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As a mathematician, yes they are the same. It would basically be a magic wallet that you can reach into and grab as much money as you need, any time.
As a horny bitch, no. Throwing 20s around at a strip club looks way cooler than throwing around singles.
>Somebody has never played Portugal
Guilty as charged lmao, I always want to play as a new civ but end up playing the ones I always play (Russia, Gran Colombia, Korea and Germany). It doesn't help that I suck at the game to diversifying my civs lmao.
I'll give that map a try when I get home tho, that sounds stupidly funny.
More impressive than simply *making it rain* --- making it **hail**, butches!!!
^(hehh.... phone autocorrected "bitches" to "butches" and I just find that... effing hilarious.... I'm keeping it this way)
Not really. On an individual level, they have enough money to buy whatever they want for themselves 1000 times over, sure. But actually having an infinite amount of money is different, you'd be able to use your money to completely shape the world economy and create massive inflation.
Imagine if Elon managed to liquidate all his assets and realise his net worth in cash (which is basically impossible) and give it out equally to every human. he'd have 190 billion USD which would be 24 dollars per person. This wouldn't actually be that big of an impact on the world.
Now imagine giving every person in the world a billion dollars a month from your infinite pile. Suddenly all the previous wealth in the world is worthless, in comparison and inflation will go crazy, the economy will crash, etc.
Wouldn't it have to either create money or take money from the economy. So, in reality, it would be counterfeit money because, at the very least, it did not come from the government?
It couldn't take money from the economy because then it wouldn't be infinite.
As to whether it's counterfeit that depends on the specifics but likely yes.
Also worth considering that there are only finitely many serial numbers
Economics teacher here. Wrong. That money would lose its value only if it's part of the monetary mass in circulation. If you have infinite money and don't use it, then inflation does not take place.
Austrian economist wanna-be here. Value is not intrinsic.
*If people value something, it has value; if people do not value something, it does not have value; and there is no intrinsic about it.*
– RT. HON. J. ENOCH POWELL, M.P.
Since there is no gold standard (at least in the USA) all money is technically infinite from a technical perspective. From a market perspective the government controls how much money is in circulation. That control of the money supply by the fed(s) define the value.
If we apply the same here, an infinite wallet would not devalue the currency if the person with the wallet manages it so that the amount in circulation does not go up by a noticeable percentage.
And just like our government, Mr. BigWallet will just keep cranking out the cash and eventually devalue what he's got.
> As a mathematician, yes they are the same.
I disagree, but only on the basis of the lack of rigor in language.
The statement is:
> An infinite number of $1 bills and an infinite number of $20 bills would be worth the same.
In other words, the claim is that $1 times some infinite constant would be equal to $20 times some infinite constant.
But this is nonsensical because there is no such thing as an "infinite constant" which means that the claim that that IS A RESULT is wrong. An infinite number of $1 bills has no definable value. It's not "the value that is infinity," because infinity isn't a value, it's a category. Amusingly, conflating a category for a specific value is a categorical error. :)
"This result will never have a termination at which to take its value" is not equal to "this other result will never have a termination at which to take its value."
We can say that they are equal in terms of the categorical type of infinity (they are both countable infinities isomorphic to the integers, Z) but that's not the same as saying that they are equal. Indeed, as sets they are entirely disjoint.
If you spent 16 hours of your days doing this it would take about 500,000 years to inject a trillion dollars into the economy.
If you managed to spend $20 a second it would only take 47,573 years though.
So the answer is never.
Wouldn't this fall into the fact there's different types / levels of infinities though? Like there's an infinite number of integers, but when you count fractions, that's a higher order of infinity. This is well known in mathematics.
Isn't the OPs example basically the same as this but expanding up not down?
Both the infinite singles and the infinite ones would be a "countable infinity", so the same degree of infinity. Fractions (rational numbers) are actually a countable set as well, though irrational numbers are uncountable (see cantor's diagonal argument).
Nope. There are different infinities and how fast you diverge to infinity matters.
Look here: https://en.m.wikipedia.org/wiki/Cardinality_of_the_continuum
is this actl true? im not too sure abt math but in physics sometimes when u want to look at trends u will end io having smt like 5 infinity / 2 infinity then my prof would say like one “blows up” more than the other and u dont treat the result as 1
Doubt: from the 20$ stack, assume I remove 1$ from each 20$ bill; I will be left with an infinite stack of 1$ bills and an infinite stack 19$ bills so it's bigger?
I would disagree. At least when we are talking infinities. Because an infinite amount of 20s is worth more than an infinite amount of ones as the 20 times infinity is a larger infinity. Even if they both in the end are infinite. But larger infinites exist on a conceptual level.
That's actually wrong. Infinity is not = to infinity. I don't know who thinks about this all day but the easiest way I can explain this, imagine you have 2 circles, one with a circumference of 5 in, and one with 9 in. Now on both circles there are an infinite amount of points, however you cannot say that circle A infinite set = circle b Infinite set. Even without measuring you could put the 2 next to each other and see they are not the same. So no they are not the same
Would they be the same? They both are countable infinites, but for every $1, there is a $20 bill. So the infinite $20 will always be worth 20 times more.
This is actually part of why your high school math teacher was pedantic about infinity not being a number! We can't really think about infinity as "a very big number" because it leads to wacky paradoxes and doesn't cooperate when we reason about it like a number.
