I was watching Veritasium’s video about math having a fatal flaw. He explained that if we make a set of all the numbers in between 0 and 1, then added one to the first digit of the first number and added one to the second digit of the second number etc, we would always have a new number. He said this proved that there were more numbers in between 0 and 1 than natural numbers.
I was confused as to why you can’t do this with natural numbers, and how that proved one infinity was smaller than another.
In: Mathematics
I saw a guy on an old NOVA documentary put it something like this:
There are an infinite number of irrational numbers between zero and one. If you create a set of numbers that includes whole numbers going to infinity *and* the irrational numbers between each of the whole numbers, the irrational numbers, “get to infinity” way sooner, and that infinity is always many infinities larger than the infinity of whole numbers in the same set. One of the infinities seems to be a multiple of the other. One sub-set will always be infinitely larger than the other until they both somehow become infinite.
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