Because the number of dollars is not the only factor in determining which is better. If I have the choice between a wallet that never runs out of $1 bills or one that never runs out of $100 bills, I’ll take it in units of $100 for sure. When I buy SpaceX or a Supreme Court justice or Australia or whatever, I don’t want to spend 15 years pulling bills out of my wallet.
I’m not mathmatician but I got explained once that there are “levels” of Infinity, and some can be larger than others, but this case is supposed to be the same level.
I dont really know much about this topic so take it with a grain of salt.
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.
Yes! The difference between these two types of infinities (the set of non-negative integers and the set of non-negative real numbers) is countability. Basically, our real numbers contain rational numbers, which are countable, and irrational numbers, which are not. Each irrational number is its own infinity, and you can tell this because you cannot write one exactly as a number (it takes an infinite numbers of decimals to write it, otherwise you’ve written a ratio :) ). So, strictly speaking, the irrational numbers are the bigger infinity between the two.
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).
Infinity aside, the growth rate of number of bills vs the value of those bills has nothing to do with the original scenario though. It’s like arguing that a kilogram of feathers weighs less than a kilogram of bowling balls because the scale goes up less for every feather I put on the scale compared to every bowling ball I put on the scale.
Edit: Though if you want to talk about how weird infinity really is, here are some fun facts for you:
There are just as many even numbers as rational numbers, even though all even numbers are rational but not all rational numbers are even. This is because both sets are countably infinite.
There are more irrational numbers than rational numbers. This is because even though both sets are infinite, the set of irrational numbers is uncountably infinite.
It’s like arguing that a kilogram of feathers weighs less than a kilogram of bowling balls because the scale goes up less for every feather I put on the scale compared to every bowling ball I put on the scale.
I’m arguing that infinity bowling balls weighs more than infinity feathers, though
Try thinking of it like this: If I have an infinite amount of feathers, I can balance a scale that has any number of bowling balls on it. Even if there was an infinite number of bowling balls on the other side, I could still balance it because I also have infinite feathers that I can keep adding until it balances. I don’t need MORE than infinite feathers just because there’s infinite bowling balls. In the same way if my scale had every rational number on one side I could add enough even numbers to the other side to make it balance, but if I had all the irrational numbers on one side of the scale then I would never have enough rational numbers to make it balance out even though they are also infinite.
Edit: I suppose the easiest explaination is that it’s already paradoxical to even talk about having an infinite number of objects in reality just like it would be paradoxical to talk about having a negative number of objects. Which weighs more, -5 feathers that weigh 1 gram each or -5 bowling balls that weigh 7000 grams each? Math tells us in this case that the feathers now weigh more than the bowling balls even though we have the same amount of each and each bowling ball weighs more than each feather. In reality we can’t have less than zero of either.
The conventional view on infinity would say they’re actually the same size of infinity assuming the 1 and the 100 belong to the same set.
You’re right that one function grows faster but infinity itself is no different regardless of what you multiply them by. The infinities both have same set size and would encompass the same concept of infinity regardless of what they’re multiplied by. The set size of infinity is denoted by the order of aleph (ℵ) it belongs to. If both 1 and 100 are natural numbers then they belong to the set of countable infinity, which is called aleph-zero (ℵ₀). If both 1 and 100 are reals, then the size of their infinities are uncountably infinite, which means they belong to aleph-one (ℵ₁).
That said, you can definitely have different definitions of infinity that are unconventional as long as they fit whatever axioms you come up with. But since most math is grounded in set theory, that’s where this particular convention stems from.
Anyways, given your example it would really depend on whether time was a factor. If the question was “would you rather have 1 • x or 100 • x dollars where x approaches infinity every second?” well the answer is obvious, because we’re describing something that has a growth rate. If the question was “You have infinity dollars. Do you prefer 1 • ∞ or 100 • ∞?” it really wouldn’t matter because you have infinity dollars. They’re the same infinity. In other words you could withdraw as much money as you wanted and always have infinity. They are equally as limitless.
Now I can foresee a counter-argument where maybe you meant 1 • ∞ vs 100 • ∞ to mean that you can only withdraw in ones or hundred dollar bills, but that’s a synthetic constraint you’ve put on it from a banking perspective. You’ve created a new notation and have defined it separately from the conventional meaning of infinity in mathematics. And in reality that is maybe more of a physics question about the amount of dollar bills that can physically exist that is practical, and a philosophical question about the convenience of 1 vs 100 dollar bills, but it has absolutely nothing to do with the size of infinity mathematically. Without an artificial constraint you could just as easily take out your infinite money in denominations of 20, 50, 1000, a million, and still have the same infinite amount of dollars left over.
Certain infinities can grow faster than others, though. That’s why L’Hôpital’s rule works.
For example, the area of a square of infinite size will be a “bigger” infinity than the perimeter of an infinite square (which will in turn be a bigger infinity than the infinity that is the side length). “Bigger” in the sense that as the side length of the square approaches infinity, the perimeter scales like 4*x but the area scales like x^2 (which gets larger faster as x approaches infinity).
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.
but in this case we are comparing the growth rate of two functions
oh, you mean like taking the ratio of the derivatives of two functions?
it’s like the elementary school playground argument saying “infinity + 1” there is no “infinity + 1”, it’s just infinity
but that’s not the scenario. The question is whether $100x is more valuable than $1x as x goes to infinity. The number of bills is infinite (and you are correct that adding one more bill is still infinity bills), but the value of the money is a larger $infinity if you have $100 bills instead of $1 bills.
