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moomerator t1_j29chbu wrote

Reply to comment by farineziq in There’s just as many numbers between 0 and 1 as there is from 0 to infinity. by Resinate1

So as somebody relatively well versed in math but definitely not full blown mathematician (engineer with a math minor) - I accept that this is a conclusion that a lot of people smarter than myself have agreed on but it still always bothered me.

I’m not disagreeing that it’s how the math falls out but I feel like there’s something fundamentally wrong with it. It’s like (and more or less related to) our understanding of quantum mechanics.. it feels like our current understandings are a case of the kid who got the right answer despite solving it incorrectly and like there’s a significant error/piece missing.

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farineziq t1_j29u4qj wrote

I think finding a one to one relation between elements of one set to elements of another set is a convincing way to know if they have the same amount of elements without counting them. Like if you have two heaps of rocks, you could make two lines of rocks where rocks are side by side and if both lines end together, both sets have the same amount of rocks.

Doing the same with infinite sets is useful because we can't count the elements, but might be able to find a function that associates each element of one set to one and only one element of the other. (Regarding the rocks example, every possible way to put them side by side is such a function.)

For example, there are as many positive integers as positive and negative integers because f(x) = (x % 2) * (-x / 2 - 1 / 2) + (1 - x % 2) * (x / 2) from 0 to ∞ gives 0, -1, 1, -2, 2, -3, 3, ...

Now regarding uncountable infinite sets like the real numbers, someone else in the comment showed that ]0, 1] has the same amount of elements as [1, ∞ [ using f(x) = 1 / x. And regarding what we're actually looking for, let's say [0, 1] has the same amount of elements as ℝ, I didn't really pay attention but this seems convincing.

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