I was promoted to ask this question after reading this comment: https://www.reddit.com/r/meirl/comments/y5ifrs/comment/iskmj29/

It's under a post stating infinite one dollar bills are worth the same as infinite 20 dollar bills. No problem here for me. But that comment stumped me.

On the one hand, I don't see why we couldn't have a dollar for every real number. But then, what stops me from numbering all the bills? But wouldn't that mean I've just mapped one to one integers to real numbers, meaning real numbers are countable, which is false. Which would mean we can't get a dollar bill for every real number. But that just seems unintuitive to me.

So, can we get a dollar for every real number? Will that be a bigger amount of dollars, than an infinite stack of dollar bills? Can the "real number" pile of dollar bills be numbered?