It's not that the set of positive integers or the set of all integers -contain- anything other than their elements. It's that you can map each positive integer to all the integers, and vice versa. Every element taken from the first set can be mapped to a unique element in the second set, and also each element in the second set can be mapped to a unique element in the first set (Bi-directional).

Updated the proof...which I had posted the wrong link that mapped to even numbers, though that demonstrated the concept.

Here's the idea:

" Solution 1:

First, I will provide a description through a story, and then we will move on to the proof. This scenario is related to the Hilbert Grand Hotel Paradox. Imagine a hotel with an infinite number of rooms, each numbered 0,1,..., All the rooms in the hotel are already occupied. At midnight, a new group of people arrives, consisting of an infinite number of individuals numbered -1, -2,... How can you accommodate them in the hotel, considering that only one person can stay in each room?

One possible approach is to leave the person in the 0th room and the 1st room unchanged. Then, ask all the current occupants to come out and move all other individuals two rooms ahead from where the last person went. This means that the member in the second room will move to the 3rd room, the 3rd room will move to the 5th room, and so on, resulting in the even-numbered rooms (starting from 2) becoming vacant. Subsequently, the person with the negative number -n can be placed in the 2n-th room. Hence, the mapping is as follows: 0 -> 0, n -> 2n-1 for n>0, and -n -> 2n where n>0. This establishes a bijection between Z (Integers) and N (Naturals). Finally, the occupants can be shifted one place to the right to obtain the desired arrangement."

So if you think of all the set 0..n, with n going to infinity, you can define a function on n which would then also define all the integers (slightly different than the above hotel description's function using -n...n, but hopefully more easily understood):

n = 0: f(0) = 0
if n is odd (1, 3, 5, ....): f(n) = - (n+1) / 2 which gives the sequence f(1) = -1, f(3) = -2, f(5) = -3, and so on forever
if n is even (2, 4, 6, ...): f(n) = n / 2 which gives the sequence f(2) = 1, f(4) = 2, f(6) = 3, again this doesn't stop either

So what numbers comprise the set of f(n) as n goes from 0 to infinity? Well, the set of values from f(n) contains all the integers!

So the size of the cardinality is the same, every natural number can be paired with an integer. And the reverse is possible. It works because the sets are not finite but infinite...they just keep going and going, and so therefore does the mappings. Infinities are trippy this way!!

Hopefully this helps, and now I need to go count birds (finitely!!) until passed out! <3

1...2...3...zzzz