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If I have a positive x, are there more integers below x or above x?

I was discussing this with some friends and we came up with two opposing ideas:

  1. No, since you can always count one more in either direction.
  2. Yes, since the infinite amount of numbers below x is greater than the infinite amount of numbers above x.
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$\mathbb{Z}$ is countable. Hence all subsets of $\mathbb{Z}$ are countably infinite, or finite. There aren't different sizes of infinity in the subsets of $\mathbb{Z}$ -- the only way you can get two subsets of different sizes is if at least one of them is finite.

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Actually there are an equal number of integers: Denote the positive integer $n$. Then the two sets you are talking about are $S_1=\{x\mid x\in \mathbb{Z} \land x>n \}$ and $S_2=\{x\mid x\in \mathbb{Z} \land x<n \}$

Then consider the following function $f:S_1\rightarrow S_2, f(x) =2n-x$. This is a bijection because injective for obvious reason and surjective because for any $y<n$ we let $x=2n-y$ which is in our domain.

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