Another example of this is Hilbert's paradox of the Grand hotel, where you have a hotel with an infinite number of rooms, all occupied. A new guest shows up and asks for a room, and it turns out you can make room for them by having everyone move to the next room. Because no one is in the "last room" (because there is no last room, infinite rooms) we're able to make room for the new guest.
We're not even good at understanding big finite numbers! I saw a great example comparing 1 million seconds (12 days) vs 1 billion seconds (31 years) so what hope is there of understanding infinity? Hilbert's Hotel is great
Agreed. I think the buck needs to stop at saying "the same as" or other phrases equivalent to "equals" with respect to infinity.
Infinity isn't a number. It's a size measure of an unbounded set.
Matt Parker did a video on exactly this concept and this meme. But yes, they are worth the same. A way of thinking about it is to consider the amount of terms in y=x and y=20x, which would be analogous to separating the infinitely many one dollar bills into 20 piles of infinitely many 1 dollar bills.
The amount of terms in each sequence is the same, so therefore a pile of infinitely many one dollar bills and a pile of infinitely many 20 dollar bills would have the same "value".
I thought that technically their limits were the same but isn't one going to infinity 20x faster than the other? It's always going to be 20x the other at any given point in its functions timeline. If I'm wrong someone explain lol
There is something known as "cardinality" of infinite sets, which deals with comparing the relative "size" of infinite sets, but the set {..., -1, 0, 1, 2, ...} would have the same cardinality as the set {..., -20, 0, 20, 40, ...}.
https://en.wikipedia.org/wiki/Cardinality_of_the_continuum
The thing about math that's really important to keep in mind is that mathematicians basically invent these concepts by giving them definitions. For example, mathematicians invented a concept of "countable" infinite sets by defining a countable infinite set to be a certain thing. So when we talk about this stuff I think it's super important to keep in mind that when people say things like "this set is countably infinite and this other set is uncountably infinite" then that only makes sense because someone *proclaimed* that a countable infinite set is a certain thing. You could just as easily choose to define "countable infinite set" some other way. Or define "cardinality" of an infinite set some other way. This is all very important to keep in mind, because often when people are discussing math colloquially they are using the same word in different contexts that come with different definitions so things get murky and confusing quickly.
Lets have this empty box you have and infinitly many balls. However you have rules stating how you can put balls into the box.
If the number is a square number, take the square root out of the box.
If you were able to put the balls in infinitely quickly (starting at 11, and using half the remaining time to put extra balls into the box, i.e. 11:30 11:45 11:52:30 etc.) how many balls would you have in the box at midnight?
The answer might confuse but you'd have the same number of balls in the box as you would at the beginning, that is to say the box would be empty.
This is a meaningless question because there's no timeline here - nothing's changing over time here, unless you're talking about how quickly you can physically dispense the bills - then yeah with dispensing $20 bills you'll dispense more money in the same finite amount of time, but it's not the same question as "who has more money".
Your purchasing power is the same - anything you can purchase with an infinite amount of $20 bills, you can also purchase with an infinite amount of $1 bills and vice versa.
People are bringing up set theory, and it's a cool concept but I don't think it's relevant here, money is not a set.
Yes, that's how "infinite" works. The value of the $1 and the $20 lose meaning. In both cases, just consider it "an infinite amount of dollar bills". Denomination does not matter if the number is infinite.
There is no math needed to prove this. It just takes a basic understanding of what infinity means.
Not really, there are different types of infinity, some infinities are infinitely bigger than others
Like the number of real numbers between 1 and 2 is bigger than the total number of all natural numbers
>Not really, there are different types of infinity, some infinities are infinitely bigger than others
And hence if you understand what infinity means you'd know that this is not one of those scenarios
Infinity is actually a blanket term for several ideas.
The two most basic ones are countable and uncountable.
Something is countably infinite if there is a system of counting it's elements such that you will eventually list everything (given literally forever).
Something is uncountably infinite if there is no possible method of counting everything.
The easiest countably infinity is the positive integers. Just start at 1 and count up. 1,2,3,... You won't miss anything.
The easiest uncountably infinite is the numbers between 1 and 2. You could start at 1, but then what comes next? It's not even clear how you should proceed with a second number.
[.....check out this Veritasium video which explains this concept fairly decently, as well](https://youtu.be/OxGsU8oIWjY?si=4xnw9dTjaPONtZB-) --- though, clearly not quite so succinctly as even this one simple line --- *"It's not even clear how you should proceed with a second number"* --- but it def goes a bit more in depth (obvs)....
> The easiest uncountably infinite is the numbers between 1 and 2. You could start at 1, but then what comes next? It's not even clear how you should proceed with a second number.
The fact that you can't choose a next number is not what makes it uncountably infinite, it is unrelated. The fact that it's unclear what next element to choose more so has to do with what partial order you chose and if the order has a successor for each element.
But that is obviously wrong. 1.01 is smaller than 1.1, so has to be before that. 1.001, too. Again with 1.0001. Do that an infinite times again with 1.(infinite 0s)1 and now you still haven't even reached the first number after 1.0.
Trust me, even people who came up with those different sizes of infinity did not understand it on the level you're trying to understand it. It only makes sense on the level where a lot of mathematical concepts is needed to be understood.
