I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
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I understand that both sets are equally infinite (same cardinality). And I do see the plausibility in your argument that, in each case, since there's an infinite number of bills their value should be equally infinite. If your argument is correct, then I should revise my understanding of infinity. So maybe you can help me make sense of 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. 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 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.
0.9 and 0.8 are in that set, so they would add up to at least 1.7. In fact if you give me any positive number I can give you a (finite!) set of (distinct) fractions less than 1 which sum to more than that number. In other words, the sum is infinite.
I meant to say that we can infinitely divide the numbers between 0 and 1 and then match each with an integer. But I realized that the former wouldn't be rational numbers, they would be real numbers.
That aside, I see now that the original idea behind the meme was mistaken.
Um no? 3/4 + 5/6 > 1
If you mean the series 1/2 + 1/3 + 1/4 + … that also tends to infinity
I meant to say that we can infinitely divide the numbers between 0 and 1 and then match each with an integer. But I realized that the former wouldn't be rational numbers, they would be real numbers.
That aside, I see now that the original idea behind the meme was mistaken.