Math Is Fun Forum

So, if every ball can be paired once to a natural number (the counting numbers), then the set of infinite balls is countable. Right? But I'm still wondering if there is a way to take away an infinite number of balls from the pile (in order!) such that there are still some balls left over. The closest I think is the idea of arranging the balls into even and odd numbered balls, and taking away an infinite number of balls such that all of the odd ones are taken, without taking all of the even ones. If this is possible, then it seems we can conclude at least that ∞-∞=1, since after the odd row of balls is taken, there is still necessarily one even numbered ball left that has not been taken. However, I still wonder how it would be possible to take all of the odd numbered balls without taking all of the odd numbered ones as well, since there is always going to be another even numbered ball which follows the odd numbered ball.

[I am moderately familiar with cantor's arguments. I'm trying to understand the following: "consider there exist an infinite number of members in a series. If an infinite number of members of the series have occurred, then zero members are left which have not occurred." I suspect, since infinity is so strange, that there is a coherent argument that says, "If an infinite number of members of the series have occurred, then there are still some members left which have not occurred."

I'm just not quite sure what that argument should look like.