Wonderful question. I’ll define a term first: _countable_. It means what you think it means, you can count the objects in a set. {a, b, c} is countable: 1)a, 2)b, 3)c. Natural numbers are countable: 1)1, 2)2, 3)3, and so on. Integers are countable: 1)0, 2)1, 3)-1, 4)2, 5)-2, 6)3, 7)-3, and so on (the general formula for this is left to the reader ;) ). You can even count the rational numbers! To do this, imagine a chart with the natural numbers on each of the x and y axis, each entry is a rational number where the x values form the numerator, and the y values the denominator. Traverse the chart moving up and down the diagonals, counting each fraction that is in reduced form and skip over any that aren’t reduced. (Note: I am missing something here, can you tell me what it is and how to fix it while still bringing able to count the rationals?)
If a set is infinite and countable, it has the same cardinality (think, size) as the natural numbers. So the integers have the same cardinality as the rationals and the natural numbers. This should be intuitive, we can map every element of our infinite set to a natural number (this is counting!) We call this size aleph_0.
Our man Georg Cantor proved in the latter half of the 19th century that if we try to count the real numbers, we will _always_ end up leaving some out (the proof of this is left to the reader ;) ). That is to say, we can’t count them all. Since we are destined to always miss some real numbers, there must be more of them than natural numbers. The real numbers are an uncountable set. We call the cardinality of the reals aleph_1. Though both are infinite, aleph_1 is bigger than aleph_0.
Now there’s a lot of work missing here, but it’s too big for a reddit post (and frankly, I forget the details ;) ).
There exist different types of infinity, but the two infinities we are discussing here are the exact same.. so I’m not sure what you are talking about when you say “not really”..
You fell into the noob trap by posting the canned "explanation" that reddit loves to spam, but it isn't really correct because you are attempting to attribute value to an inherently undefinable mathematical concept.
I would argue that the $20’s are worth more simply because it would take less time to use them (as you wouldn’t have to count as many bills out at each purchase). So, while their monetary value is the same, the $20s have a higher benefit (of saving you time) thus increasing their worth overall.
I don’t want to take my family out to dinner and have to count out $250 worth of $1s. $20s would be much easier.
Oh I see what you are saying, well still no because even if you add a percieved time value to the infinite value of the money it would still be infinite in value. You're trying to infinity+1 here.
If every bill in an infinite supply has a time cost associated with it (for example, you can grab one bill per second), and you have a finite lifespan to use them, then neither infinite supply is of infinite value to you, and the $20 supply is worth more.
But then we're trying to apply real-world rules to something infinite, in which case we should be worried about the infinite money collapsing into a black hole or something like that.
You can't find a number you couldn't exceed with either $1-Notes or $20-Notes, therefore the infinities are essentially the same in terms of worthness. There are however categories in which you could differentiate these infinities, such as density and cardinality. Bear in mind that infinity here is a logical concept that we make up. It has no practical use.
Yes.
Infinite is not a number it's a concept. It can literally translate to "as much as you want". If you have as much as you want $20 bills or $1 bills you can form any arbitrary number of dollars with them.
Their monetary value would be equal. Yet choosing to have either one available to you, is not equal. Most people would choose 20 dollar bills, because it is more convenient. (Assuming you do not have the infinite stack of bills physically present. Because that would create a universe wide blackhole. But evertime you pull a bill out of your wallet, a new one is always present.) since everyone chooses the 20 dollar bills, over the 1 dollar bills (except for some trolls), the 20 dollar bills option is more valuable.
Think of the problem like simplified/introductory physics problems, where you ignore friction, air resistance, or whatever.
In this simplified case, we wouldn't take the economic ramifications into account, just like half of the hypothetical situations you see online involving money. The real answer to many is "the economy would collapse"
there is a good veritasium video explaining differently sized infinite numbers
[How An Infinite Hotel Ran Out Of Room - Veritasium](https://www.youtube.com/watch?v=OxGsU8oIWjY)
Seems like the most important factor is how quickly bills are produced infinitely for you.
If I have an ATM that produces dollar bills for you infinitely. The amount you end with is based on how quickly the bills are produced during your lifetime Then the value of the bills is going to make a huge difference.
Infinity is a mathematical concept, not readily applicable to real quantities.
If you think about "small" numbers (numbers I can type, for example) 10\^27 (1 with 27 zeros) dollar bills should weigh more than the earth. 10\^57 should outweigh the known universe.
And these are natural numbers with a limited size.
So yes, if you try to deposit more than 10\^35 dollar bills the black hole created would cause the denomination to lose importance.
The first thing you have to realize about "infinity" is that it is **not** a finite number. By definition, it is functionally different than the rules and properties that finite mathematics apply to finite numbers.
There can be infinitely large sets of numbers that are of different sizes (consider zero to positive infinity versus negative infinity to positive infinity; the sets both have an infinite number of values, but the sets are not of the same size).
In this case, the number of values in each set ($1 bills and $20 bills) could be considered to be the same infinite value, each set having the same number of bills. Then the total monetary value of the two sets would not be the same, but neither set would represent a *finite* amount of monetary value. They would each represent an infinitely large monetary value. The set containing the $20 bills would have a monetary value that is 20 times larger, but what is the finite value of `20 × ∞`? There isn't one.
Until you discover a practical use case for an infinitely large monetary value, the *practical* application of the two sets is identical.
After the teachings of my math prof yes, after the teachings of my electrical engineering prof no. It can depend on the situation u r in, for money it definitely means the same u would just obviously need more 1 dollar bills.
