I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
Neither is bigger. Even “∞ x ∞” is not bigger than “∞”. Classical mathematics sort of break down in the realm of infinity.
It was probably mentioned in other comments, but some infinities are “larger” than others. But yes, the product of the two with the same cardinal number will have the same
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”.
Yeah, inverting it is a good idea, truly
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.
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What about lemons?
Mathematically speaking, they should be converted to lemonade.
Screw that! I’m the man who’s gonna burn your house down! With the lemons! I’m gonna get my engineers to invent a combustible lemon that burns your house down!
Depends on if there’s any lemon stealing whores around.
This problem doesn’t involve cardinal numbers.
So it’s basically just a form of NaN?
It’s (literally) +Inf
You’re the guy in the middle by the way.
This meme was made by thr guy on the left thinking he was thr guy on the right.
An infinite number of bills would mean there’s no space to move or breathe in, right? We’d all suffocate or be crushed under the pressure?
Depends on implementation.
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!
I guess you’d need infinite space for an infinite number of bills. But it’d still be full to the brim?
We’d all suffocate or be crushed under the pressure?
hey just like regular capitalism
Really depends on whether the infinite store will break hundreds for you.
nah aah, infinity plus 1 is more, I win
Infinity plus infinity … Na na NA NA na!!! I win! Na na 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
nonono, Infinity times infinity plus one, I win! *tehee nanana Na na na! *
The people struggling with this are the same ones that think a ton of lead is heavier than a ton of feathers
You can have infinities of different sizes
You can. This is not one of them
True, but understanding that different sizes of infinity exist and applying that incorrectly is not the same as not realizing that “a ton is a ton”.
It’s a fair analogy to obscure some complexity.
Got a better one?
I’d rather have 100 dollar bills rather than 1 dollar bills.
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.
There is a lot of very confident, opposing answers here…
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.
The point of the meme format is not that the middle guy is right. It’s the opposite.
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:
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[wrong] Frankenstein is the monster.
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[average] No, Frankenstein is the doctor.
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[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.
Heres my example
- [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 think you can make an argument either way of whether it would be better to have $1 or $100 bills.
I was more interested in debating the meme format XD
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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If I had infinity $100 notes I could ask to break them into 50s and have 2x infinity $50 notes. It’s called winning.
False. There would still be a finite supply of $50 bills, so you could never have infinity $50 bills at any time.
I forgot to mention they’re forgeries.
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.
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?
once the two sets have reached their value
will weigh 100 times less
there should be the same number bills in the sets
The short answer is that none of these statements apply the way you think to infinite sets.
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 see what you did here…
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?
Is it possible for infinite numbers to be larger than others? Or are all infinite numbers equal?
There are different things which could be called “infinite numbers.” The one discussed in the other reply is “cardinal numbers” or “cardinalities,” which are “the sizes of sets.” This is the one that’s typically meant when it’s claimed that “some infinities are bigger than others,” because e.g. the set of natural numbers is smaller (in the sense of cardinality) than the set of real numbers.
Ordinal numbers are another. Whereas cardinals extend the notion of “how many” to the infinite scale, ordinals extend the notion of “sequence.” Just like a natural number always has a successor, an ordinal does too. We bridge the gap to infinity by defining an ordinal as e.g. “the set of ordinals preceding it.” So {} is the first one, called 0, and {{}} is the next one (1), and so on. The set of all finite ordinals (natural numbers) {{}, {{}}, …} = {0, 1, 2, 3, …} is an ordinal too, the first infinite one, called omega. And now clearly {omega} = omega + 1 is next.
Hyperreal numbers extend the real numbers rather than just the naturals, and their definition is a little more contrived. You can think of it as “the real numbers plus an infinite number omega,” with reasonable definitions for addition and multiplication and such, so that e.g. 1/omega is an infinitesimal (greater than zero but smaller than any positive real number). In this context, omega + 1 or 2 * omega are greater than omega.
Surreal numbers are yet another, extending both the real and hyperreal numbers (so by default the answer is “yes” here too).
The extended real numbers are just “the real numbers plus two formal symbols, “infinity” and “negative infinity”.” This lacks the rich algebraic structure of the hyperreals, but can be used to simplify expressions involving limits of real numbers. For example, in the extended reals, “infinity plus one is infinity” is a shorthand for the fact that “if a_n is a series approaching infinity as n -> infinity, then (a_n + 1) approaches infinity as n -> infinity.” In this context, there are no “different kinds of infinity.”
The list goes on, but generally, yes-- most things that are reasonably called “infinite numbers” have a concept of “larger infinities.”
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.
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.
OK thanks for your explanation.
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.
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.
For what it’s worth, people actually taking the time to explain helped me see the error in my reasoning.
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.
It’s the armchair experts in the thread trying to “umm actually…” the memer when their “experience” is a YouTube video at best.
And don’t you worry, that YouTuber with sketchy credibility and high production values has got an exclusive course just for you! Ugh. Lol
As a math PhD you should come to the conclusion that both are enumerable and this is why they are the same.
Y=x
And
Y=e^x
With lim x->(I don’t know how to type it here)
are not the same but both unbounded.
So to paraphrase, the raging person in the middle is right? I’ll take your answer no questions asked.
In short, yes.
Value is a weird concept. Even if mathematically the two stacks should have the same value, odds are some people will consider the $100 bill stack worth more, and be willing to do more in exchange for it. That effectively does make it worth more.
An infinite stack of either would devalue the currency so as to be completely worthless. Well, perhaps worth whatever you can recycle those bills into.
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.
If you manage to keep an infinitely large stack of bills a secret, sure. Once somebody notices and word gets out, I’m doubtful it doesn’t get devalued in a hurry. Since these are bills that we are assuming are valid, it’s going to seem like the central bank is printing money with abandon. Famously not great for public confidence in a currency. Why would I keep my wealth in a currency that somebody has an infinite amount of? They may not be spending it today, but who knows when that changes? I’d certainly be scrambling to convert mine to something else.
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.
So, the value is inversely correlated to the ability to spend it.
… Still sounds worthless, TBH.
The moment you bring in the concept of actually using this money to pay for things, you have to consider stuff like how easy it is to carry around, and the 100s win. If your pile is infinite then you don’t even need 1s at the strip club.
No matter which denomination you choose, the infinite motel will always have room for another bill.
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.
This is especially true if you live in a place where the 1 dollar option would mean infinite coins rather than bank notes. :P
But then you can imagine you found a pirate’s chest with all the coins inside 🤩
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
Might just be me.
You’ve solved inflation!
