EDIT: I THINK I STAND CORRECTED

Yup. Exactly this.

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.

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.

There’s no problem at all with not understanding something, and I’d go so far as to say it’s virtuous to seek understanding. I’m talking about a certain phenomenon that is overrepresented in STEM discussions, of untrained people (who’ve probably took some internet IQ test) thinking they can hash out the subject as a function of raw brainpower. The ceiling for “natural talent” alone is actually incredibly low in any technical subject.

There’s nothing wrong with meming on a subject you’re not familiar with, in fact it’s often really funny. It’s the armchair experts in the thread trying to “umm actually…” the memer when their “experience” is a YouTube video at best.

I don't know why you see it as an error. It's the format of the meme. The guy in the middle is right, the guy on the left is wrong. That's just how this meme works. But the punchline in this meme format is the the guy on the right agrees with the wrong guy in an unexpected way. I'm with the guy on the right and no appeals to Schröder–Bernstein theorem is going to change my mind.

In short, yes.

I admit the only time I've encountered the word utility as an algebraist is when I had to TA Linear Optimisation & Game Theory; it was in the sections of notes for the M level course that wasn't examinable for the Bachelors students so I didn't bother reading it. My knowledge caps out at equilibria of mixed strategies. It's interesting to see that there's some rigorous way of codifying user preference. I'll have to read about it at some point.

Sorry if you've seen this already, as your comment has just come through. The two sets are the same size, this is clear. This is because they're both countably infinite. There isn't such a thing as different sizes of countably infinite sets. Logic that works for finite sets ("For any finite a and b, there are twice as many integers between a and b as there are even integers between a and b, thus the set of integers is twice the set of even integers") simply does not work for infinite sets ("The set of all integers has the same size as the set of all even integers").

So no, it isn't due to lack of knowledge, as we know logically that the two sets have the exact same size.

If we’re adding real world hypotheticals you would be paying for your shit on card anyway. You just go to the bank with how ever many truck loads of $100 bills when ever you needed a top up. Secondly you wouldn’t be doing it yourself, you pay someone else to do it. Thirdly as soon as the government found out you effectively had a money printer they would put you in prison or disappear you to prevent you from collapsing their money system, not to mention the serial numbers on the notes would have to be fraudulent because they wouldn’t match up with mints. And finally any physical object with an infinite quantity would be the size of the universe, likely causing either black hole or destroying the universe and us along with it. So in closing what sounds like a great situation is probably worth any potential risk

Yeah, this is what it comes down to. In calculus, infinity doesn't exist, you just approach it when you take the limit. You'll approach it "quicker" with the 100 dollar bills, so to speak

You could make an argument that infinite $100 bills are more valuable for their ease of use or convenience, but infinite $100 bills and infinite $1 bills are equivalent amounts of money. Don't think of infinity as a number, it isn't one, it's infinity. You can map 1000 one dollar bills to every single 100 dollar bill and never run out, even in the limit, and therefore conclude (equally incorrectly) that the infinite $1 bills are worth more, because infinity isn't a number. Uncountable infinities are bigger than countable ones, but every countable infinity is the same.

Another thing that seems unintuitive but might make the concept in general make more sense is that you cannot add or do any other arithmetic on infinity. Infinity + infinity =/= 2(infinity). It's just infinity. 10 stacks of infinite bills are equivalent to one stack of infinite bills. You could add them all together; you don't have any more than the original stack. You could divide each stack by any number, and you still have infinity in each divided stack. Infinity is not a number, you cannot do arithmetic on it.

100 stacks of infinite $1 bills are not more than one stack of infinite $1 bills, so neither is infinite $100 bills.

I don't see what you are trying to say. You can also match 200 $1 bills with each $100 bill. The correspondence does not need to be one-to-one.

but each $100 bill is worth more.

But the meme doesn't talk about the value of each $100 bill; it talks about the value of the bills collectively.

I think you're misunderstanding the math a bit here. Let me give an example.

If you took a list of all the natural numbers, and a list off all multiples of 100, then you'll find they have a 1 to 1 correspondence.
Now you might think "Ok, that means if we add up all the multiples of 100, we'll have a bigger infinity than if we add up all the natural numbers. See, because when we add 1 for natural numbers, we add 100 in the list of multiples of 100. The same goes for 2 and 200, 3 and 300, and so on."
But then you'll notice a problem. The list of natural numbers already contains every multiple of 100 within it. Therefore, the list of natural numbers should be bigger because you're adding more numbers. So now paradoxically, both sets seem like they should be bigger than the other.

The only resolution to this paradox is that both sets are exactly equal. I'm not smart enough to give a full mathematical proof of that, but hopefully that at least clears it up a bit.

Adding up 100 dollar bills infinitely and adding up 1 dollar bills infinitely is functionally exactly the same as adding up the natural numbers and all the multiples of 100.

The only way to have a larger infinity that I know of us to be uncountably infinite, because it is impossible to have a 1 to 1 correspondence of a countably infinite set, and an uncountably infinite set.