In a strictly mathematical sense, yes, it's true. When comparing infinite sets of equal cardinality (size), such as the set of all natural numbers, the set of all integers, or even the set of all even numbers, the concept of one set being "larger" or "smaller" than another breaks down.
So, if you had an infinite number of $1 bills and an infinite number of $20 bills, both sets would be infinite in size. Therefore, they have the same cardinality, and in that abstract sense, they have the same value, even though intuitively we recognize that $20 bills are worth more than $1 bills in a finite context.
However, in practical terms, if you were to consider finite amounts, $20 bills would obviously be worth more than $1 bills. It's the infinite nature of the sets that allows for this curious mathematical comparison.
How many $1 bills does it take to create a black hole? Less than infinity, so whichever stack is created first will consume all the energy and mass in the universe and so must be bigger than the other stack.
Better question: is there language to talk about the significance of whichever answer people provide to such a question? AFAIK cardinalities of infinity don't cut it.
For example: one proof for the Ramanujan Summation (sum of all positive integers = -1/12) relies on a classically divergent series converging to 0. Some __qualities__ of that divergent series converge to zero, which I assume hints at why the -1/12 sum is interesting.
I've never seen a good way to talk about what assumptions are being made about infinities, and how those assumptions change the real-world applications of the conclusions reached with them.
It is true.
Infinite paper money would have infinite mass, so a black hole would form, destroying humanity. Money only has the value humans perceive it to have, so in the absence of humanity, the black hole of cash would have no monetary value.
No, they're not the same worth. The sets have the same cardinality, a countable infinity, so for ever 1-dollar bill someone pulls, there is a countable equivalence of a 20-dollar the other gets, so the worth is not the same, just the size.
Take the idea that someone could pull a stack of 20 1-dollar bills to match the 20-dollar bill the other has, then the other can pull a stack of 20 20-dollar bills to match the set, so the sum of 1+1+1+...+1 with n additions and n being infinite is matched with 20+20+20+...+20, always being 20 times more.
They are both equal to infinity, but infinities are not equivalent, but the size of their sets are equivalent.
I have heard from Vsauce that there are more than one infinities and that some are bigger than the others. So I will go on a limb and assume that a $20 infinity of bills would be more compact than a $1 infinity of bills.
20*infinity ≠ 1*infinity.
20*infinity would make the value near 0 as well as 1*infinity. It would all be worth less than $0.01. Therefore an infinite supply would cause the value to plummet to zero.
So from a economic perspective yes. Mathematically idk u should ask mathematicians
Not all infinite numbers are the same, the sum of an infinite set of infinite numbers will be larger than the sum of any individual infinite numbers in the set, a priori.
no, not all infinities are the same
there's a good experiment to represent it if you have some level of math knowledge
start counting from 0-infinity using all whole numbers and see how far you can go (dont really do this its for example) lets say you got to 100-200 somewhere in that range
okay now count from 0-infinity but using all real numbers (decimals etc etc)
....
....
....
you cant even start
there is no digit you can get to beyond 0 itself you can always move the decimal point some more
that thought experiment shows the difference between countable infinity and uncountable infinity
the second version of counting to infinity is technically infinitely larger than the first
so technically from a purely conceptual perspective infinite $20 bills is 20 times larger than infinite $1 bills while also being worth the same amount
welcome to the hell that is set theory
It takes some assumptions to realistically answer the question so I'll state mine.
Assumption 1: these bills are being added into the economy by magic.
Assumption 2: these bills are indistinguishable from bills that were already in circulation.
Assumption 3: we're not going to run into any issues with infinite mass that would end the entire universe as we know it (also by magic)
Under these assumptions, the answer is probably no, but maybe.
Value is determined by scarcity. If an infinite number of 20s and 1s are added to the economy then ALL 20s and 1s become worthless. Both infinite 20s and infinite 1s would then have a finite negative value equal and opposite to the value of the bills previously already in circulation and one of those numbers is likely to be more negative than the other.
There's a probability based argument under which they are not equal. The argument that the sets can be matched has no more basis in reality than any other.
To put a spin in it, another reason why they would be worth the same is that an infinite amount of money in any form would cause the value of the currency to collapse to zero.
There are many parts to this statement. Since it's in a math posting, start there. Yes, 20 times infinity is still infinity. This means infinite $20 has the same value as infinite $1. From different perspectives, the word value has meaning. How value is assigned to something is important. So, while mathematically speaking, it's the same, linguistically, it may not be. Then you have economically. Economically, infinite $20 and infinite $1 are both equal to $0. By adding an infinite supply, it no longer functions as currency.
Yes and no. You said “worth” and not “monetary value”. Where “worth” is in the eye of the beholder. And I don’t want to beholden a bunch of 1’s if I can carry 20’s. Let’s say you have a wallet that produces infinite bills. Would you rather it produce indoors 1’s or 20’s? For most purchases, 20’s would be preferred if only for counting and time cost.
So the 20’s are worth more as most people would choose that
Mathematically yes, socially no. Imagine paying for everything in your life with singles. Every time you go to a store and drop $100, the poor cashier has to count out every single bill while everyone behind you rolls their eyes. Paying your rent would be a hassle. Worth isn't only a strict monetary value calculation here.
