeah, that's what it sounds like. Basically, any set of numbers that you can put in any kind of list is going to be exactly the same size as any other set with that quality. So the whole numbers and odd numbers are obviously like that. The integers are a bit less obviously like that, because you can list them like 0, 1, -1, 2, -2... The fractions are not at all obviously like that. For that ordering, you want to make a chart with the integers hanging out on the X and Y axis, and then the entries in the chart use the X axis as the numerator and the Y as the denominator (the Y axis excludes zero). Then you make this big zig zagging line that goes through every entry in the list, remove any repeats, and you have a full ordering of the rationals. Even less obvious is the set of algebraic numbers. Those are the real numbers that you can write as a finite algebraic expression. So, like the square root of 2 would be an algebraic number. Phi is an algebraic number too. But something like pi cannot be expressed that way, and so is not algebraic. The ordering there is relatively straightforward. You can kinda treat each part of the expression as a digit, and then remove illogical combinations. The reason all these sets have the same size is because you can easily create a mapping of the sort I mentioned. All the elements are already in a list, so there's a first element, a second element, a third element, and so on. All you need to do is pair the nth element of set A to the nth element of set B and the mapping is done. The real numbers cannot be listed this way, and so that is a bigger set.
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