Wouldn’t an infinite number of anything with physical mass collapse the universe as we know it and challenge our models of physics? But yeah sign me up for the Benjamin’s.
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
This is wrong. Having an infinite amount of something is like dividing by zero - you can’t. What you can have is something approach an infinite amount, and when it does, you can compare the rate of approach to infinity, which is what matters.
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 was just over here thinking this was about the practical utility of a $100 bill versus a wad of 100 $1 bills making an infinite quantity of the former preferable in comparison to (i.e. “worth more than”) the latter…
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.”
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”.
This kind of thread is why I duck out of casual maths discussions as a maths PhD.
The two sets have the same value, that is the value of both sets is unbounded. The set of 100s approaches that value 100 times quicker than the set of singles. Completely intuitive to someone who’s taken first year undergraduate logic and calculus courses. Completely unintuitive to the lay person, and 100 lay people can come up with 100 different wrong conclusions based on incorrect rationalisations of the statement.
I’ve made an effort to just not participate since back when people were arguing Rick and Morty infinite universe bollocks. “Infinite universes means there are an infinite number of identical universes” really boils my blood.
It’s why I just say “algebra” when asked what I do. Even explaining my research (representation theory) to a tangentially related person, like a mathematical physicist, just ends in tedious non-discussion based on an incorrect framing of my work through their lens of understanding.
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.
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.
Yeah I sell cabinets and sometimes people are like “How much would a 24 inch cabinet cost?”
It could cost anything!
Then there are customers like “It’s the same if I just order them online right?” and I say “I wouldn’t recommend it. There’s a lot of little details to figure out and our systems can be error probe anyway…” then a month later I’m dealing with an angry customer who ordered their stuff online and is now mad at me for stuff going wrong.
The two sets have the same value, that is the value of both sets is unbounded. The set of 100s approaches that value 100 times quicker than the set of singles.
Hey. Sorry, I’m not at all a mathematician, so this is fascinating to me. Doesn’t this mean that, once the two sets have reached their value, the set of 100 dolar bills will weigh 100 times less (since both bills weigh the same, and there are 100 times fewer of one set than the other)?
If so, how does it reconcile with the fact that there should be the same number bills in the sets, therefore the same weight?
How would you even weigh an infinite number of dollar bills? You’d need an infinite number of assistants to load the dollar bills onto the scale for you, and months to actually do it!
I like this comment. It reads like a mathematician making a fun troll based on comparing rates of convergence (well, divergence considering the sets are unbounded). If you’re not a mathematician, it’s actually a really insightful comment.
So the value of the two sets isn’t some inherent characteristic of the two sets. It is a function which we apply to the sets. Both sets are a collection of bills. To the set of singles we assign one value function: “let the value of this set be $1 times the number of bills in this set.” To the set of hundreds we assign a second value function: “let the value of this set be $100 times the number of bills in this set.”
Now, if we compare the value restricted to two finite subsets (set within a set) of the same size, the subset of hundreds is valued at 100 times the subset of singles.
Comparing the infinite set of bills with the infinite set of 100s, there is no such difference in values. Since the two sets have unbounded size (i.e. if we pick any number N no matter how large, the size of these sets is larger) then naturally, any positive value function applied to these sets yields an unbounded number, no mater how large the value function is on the hundreds “I decide by fiat that a hundred dollar bill is worth $1million” and how small the value function is on the singles “I decide by fiat that a single is worth one millionth of a cent.”
In overly simplified (and only slightly wrong) terms, it’s because the sizes of the sets are so incalculably large compared to any positive value function, that these numbers just get absorbed by the larger number without perceivably changing anything.
The weight question is actually really good. You’ve essentially stumbled upon a comparison tool which is comparing the rates of convergence. As I said previously, comparing the value of two finite subsets of bills of the same size, we see that the value of the subset of hundreds is 100 times that of the subset of singles. This is a repeatable phenomenon no matter what size of finite set we choose. By making a long list of set sizes and values “one single is worth $1, 2 singles are worth $2,…” we can define a series which we can actually use for comparison reasons. Note that the next term in the series of hundreds always increases at a rate of 100 times that of the series of singles. Using analysis techniques, we conclude that the set of hundreds is approaching its (unbounded) limit at 100 times the rate of the singles.