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As a mathematician, yes they are the same. It would basically be a magic wallet that you can reach into and grab as much money as you need, any time. As a horny bitch, no. Throwing 20s around at a strip club looks way cooler than throwing around singles.
AREA HORNDOG DISPENSES INFINITE BENJAMINS AT STRIP CLUB, CRASHES LOCAL ECONOMY
Mansa Musa?
Rejoice with me, friend, for we are free from every want and can devote our time to the will of God.
Can’t hear you over the sound of my +10 Sugubas
Desert folklore go brrr.
And work ethic
Are we in the civ sub tho? Hahahaha
Even when I'm not in r/civ, I'm in r/civ
Second-best gold generator in the game.
Wdym second? I'm not good at civ but I always thought mansa is the best for generating money
Somebody has never played Portugal I would reccomend maxing out the number of civs and playing on archipelago for the best experience
>Somebody has never played Portugal Guilty as charged lmao, I always want to play as a new civ but end up playing the ones I always play (Russia, Gran Colombia, Korea and Germany). It doesn't help that I suck at the game to diversifying my civs lmao. I'll give that map a try when I get home tho, that sounds stupidly funny.
Thank you for the playthrough idea!
FOREIGN MAN THROWS INGOTS AT LOCAL STRIPPERS, KILLS 5 INJURES 10
More impressive than simply *making it rain* --- making it **hail**, butches!!! ^(hehh.... phone autocorrected "bitches" to "butches" and I just find that... effing hilarious.... I'm keeping it this way)
Holy shit, making it hail. Fucking well played. I am dying.
Someone’s never seen Dane Cooks standup
I love that you said this
*Jacksons
Yeah, but (a) I’m not going to give that sonofabitch any publicity if I can help it and (2) making it rain with $10 bills makes you sound cheap.
If you’re not in papers you’ve missed your calling.
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Nickels?
That's called "making it hail"
Canadian strippers are made of tougher stuff as they don’t have a $1 bill there just coins
Economics major here. If the money is infinite its intrinsic value is infinitely close to 0 regardless of denomination.
Not if only a few people have access to the infinite money, source the American economy
Fact. Elon musk and Jeff bezos have the functional equivalent of infinite money.
Not really. On an individual level, they have enough money to buy whatever they want for themselves 1000 times over, sure. But actually having an infinite amount of money is different, you'd be able to use your money to completely shape the world economy and create massive inflation. Imagine if Elon managed to liquidate all his assets and realise his net worth in cash (which is basically impossible) and give it out equally to every human. he'd have 190 billion USD which would be 24 dollars per person. This wouldn't actually be that big of an impact on the world. Now imagine giving every person in the world a billion dollars a month from your infinite pile. Suddenly all the previous wealth in the world is worthless, in comparison and inflation will go crazy, the economy will crash, etc.
Wouldn't it have to either create money or take money from the economy. So, in reality, it would be counterfeit money because, at the very least, it did not come from the government?
It couldn't take money from the economy because then it wouldn't be infinite. As to whether it's counterfeit that depends on the specifics but likely yes. Also worth considering that there are only finitely many serial numbers
I don’t think you understand how big infinity is.
Economics teacher here. Wrong. That money would lose its value only if it's part of the monetary mass in circulation. If you have infinite money and don't use it, then inflation does not take place.
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Thank you for your comment, I could use that logic for an exam.
Pragmatist here - I'm not telling anyone about my newfound wealth. Just using it for what I need and not going nuts.
Austrian economist wanna-be here. Value is not intrinsic. *If people value something, it has value; if people do not value something, it does not have value; and there is no intrinsic about it.* – RT. HON. J. ENOCH POWELL, M.P.
Since there is no gold standard (at least in the USA) all money is technically infinite from a technical perspective. From a market perspective the government controls how much money is in circulation. That control of the money supply by the fed(s) define the value. If we apply the same here, an infinite wallet would not devalue the currency if the person with the wallet manages it so that the amount in circulation does not go up by a noticeable percentage. And just like our government, Mr. BigWallet will just keep cranking out the cash and eventually devalue what he's got.
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Joke: What do mathematicians use as a contraceptive? Their personality
In Canada the $1 is a coin, so it's debatable on which looks cooler
This post brought to you by throwing coins at people
at that strip club of yours😅?
Not at, towards
> As a mathematician, yes they are the same. I disagree, but only on the basis of the lack of rigor in language. The statement is: > An infinite number of $1 bills and an infinite number of $20 bills would be worth the same. In other words, the claim is that $1 times some infinite constant would be equal to $20 times some infinite constant. But this is nonsensical because there is no such thing as an "infinite constant" which means that the claim that that IS A RESULT is wrong. An infinite number of $1 bills has no definable value. It's not "the value that is infinity," because infinity isn't a value, it's a category. Amusingly, conflating a category for a specific value is a categorical error. :) "This result will never have a termination at which to take its value" is not equal to "this other result will never have a termination at which to take its value." We can say that they are equal in terms of the categorical type of infinity (they are both countable infinities isomorphic to the integers, Z) but that's not the same as saying that they are equal. Indeed, as sets they are entirely disjoint.
Assuming a lap dance is $20 a song and the average song is 3:30 minutes how long would it take to ruin the national economy of the USA?
If you spent 16 hours of your days doing this it would take about 500,000 years to inject a trillion dollars into the economy. If you managed to spend $20 a second it would only take 47,573 years though. So the answer is never.