You're right that they're the same size but you're mistaken when you try to assign a total value to the stack. Consider breaking each $100 bill into 100 $1 bills. The value is the same, clearly. So for each pair, you have a $1 bill and a small stack of 100 $1 bills. Now combine all singles back together in an infinite stack. Then combine all stacks of 100 into an infinite stack.

And you know what? Both infinite stacks are identical. They have the same value.

each $100 bill is worth more.

But the creation of each additional bill devalues the currency. At some point the value of all this paper money is negative because it’s not worth keeping and storing. The point at which they cross from positive to negative value would give them zero value, and they’d be equal.

You likely can see the point I’m making now; there are just as many $1 bills as there are $100 bills, but each $100 bill is worth more.

But the monetary value of the bills in each stack still adds up to infinity for both. It's like having an uncapped Internet connection at 56 KBit/s versus 100 MBit/s: You can download all the things with both, but that alone doesn't make them equal.

I think it would be easier to go to thebank now and then to exchange some of your $100 for lower value bills than to do the opposite.

Besides, if there are places that don't take bills less than $20, there are a lot of places that aren't going to let you pay with only $1 bills. I can't imagine a car dealership letting you do that. Or if you want to buy a home, etc.

Yes, uncountably infinite sets are larger than countably infinite sets.

But these are both a countably infinite number of bills. They're the same infinity.

I think quite some people heard of the concept of different kinds of infinity, but don't know much about how these are defined. That's why this meme should be inverted, as thinking the infinities described here are the same size is the intuitive answer when you either know nothing or quite something about the definition whereas knowing just a little bit can easily lead you to the wrong answer.

As the described in the wikipedia article in the top level comment, the thing that matters is whether you can construct a mapping (or more precisely, a bijection) from one set to the other. If so, the sets/infinities are of the same "size".

What about lemons?

It's (literally) +Inf

Imagine being the guy with an infinite money glitch but having to pay for EVERYTHING in one dollar bills. Buying even a modest sedan would be a pain in the ass. "What do you do for a living?" "I'm 47 strippers."

nonono, Infinity times infinity plus one, I win! *tehee nanana Na na na! *

Infinity plus infinity, plus +1 to whatever anyone else says, recurring. No returns!!

I think you'll find that's sealed tight now and that I win. Na NAAAA!

/runs off maniacally

Approaching infinite mass means you'll probably form a black hole or end the universe before being able to weigh either of them

edit: meant to reply to the parent comment

Both are equally easy to pay with since ∞ - X = ∞ if we disregard that 100s are more convenient way to pay for cars.

Except subtraction is not applicable in this case

Not true. 2∞ is not bigger or smaller than ∞. This is explained by Hilbert's hotel. And subtracting infinity from infinity is undefined so you do not get 0 = ∞.

even if that's not how you can write it, one gets the same issue in yours subtracting infinity from both sides

Yes and no. If you spend that infinite money, then yes. The currency would be massively devalued as you would be adding money into the economy.

If you sat on it, nothing would happen. I imagine that the Federal Bank doesn't know about your infinite stash and therefore isn't taking into account any equation.

The money only devalues based on how much is in circulation. You'll only devalue the currency as you spend it and you'd have to spend a trillion to have a non-minor effect.

There is an infinite amount of possible values between 0 and 1. But factorially it means measuring a coastline will lead towards infinity the more precise you get.

And up all the values between 0 and 1 with an infinite number of decimal places and you get an infinite value.

Or there's the famous frog jumping half the distance towards a lilly pad, then a quarter, than an eighth. The distance halfs each time so it looks like they'll never make it. An infinitesimally decreasing distance until the frog completes an infinite number of jumps.

Then what most people understand by infinity. There are an infinite number of integers from 0 to infinity. Ultimately this infinity we tend to apply in real world application most often to mean limitless.

These are mathematically different infinities. While all infinity, some infinities have limits.

for $1 bills: lim(x->inf) 1*x

for $100 bills: lim(x->inf) 100*x

Using L'Hôpital's rule, we take the derivative of each to get their ratio, ie: 100/1, so the $100 bill infinity is bigger (since the value of the money grows faster as the number of bills approaches infinity, or said another way: the ratio of two infinities is the same as the ratio of their rates of change).

Something to do with the...Greek? Hebrew? Klingon? Letter Aleph

Those are all aleph 0 infinities. There's is a mathematical proof that shows the square of infinity is still infinity. The same as "there is the same number of fractions as there is integers" (same size infinities).

It might give use different growth rate but Infinity is infinite, it's like the elementary school playground argument saying "infinity + 1" there is no "infinity + 1", it's just infinity. Infinity is the range of all the numbers ever, you can't increase that set of numbers that is already infinite.

stahp

Most places where you live. This isn't a problem for the entire globe. Some of it, sure. But not all of it. I pay with hundreds all the time in Australia and noone gives a shit.

The point of the meme format is not that the middle guy is right. It's the opposite.

But then you can imagine you found a pirate's chest with all the coins inside 🤩

You can. This is not one of them

I've been a fool

There is, though. Maybe not wide in your inner circle, but very, very wide if you actually look around.