The reason we cannot make such comparisons for the unbounded sets is that they’re unbounded. What is the weight of an unbounded number of hundreds? What is the weight of an unbounded number of collections of 100x singles?
Yes, there are infinities of larger magnitude. It’s not a simple intuitive comparison though. One might think “well there are twice as many whole numbers as even whole numbers, so the set of whole numbers is larger.” In fact they are the same size.
Two most commonly used in mathematics are countably infinite and uncountably infinite. A set is countably infinite if we can establish a one to one correspondence between the set of natural numbers (counting numbers) and that set. Examples are all whole numbers (divide by 2 if the natural number is even, add 1, divide by 2, and multiply by -1 if it’s odd) and rational numbers (this is more involved, basically you can get 2 copies of the natural numbers, associate each pair (a,b) to a rational number a/b then draw a snaking line through all the numbers to establish a correspondence with the natural numbers).
Uncountably infinite sets are just that, uncountable. It’s impossible to devise a logical and consistent way of saying “this is the first number in the set, this is the second,…) and somehow counting every single number in the set. The main example that someone would know is the real numbers, which contain all rational numbers and all irrational numbers including numbers such as e, π, Φ etc. which are not rational numbers but can either be described as solutions to rational algebraic equations (“what are the solutions to “x^2 - 2 = 0”) or as the limits of rational sequences.
Interestingly, the rational numbers are a dense subset within the real numbers. There’s some mathsy mumbo jumbo behind this statement, but a simplistic (and insufficient) argument is: pick 2 real numbers, then there exists a rational number between those two numbers. Still, despite the fact that the rationals are infinite, and dense within the reals, if it was possible to somehow place all the real numbers on a huge dartboard where every molecule of the dartboard is a number, then throwing a dart there is a 0% chance to hit a rational number and a 100% chance to hit an irrational number. This relies on more sophisticated maths techniques for measuring sets, but essentially the rationals are like a layer of inconsequential dust covering the real line.
If we only consider the monetary value, both “briefs” have the same value. Otherwise if we incorporate utility theory with a concave bounded utility curve over the monetary value and factor in other terms such as ease of payments, or weight (of the drawn money) then the “worth” of the 100 dollar bills brief could be greater for some people. For me, the 1 dollar bills brief has more value since I’m considering a potential tax evasion prosecution. It would be very suspicious if I go around paying everything with 100 dollar bills, whereas there’s a limit on my daily spending with the other brief (how many dollars I can count out of the brief and then handle to the other person).
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.
Correct me if I’m wrong, but isn’t it that a simple statement(this is more worth than the other) can’t be done, since it isn’t stated how big the infinities are(as example if the 1$ infinity is 100 times bigger they are worth the same).
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.
The reason infinity $100 bills is more valuable than infinity $1 bills: it takes less effort to utilize the money.
Let’s say you want to buy a $275,000 Lamborghini. With $1 bills, you have to transport 275,000 notes to pay for it. That will take time and energy. With $100 bills, you have to transport 2750 notes. That’s 100x fewer, resulting in a more valuable use of time and energy.
Even if you had a magical wallet that weighed the same as a standard wallet and always a had bills of that type available to pull out when you reach in. It’s less energy to reach in a fewer number of times.
Let’s toss in the perspective of the person receiving the money, too. Wouldn’t you rather deal with 2750 notes over 275,000, if it meant the same monetary value? If you keep paying in ones, people will get annoyed. Being seen favorably has value.
You couldn’t buy that car with either denomination since money would have zero value now… and also the universe would collapse in on itself… but mainly the zero value thing.
Fair, but there’s also a lot of businesses that don’t accept $100 bills which would make paying for smaller everyday things annoying, and realistically I don’t think any car dealership would want to deal with 2,750 $100 bills either. Besides, with infinite money you could hire people to count and move the money for you, if it’s $1 bills instead of $100 bills you simply hire 100 times as many people!
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.
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
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