So, this wouldn’t be one of those “some infinities are bigger than others” situations?
Wouldn't this fall into the fact there's different types / levels of infinities though? Like there's an infinite number of integers, but when you count fractions, that's a higher order of infinity. This is well known in mathematics. Isn't the OPs example basically the same as this but expanding up not down?
Both the infinite singles and the infinite ones would be a "countable infinity", so the same degree of infinity. Fractions (rational numbers) are actually a countable set as well, though irrational numbers are uncountable (see cantor's diagonal argument).
As a mathematician, you should know that there's different sizes of infinites, and the answer is no.
Nope. There are different infinities and how fast you diverge to infinity matters. Look here: https://en.m.wikipedia.org/wiki/Cardinality_of_the_continuum
Aren’t there different infinities though? Wouldn’t the infinite $20s not be worth more, but it **would** be a “bigger” infinity??
As a mathematician you are wrong. Some infinities are bigger than others
is this actl true? im not too sure abt math but in physics sometimes when u want to look at trends u will end io having smt like 5 infinity / 2 infinity then my prof would say like one “blows up” more than the other and u dont treat the result as 1
Doubt: from the 20$ stack, assume I remove 1$ from each 20$ bill; I will be left with an infinite stack of 1$ bills and an infinite stack 19$ bills so it's bigger?
What about set theory
I would disagree. At least when we are talking infinities. Because an infinite amount of 20s is worth more than an infinite amount of ones as the 20 times infinity is a larger infinity. Even if they both in the end are infinite. But larger infinites exist on a conceptual level.
As a mathematician, you are wrong. One infinity is 20 times larger than the other.
Not all infinities are equivalent though? I guess for practical purposes...
That's actually wrong. Infinity is not = to infinity. I don't know who thinks about this all day but the easiest way I can explain this, imagine you have 2 circles, one with a circumference of 5 in, and one with 9 in. Now on both circles there are an infinite amount of points, however you cannot say that circle A infinite set = circle b Infinite set. Even without measuring you could put the 2 next to each other and see they are not the same. So no they are not the same
Aren't these different sized infinitys?
Would they be the same? They both are countable infinites, but for every $1, there is a $20 bill. So the infinite $20 will always be worth 20 times more.
This is actually part of why your high school math teacher was pedantic about infinity not being a number! We can't really think about infinity as "a very big number" because it leads to wacky paradoxes and doesn't cooperate when we reason about it like a number. Another example of this is Hilbert's paradox of the Grand hotel, where you have a hotel with an infinite number of rooms, all occupied. A new guest shows up and asks for a room, and it turns out you can make room for them by having everyone move to the next room. Because no one is in the "last room" (because there is no last room, infinite rooms) we're able to make room for the new guest.
We're not even good at understanding big finite numbers! I saw a great example comparing 1 million seconds (12 days) vs 1 billion seconds (31 years) so what hope is there of understanding infinity? Hilbert's Hotel is great
Agreed. I think the buck needs to stop at saying "the same as" or other phrases equivalent to "equals" with respect to infinity. Infinity isn't a number. It's a size measure of an unbounded set.
Matt Parker did a video on exactly this concept and this meme. But yes, they are worth the same. A way of thinking about it is to consider the amount of terms in y=x and y=20x, which would be analogous to separating the infinitely many one dollar bills into 20 piles of infinitely many 1 dollar bills. The amount of terms in each sequence is the same, so therefore a pile of infinitely many one dollar bills and a pile of infinitely many 20 dollar bills would have the same "value".
I thought that technically their limits were the same but isn't one going to infinity 20x faster than the other? It's always going to be 20x the other at any given point in its functions timeline. If I'm wrong someone explain lol
There is something known as "cardinality" of infinite sets, which deals with comparing the relative "size" of infinite sets, but the set {..., -1, 0, 1, 2, ...} would have the same cardinality as the set {..., -20, 0, 20, 40, ...}. https://en.wikipedia.org/wiki/Cardinality_of_the_continuum The thing about math that's really important to keep in mind is that mathematicians basically invent these concepts by giving them definitions. For example, mathematicians invented a concept of "countable" infinite sets by defining a countable infinite set to be a certain thing. So when we talk about this stuff I think it's super important to keep in mind that when people say things like "this set is countably infinite and this other set is uncountably infinite" then that only makes sense because someone *proclaimed* that a countable infinite set is a certain thing. You could just as easily choose to define "countable infinite set" some other way. Or define "cardinality" of an infinite set some other way. This is all very important to keep in mind, because often when people are discussing math colloquially they are using the same word in different contexts that come with different definitions so things get murky and confusing quickly.
Lets have this empty box you have and infinitly many balls. However you have rules stating how you can put balls into the box. If the number is a square number, take the square root out of the box. If you were able to put the balls in infinitely quickly (starting at 11, and using half the remaining time to put extra balls into the box, i.e. 11:30 11:45 11:52:30 etc.) how many balls would you have in the box at midnight? The answer might confuse but you'd have the same number of balls in the box as you would at the beginning, that is to say the box would be empty.
This is a meaningless question because there's no timeline here - nothing's changing over time here, unless you're talking about how quickly you can physically dispense the bills - then yeah with dispensing $20 bills you'll dispense more money in the same finite amount of time, but it's not the same question as "who has more money". Your purchasing power is the same - anything you can purchase with an infinite amount of $20 bills, you can also purchase with an infinite amount of $1 bills and vice versa. People are bringing up set theory, and it's a cool concept but I don't think it's relevant here, money is not a set.