Edit: just for clarity, the original comment i replied to said
Lhopital’s rule doesn’t fucking apply when it comes to infinity. Why are so many people in this thread using lhopital’s rule. Yes, it gives us the limit as x approaches infinity but in this case we are comparing the growth rate of two functions that are trying to make infinity go faster, this is not possible. 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.
I love how people here try to put this in practical terms like “when you need to pay something 100 is better”. It’s infinite. Infinite. The whole universe is covered in bills. We all would probably be dead by suffocation. It makes no sense to try to think about the practicality of it. Infinite is infinite, they are the same amount of money, that’s all.
I think it’s because there is nuance in the wording. It doesn’t say “dollar amount”, it says “worth”, and the worth of a thing can be more than its dollar amount.
Infinite hundreds is “worth more” in a sense because it’s easier to use, and that is added value!
There’s a hierarchy called cardinality, and any two infinitives that can be cleanly mapped 1:1 are considered equal even if one “looks” bigger, like in the example from OP where you can map 100x 1 dollar bills to each 100 dollar bill into infinity and not encounter any “unmappable” units, etc.
So filling an infinite 3D volume with paper bills is practically equivalent to filling a line within the volume, because you can map an infinite line onto a growing spiral or cube where you keep adding more units to fill one surface. If you OTOH assumed bills with zero thickness you can have some fun with cardinalities and have different sized of infinities!
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 = ∞.
As a mathematician, I tend to disagree with this common truism because it limits one’s ability to think about transfinites and suggests there’s a widely agreed upon technical definition of “number”.
I’m super confused. It seems most of this conversation misses the meme format. Everyone is agreeing with the middle guy. That’s how the meme works. The middle guy is right, but that’s not the point. It’s like I’m taking crazy pills.
While I wish I didn’t just learn this meme’s far-right origins, I have to disagree with you. The middle guy is supposed to give a correct and boring answer. The left guy is supposed to give a common and wrong answer, and the guy on the right is supposed to have an enlightened view of the wrong guy’s answer. The best example I’ve seen is the one:
[wrong] Frankenstein is the monster.
[average] No, Frankenstein is the doctor.
[enlightened] Frankenstein is the monster.
It’s still wrong to say Frankenstein was the monster. It’s deliberately misleading. So yeah, the money is the same amount, but I’d rather have a secret endless stash of hundreds than singles.
[wrong] police can get away with any crimes they want (child POV)
[average] police can’t break the law even though they are police (the way it ‘should be’)
[enlightened] police can get away with any crimes they want (reality)
The middle person thinks they are smarter than the left person. But it turns out the left person was naive but correct, as shown by the right/enlightened person having the same conclusion. IMO anyway.
EDIT: also, the point of the example you gave is that Frankenstein the Doctor IS the monster (metaphorically) because he created the actual monster - I think you have misinterpreted it.
If you take the average interpretation of “police can’t …” to mean “it’s illegal for the police to …”, then yes, the average guy is right in a boring (and tautological) way. I don’t think that’s an unreasonable way for the average person to interpret the [average] line. The meme hopes to get you thinking about the last line.
Now this money example is particularly hard to argue since if you have the interpretation that the dollar will collapse, or if you think you have to store this money, then, all resolutions suck.
If a genie were to say to me, “Anytime you need money, you can reach in your pocket and pull out a bill. Would you like it to always be $1 or always be $100?”, I think I would agree with the enlightened guy here, even though I know the boring answer is right.
I confused though. You seem to think I didn’t quite get the format, but I feel I’ve explained how I see it, and don’t see any contrast with the meme you posted. So far that’s three in the format as I understand it.
So, if your answer to the genie question I asked a bit ago is “$100s please”, then the meme would speak to you. If you truly don’t care, the guy on right is wrong in your book. I’m in the “$100s please” camp.
Dr. Frankenstein is not a monster for creating an actual monster. He is a monster for abandoning his creation and escaping the responsibility of care for his creation.
I thought we were having a friendly chat, but you really insulted my intelligence there. Do you think I would have found that meme remotely interesting if I thought Frankenstein had terminals on his neck?
Sorry, I didn’t mean to insult your intelligence in any way. Just trying to have a friendly chat also. Sometimes tone can come across wrong in text. My bad
It depends what worth entails - if it’s just the monetary value then yeah they’re the same, but if the worth also comes from desirability and convenience, then infinite stack of 100 dollar bills would be way more desirable when compared to 1 dollar bills.
Less space needed to carry the money around (assuming it’s stored in some negative space and you just grab a bunch of bills when you wanna buy something), faster to take the bills for higher value items and easier to count as well.
I wouldn’t. Most places refuse to take $100s due to rampant counterfeiting, and banks don’t bat an eye at a huge stack of ones as a deposit. To just flow through life, a limitless supply of ones is far easier to deal with than any amount of crisp $100 bills. Inflation might change this, but probably not in my lifetime.
Most places where you live. This isn’t a problem for the entire globe. Some of it, sure. But not all of it. I pay with hundreds all the time in Australia and noone gives a shit.
Tbh I think this is correct. Not necessarily mathematically, idk maths, but realistically paying with infinite 100$ Bills is easier than with 1$ Bills. Therefore it saves time and so the .infinite 100$ Bills are worth more
I’d argue that infinite 1 bills are worth less than infinite 100 bills. Because infinite 100 is infinite 1 times infinite 100. Even though they effectively turn into the same amount that is infinity.
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