“Some infinities are bigger than other infinities” – Hazel Grace Lancaster, in “The Fault in Our Stars,” by John Green.
I disagree with the idea that there are not more numbers between 0 and 2 than there are between 0 and 1.
Here is my logic:
"You can relate every number between 0 and 1 to its double, giving you every number between 0 and 2."
1. Some of those numbers will combine to doubles of themselves that fall under the number one (.34 plus .34 being .64).
2. Half will double up at 1 and over (starting from .5) and half will double up under 1 (from everything below .5), so when we compare the number of doubles UNDER 1 to their reps in 1 – 2, plus doubles INSIDE 0-1 (since this dude kind of conveniently forgot about them), we can know that 3/4ths of them are 0 – 2, and only 1/4th from 0 – 1, because we forgot all about the numbers between 0 and 1 who's doubles do not appear between 1 and 2.
3. In addition to those doubles found by those doubled at .5 and over, we still must include those numbers that doubled from below .5 and under, PLUS those numbers found between 1 and 2 that were not JUST a double but were still greater than those used in 0 to 1.
4. By this logic, there are more numbers between 0 and 2 than there are between 0 and 1.
"But we've already drawn a line from every integer, so there's no one left to be this number's partner."
1. You've drawn a line from every number in an infinite set?
2. There's no one left in an infinite set?
Come on, now. It's infinite.
By this logic, infinity == infinity, not infinity > infinity.
Infinity as a concept can be greater than itself, but not by this logic.
Cantor already showed that…
"…no list is possible for the set of real numbers R, and so it is a 'larger' infinite set, one with a 'higher order of infinity' than the infinity of the set of fractions Q."
– Tony Crilly, 50 Mathematical Ideas You Really Need To Know"
Yes, every sequence of numbers between two numbers — given that the range is great enough — can be related back. This is just a matter of division, as in 2/2 is = to 1/1.
Also, as for the video ( which I didn't care to watch ), infinity is merely a concept; there's no such thing as infinity in a deterministic world, just as there is nothing "random", other than what is subjectively unknown.
However, when comparing 0 to 2 with 0 to 1, just because you can map every number between 0 and 1 to it's double does not mean you should not ALSO INCLUDE that number itself when considering the number of numbers between 0 and 2. This means there are more between 0 and 2 then there are between 0 and 1. You don't need to draw a map at all.
If there are no numbers left out in an infinite set, than how can one be found that disagrees with some number in some decimal position as described?
It's an infinite set; a number will be found that matches. That hypothesis is stretching the bounds of possibility out to absurdity.
It is absurd to posit that you can list an infinite set (not that you can't represent it), first of all, and even more absurd that if you could do that, that you could then find a number in that set not already represented. That is absurd. It is infinite!
I don't really agree with the way the video presented the topic. In truth, there are two kinds of infinity:
Countable infinity and Uncountable infinity.
The set of all real numbers is an uncountable infinity, while the set of all integers is a countable infinity.
All you can really say about this is that if a set is countable, like the set of all integers (even though it is infinite) certain properties hold that allow specific operations to be done to it, while these would not hold for an uncountable set.
I bet he plays a mean Draw Something…