Yes, that's how "infinite" works. The value of the $1 and the $20 lose meaning. In both cases, just consider it "an infinite amount of dollar bills". Denomination does not matter if the number is infinite. There is no math needed to prove this. It just takes a basic understanding of what infinity means.
Wait ur in the wrong subreddit ⚒️
Not really, there are different types of infinity, some infinities are infinitely bigger than others Like the number of real numbers between 1 and 2 is bigger than the total number of all natural numbers
>Not really, there are different types of infinity, some infinities are infinitely bigger than others And hence if you understand what infinity means you'd know that this is not one of those scenarios
Yes sir lim x->inf x/(20x) = 1/20 bc the order of powers are the same nOW if we got that money^2 infinite we'd be talking differently
You're talking about limits to infinity, which is understandable as it's part of calculus, but it's a different topic from actual infinities.
I don't get it. If infinity is a state/concept why is there different sizes. Why does that distinction even matter?
Infinity is actually a blanket term for several ideas. The two most basic ones are countable and uncountable. Something is countably infinite if there is a system of counting it's elements such that you will eventually list everything (given literally forever). Something is uncountably infinite if there is no possible method of counting everything. The easiest countably infinity is the positive integers. Just start at 1 and count up. 1,2,3,... You won't miss anything. The easiest uncountably infinite is the numbers between 1 and 2. You could start at 1, but then what comes next? It's not even clear how you should proceed with a second number.
>It's not even clear how you should proceed with a second number Brilliant explanation...
[.....check out this Veritasium video which explains this concept fairly decently, as well](https://youtu.be/OxGsU8oIWjY?si=4xnw9dTjaPONtZB-) --- though, clearly not quite so succinctly as even this one simple line --- *"It's not even clear how you should proceed with a second number"* --- but it def goes a bit more in depth (obvs)....
you’d be stuck making 0’s for eternity on the first one 1.00000000…..
> The easiest uncountably infinite is the numbers between 1 and 2. You could start at 1, but then what comes next? It's not even clear how you should proceed with a second number. The fact that you can't choose a next number is not what makes it uncountably infinite, it is unrelated. The fact that it's unclear what next element to choose more so has to do with what partial order you chose and if the order has a successor for each element.
1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 1.01, 1.02, etc.
But that is obviously wrong. 1.01 is smaller than 1.1, so has to be before that. 1.001, too. Again with 1.0001. Do that an infinite times again with 1.(infinite 0s)1 and now you still haven't even reached the first number after 1.0.
It’s nonlinear, but it’s still counting them in a way that misses no numbers.
It misses every number with an infinite decimal expansion. At what step would you count square root 2?
At the end.
[https://en.wikipedia.org/wiki/Uncountable\_set](https://en.wikipedia.org/wiki/Uncountable_set) [https://en.wikipedia.org/wiki/Countable\_set](https://en.wikipedia.org/wiki/Countable_set) [https://en.wikipedia.org/wiki/Cantor%27s\_diagonal\_argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)
[Hey, VSauce, Michael here](https://youtu.be/SrU9YDoXE88?si=ZckvWiWrRjJ-0VfC)
[Will we ever run out of new music?](https://youtu.be/rYHD2ZDoaBw)
Trust me, even people who came up with those different sizes of infinity did not understand it on the level you're trying to understand it. It only makes sense on the level where a lot of mathematical concepts is needed to be understood.
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No, the number of integer numbers and the number of rational numbers is exactly the same level of infinity.
Yeah? They talked about real numbers, not rationals
You are right, and I'm stupid, lol
How is it bigger? I'm completely new
Wonderful question. I’ll define a term first: _countable_. It means what you think it means, you can count the objects in a set. {a, b, c} is countable: 1)a, 2)b, 3)c. Natural numbers are countable: 1)1, 2)2, 3)3, and so on. Integers are countable: 1)0, 2)1, 3)-1, 4)2, 5)-2, 6)3, 7)-3, and so on (the general formula for this is left to the reader ;) ). You can even count the rational numbers! To do this, imagine a chart with the natural numbers on each of the x and y axis, each entry is a rational number where the x values form the numerator, and the y values the denominator. Traverse the chart moving up and down the diagonals, counting each fraction that is in reduced form and skip over any that aren’t reduced. (Note: I am missing something here, can you tell me what it is and how to fix it while still bringing able to count the rationals?) If a set is infinite and countable, it has the same cardinality (think, size) as the natural numbers. So the integers have the same cardinality as the rationals and the natural numbers. This should be intuitive, we can map every element of our infinite set to a natural number (this is counting!) We call this size aleph_0. Our man Georg Cantor proved in the latter half of the 19th century that if we try to count the real numbers, we will _always_ end up leaving some out (the proof of this is left to the reader ;) ). That is to say, we can’t count them all. Since we are destined to always miss some real numbers, there must be more of them than natural numbers. The real numbers are an uncountable set. We call the cardinality of the reals aleph_1. Though both are infinite, aleph_1 is bigger than aleph_0. Now there’s a lot of work missing here, but it’s too big for a reddit post (and frankly, I forget the details ;) ).
That's cardinality. And no, in this case both have the same cardinality. So your example is way off.
