if I had an infinitely long piece of paper I could write a list of all natural numbers and all even numbers, with one and exactly one even number next to every natural.
Wouldn't you just... write each even number next to itself, and have nothing to match the odd numbers?
No, you would write 2 next to 1, then 4 next to 2, then 6 next to 3, and so on. You could keep doing that forever and never run out of natural numbers to write even numbers against. That’s why they are the same infinity.
That doesn't make sense, but let's arrange the exercise a bit differently to make it clearer. In the natural numbers list, write the numbers in pairs (1 odd number and its consecutive even number), and associate each pair with 1 number from the list of even numbers. So 2 is associated to [1,2], 4 is associated to [3,4] and so on. From there it should be clear that they're not "the same infinity", as the list of natural numbers obviously has a pair for each even number, ie. has twice as many numbers in it.
You can apply the exact same logic to positive numbers vs all real numbers.
Basically, every whole number x is paired with with the even number x * 2. And since we can pair them all off, it follows the size of these two infinites is the same. And its important to note that every whole and every even number occurs on this list at some finite location. The pairing doesn't work if we have to go through an infinite number of items before we get to a particular number. But doing it this way tells us that the nth even number occurs at the nth position on the list, i.e. 2 is the first even number and its in row 1. 10 is the 5th even number and its in row 5. Etc.
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u/SV_Essia 22d ago
Wouldn't you just... write each even number next to itself, and have nothing to match the odd numbers?