memes

~~I was thinking that, bill for bill, the $100 bill will always be greater value. But I can see the plausibility in your argument that, when we're counting both the value of the members of each set, the value of the $100 bill pile can always be found somewhere in the series of $1 bills. The latter will always "catch up" so to speak. But, if this line of reasoning is true, it should apply to other countably infinite sets as well. Consider the following two examples.

First, the number of rational numbers between 0 and 1 is countably infinite. That is, we can establish a 1-1 correspondence between the infinite set of fractions between 0-1 and the infinite set of positive integers. So the number of numbers is the same. But clearly, if we add up all the infinite fractions between 0 and 1, they would add up to 1. Whereas, adding up the set of positive integers will get us infinity.

Second, there are equally many positive integers as there are negative integers. There is a 1-1 correspondence such that the number of numbers is the same. However, if we add up the positive integers we get positive infinity and if we add up all the negative integers we get negative infinity. Clearly, the positive is quantitatively greater than the negative.

In these two cases, we see that a distinction needs to be made between the infinite number of members in the set and the value of each member. The same arguably applies in the case of the dollar bills.~~

EDIT: I see now that I was mistaken.