I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
this post was submitted on 06 Jan 2024
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I got tired of reading people saying that the infinite stack of hundreds is more money, so get this :
Both infinites are countable infinites, thus you can make a bijection between the 2 sets (this is literally the definition of same size sets). Now use the 1 dollar bills to make stacks of 100, you will have enough 1 bills to match the 100 bills with your 100 stacks of 1.
Both infinites are worth the same amount of money... Now paying anything with it, the 100 bills are probably more managable.
You could also just divide your infinite stack of $1 bills into 100 infinite stacks of $1 bills. And, obviously, an infinite stack of $100 bills is equivalent to 100 infinite stacks of $1 bills.
(I know this is only slightly different than what you're getting at, which is that infinitely many stacks of 100 $1 bills is equivalent to an infinite stack of $100 bills)
I'd take the 1's just because almost everywhere I spend money has signs saying they don't take bills higher than $20.
Yup. Exactly this.
They can spend the same amount of money, but at any moment the one with 100s has more money. If you have 2 people each picking up 1 bill at the same rate at any singular moment the person picking up the 100s will have more money.
Since we're talking about a material object like dollar bills and not a concept like money we have to take into consideration it's utility and have to keep in mind the actual depositing and spending would be at any individual moment. The person with 100s would have a much easier/quicker time using the money therefore the 100s have more utility.
We're definitely not talking about this like a material object at the same time, though. There's no way for a single person to store and access an infinite pile of bills.
You can spend a 100 dollar bill faster than a 1 dollar bill, sure, but both stacks would have the same money in the bank.
Except you're given an infinite amount of bills, not money in the bank. So even when moving the money to the bank you'd be able to access it quicker with the 100s
Alternatively for small brains like me:
Imagine you have an infinite amount of $1 bills are laid out in a line. Right next to it is a line of $100 bills.
As you go down the line, count how much money you have at any given point.
Which total is worth more?
Your example introduces the axis of time which is not in consideration when discussing infinity. You're literally removing infinity from the equation by doing that because "at any given point" by definition is not infinity. Let's say that point is 1 million bills down the line. Now you're comparing 1,000,000 x 100 vs 1,000,000 x 1, nothing to do with infinity
Imagine the line of 1s is stacked like pages in books on a shelf, but the line of 100s is placed in a row so they're only touching on the sides. You could probably fit a few hundred 1s in the space of one 100. Both lines still have infinite bills in them, but now as you go along, you're seeing a lot more 1s at a time.
That's the thing about infinities, you can squish and stretch them, and they're still infinite.