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Dr0110111001101111 t1_iyezsop wrote

Reply to comment by mysticpolka in There's the same amount of numbers between -∞ to 0 and -∞ to ∞ by [deleted]

I mentioned it in another comment. When it comes to infinite sets, we have two sizes: countable and uncountable.

A countable set is a set where you can make a rule that binds each number to a number in the set of natural (counting) numbers. So for example, the set of positive, even numbers is countable because we can say "the first one is 2, the second one is 4, the third one is 6..." etc. There's an organized way to do this.

We can also make a similar rule for all even numbers: the first one is zero, second is 2, third is -2, fourth is 4, fifth is -4, and so on. This means that the set of all even numbers and the set of positive even numbers are both countable and therefore the same "size".

Things get weird when you try to do this for the real numbers. You can say the first one is 0, but then what comes next? What rule can you produce in the way I did above to be sure that you can "count" all of them? It turns out it's impossible. It's also impossible to do it just with the real numbers between 0 and 1. You can't even do it with the real numbers between 0 and 0.000001. As a result we say that these sets are all "uncountable" and therefore the same size.

As an interesting side note, it turns out that the rational numbers are countable, which surprised a lot of people. A mathematician named Cantor proved this with an ingenious strategy known as the diagonalization argument.

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mysticpolka t1_iyf9cxh wrote

I think I understand the idea of countable and uncountable, but I don’t see why those would be the only two possible sizes. If two sets of infinities are countable, why wouldn’t one set be larger than the other? It seems perfectly logical to me that the infinity between 0-2 would be twice the size as infinity between 0-1.

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splitlikeasea t1_iyf5exa wrote

I think this is a bit complex for someone who made such a comment. Basically: If you slice up as part of your number and you get a finite amount of numbers, it's a countable infinity.

Take 15-30 range from -infinity to +infinity range of integes and you have finite amount of integer.

If you slice up a part of your numbers and there is still infinite amount of numbers in it, it's an uncountable infinity.

Take 1-2 on rational set and pick any part of it, say from 1.2 to 1.3. there are are infinite amount of numbers you can choose as you can just add another decimal to end.

Uncountable infinities are infinitely denser than countable infinities. And since every uncountable infinity is infinitely dense, all of them are incomparable to each other when it comes to size. Infinity is not a number but a virtual concept so it can go into unimaginable areas.

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