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
>… Why you can’t do this with natural numbers
Well, in order to do the trick, you need an ordered list first. You need something that clearly comes after another thing, so that you can assing an ‘ID’ to a natural number the same way Vertitasium uses a natural number as an ID for the real ones.
But the thing is, what else is there but natural numbers? Nothing else is ‘naturally ordered’. You could use the alphabet, but there is no reason why it should be ordered the way it is in the first place. A being first is arbitrary.
But 1 coming before 2 isn’t, and it’s the only thing we can all agree on. It’s also the reason why it’s called a countable infinity.
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