Why ℕ ~ {2x|x∈ℕ}?
I actually understand this but I can’t explain this to my father who says that ‘for each even natural number there are 2 natural numbers’ – How can I explain this concept to him in a way that he will understand? (not saying he is stupid or anything he just cant figure it out)
In: Mathematics
Perhaps an important piece of context for the correct and good proofs being offered here is that they’re all oriented around the ways we *choose* to make infinity mathematically comprehensible. Math is a constructed system of definitions. This is likely a case where something is true according to a mathematical definition, but the definition itself feels “wrong.”
The very intuitive approach to comparing the “size” of two infinite sets is to truncate them both at the “same place” and compare the size of the two resulting finite sets. I’d wager that’s what your father is doing in his head. The problem with this is that we’ve jettisoned any notion that the sets are infinite and so aren’t really solving the problem we set out to solve.
Instead mathematicians have come up with a different way of comparing the size of infinite sets that involves asking the question of whether you can map the elements of one onto the elements of the other. Using this definition of “size,” the natural numbers and the even natural numbers have the same size. This is quite counterintuitive, and one might even say that’s a problem with the definition. However, the definition is useful in other contexts, like showing how and why the real numbers are “bigger” than the natural numbers, so we continue to use it.
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