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
It happens because the amount of real numbers between 0 and 1 is infinite and it’s the same between 1 and 2. So what you get is this definition of real numbers, “The amount of real numbers is equal to the number of natural numbers times infinity.” So no matter how many natural numbers you have there will always be that number times infinity real numbers, even if there are an infinite number of natural numbers. Which there is. It’s all just based on definitions and word play.
The number of real numbers=Xinfinity, where X=the number of natural numbers.
The number of natural numbers=infinity.
So…
The number of real numbers=infinity times infinity.
Or infinity^2
When the number of natural numbers is only “infinity.”
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