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I don't see how your point about countability invalidates what I said. Irrationals are uncountable, but there are fewer of them between 0 and 1 than reals betwe

I don't see how your point about countability invalidates what I said. Irrationals are uncountable, but there are fewer of them between 0 and 1 than reals between 0 and 1, by virtue of the fact that all irrationals are reals but not all reals are irrationals. I admit I'm using "subset" in the colloquial sense, meaning "a set that's missing some of the elements in its superset", which is not quite the correct definition (a subset doesn't have to be missing anything, i.e., every set is a subset of itself). So technically a subset does not imply fewer elements. Is that what you're getting at? I'm not particularly well versed in set theory, so maybe I'm just misunderstanding your point about countability.

roywiggins 3y ago
You didn't say anything entirely incorrect, I think! But the important bit when it comes to talking about sizes of infinity is that, while both irrationals and rationals are smaller than the reals in one sense (they're both proper subsets), they're very different in another sense: you actually can index the rationals by the natural numbers whereas you can't do the same with the irrationals. That's the much more interesting sense in which the rationals are smaller (or, alternatively, that reals are bigger). The rationals are smaller than the irrationals in this sense as well, despite not being a subset of them at all.
- roarcher 3y ago
  I think I still don't understand 100%, but that's probably my lack of formal knowledge about this stuff. I appreciate the explanation.
ska 3y ago
Consider it from a measure theory point of view, at least colloquially. If you could assign a mass of 1 (lb/kg/g/whatever) to the reals and separate them into two piles: rational and irrational then weigh again. The irrational pile would weigh 1 and the rationals, 0. So yes you took something out, but also in some sense, it doesn't matter.
tsimionescu 3y ago
> Irrationals are uncountable, but there are fewer of them between 0 and 1 than reals between 0 and 1, by virtue of the fact that all irrationals are reals but not all reals are irrationals. This is the important point where you are wrong. For a simpler to explain example : the naturals are a subset of the integers, and yet the set of all integers is exactly as big as the set of all naturals. We say this because we can actually find a natural number that corresponds to every integer number: 0:0, 1:+1, 2:-1, 3:+2, 4:-2,... A slightly more complicated construction exists for the naturals compared to the rationals: they are also the same size. However, it can be shown that the same can't be done between the naturals and the reals, or even any real interval. So, by this definition, there are more real numbers between 0 and 1 than there are in the entire set of natural numbers.

roywiggins 3y ago

You didn't say anything entirely incorrect, I think! But the important bit when it comes to talking about sizes of infinity is that, while both irrationals and rationals are smaller than the reals in one sense (they're both proper subsets), they're very different in another sense: you actually can index the rationals by the natural numbers whereas you can't do the same with the irrationals. That's the much more interesting sense in which the rationals are smaller (or, alternatively, that reals are bigger). The rationals are smaller than the irrationals in this sense as well, despite not being a subset of them at all.

roarcher 3y ago
I think I still don't understand 100%, but that's probably my lack of formal knowledge about this stuff. I appreciate the explanation.

ska 3y ago

Consider it from a measure theory point of view, at least colloquially. If you could assign a mass of 1 (lb/kg/g/whatever) to the reals and separate them into two piles: rational and irrational then weigh again. The irrational pile would weigh 1 and the rationals, 0. So yes you took something out, but also in some sense, it doesn't matter.

tsimionescu 3y ago

> Irrationals are uncountable, but there are fewer of them between 0 and 1 than reals between 0 and 1, by virtue of the fact that all irrationals are reals but not all reals are irrationals. This is the important point where you are wrong. For a simpler to explain example : the naturals are a subset of the integers, and yet the set of all integers is exactly as big as the set of all naturals. We say this because we can actually find a natural number that corresponds to every integer number: 0:0, 1:+1, 2:-1, 3:+2, 4:-2,... A slightly more complicated construction exists for the naturals compared to the rationals: they are also the same size. However, it can be shown that the same can't be done between the naturals and the reals, or even any real interval. So, by this definition, there are more real numbers between 0 and 1 than there are in the entire set of natural numbers.