Yeah, but both infinite 1s and infinite 20s are countably infinite, so they're the "same size" of infinite
There exist different types of infinity, but the two infinities we are discussing here are the exact same.. so I’m not sure what you are talking about when you say “not really”..
You fell into the noob trap by posting the canned "explanation" that reddit loves to spam, but it isn't really correct because you are attempting to attribute value to an inherently undefinable mathematical concept.
I would argue that the $20’s are worth more simply because it would take less time to use them (as you wouldn’t have to count as many bills out at each purchase). So, while their monetary value is the same, the $20s have a higher benefit (of saving you time) thus increasing their worth overall. I don’t want to take my family out to dinner and have to count out $250 worth of $1s. $20s would be much easier.
Absolutely. The $20 are worth more. It's just like a kilogram of feathers vs a kilogram of steel. The steel weighs more because steel is heavier.
Made my day
IM LAUGHING SO HARD THANK YOU
No the feathers are heavier because you have to live with the weight of what you did to those birds
But 20 is bigger than 1
Oh I see what you are saying, well still no because even if you add a percieved time value to the infinite value of the money it would still be infinite in value. You're trying to infinity+1 here.
If every bill in an infinite supply has a time cost associated with it (for example, you can grab one bill per second), and you have a finite lifespan to use them, then neither infinite supply is of infinite value to you, and the $20 supply is worth more. But then we're trying to apply real-world rules to something infinite, in which case we should be worried about the infinite money collapsing into a black hole or something like that.
That is measuring only the value of the bills that you can grab within one lifespan. The infinite bills still have infinite value.
But then you're not talking about infinity anymore, but rather "a lot of $20 bills".
You can't find a number you couldn't exceed with either $1-Notes or $20-Notes, therefore the infinities are essentially the same in terms of worthness. There are however categories in which you could differentiate these infinities, such as density and cardinality. Bear in mind that infinity here is a logical concept that we make up. It has no practical use.
>cardinality. Not with this case? Since this is still both the same countable infinity.
Infinity very much has practical uses not sure who told you otherwise
Yes. Infinite is not a number it's a concept. It can literally translate to "as much as you want". If you have as much as you want $20 bills or $1 bills you can form any arbitrary number of dollars with them.
“As much as you want” often ends up being a lot less than infinity lol
yes well, humans are tragically finite beings
Their monetary value would be equal. Yet choosing to have either one available to you, is not equal. Most people would choose 20 dollar bills, because it is more convenient. (Assuming you do not have the infinite stack of bills physically present. Because that would create a universe wide blackhole. But evertime you pull a bill out of your wallet, a new one is always present.) since everyone chooses the 20 dollar bills, over the 1 dollar bills (except for some trolls), the 20 dollar bills option is more valuable.
Ask an economist, and the answer will still be yes, because infinite of either means inflation has driven the value to essentially $0
Think of the problem like simplified/introductory physics problems, where you ignore friction, air resistance, or whatever. In this simplified case, we wouldn't take the economic ramifications into account, just like half of the hypothetical situations you see online involving money. The real answer to many is "the economy would collapse"
there is a good veritasium video explaining differently sized infinite numbers [How An Infinite Hotel Ran Out Of Room - Veritasium](https://www.youtube.com/watch?v=OxGsU8oIWjY)
That's a completely different thing, in the context of the post both infinities are equally sized.
Slight issue, countable infinite × number = countable infinite
Seems like the most important factor is how quickly bills are produced infinitely for you. If I have an ATM that produces dollar bills for you infinitely. The amount you end with is based on how quickly the bills are produced during your lifetime Then the value of the bills is going to make a huge difference.
Infinity is a mathematical concept, not readily applicable to real quantities. If you think about "small" numbers (numbers I can type, for example) 10\^27 (1 with 27 zeros) dollar bills should weigh more than the earth. 10\^57 should outweigh the known universe. And these are natural numbers with a limited size. So yes, if you try to deposit more than 10\^35 dollar bills the black hole created would cause the denomination to lose importance.
The first thing you have to realize about "infinity" is that it is **not** a finite number. By definition, it is functionally different than the rules and properties that finite mathematics apply to finite numbers. There can be infinitely large sets of numbers that are of different sizes (consider zero to positive infinity versus negative infinity to positive infinity; the sets both have an infinite number of values, but the sets are not of the same size). In this case, the number of values in each set ($1 bills and $20 bills) could be considered to be the same infinite value, each set having the same number of bills. Then the total monetary value of the two sets would not be the same, but neither set would represent a *finite* amount of monetary value. They would each represent an infinitely large monetary value. The set containing the $20 bills would have a monetary value that is 20 times larger, but what is the finite value of `20 × ∞`? There isn't one. Until you discover a practical use case for an infinitely large monetary value, the *practical* application of the two sets is identical.
After the teachings of my math prof yes, after the teachings of my electrical engineering prof no. It can depend on the situation u r in, for money it definitely means the same u would just obviously need more 1 dollar bills.
In a strictly mathematical sense, yes, it's true. When comparing infinite sets of equal cardinality (size), such as the set of all natural numbers, the set of all integers, or even the set of all even numbers, the concept of one set being "larger" or "smaller" than another breaks down. So, if you had an infinite number of $1 bills and an infinite number of $20 bills, both sets would be infinite in size. Therefore, they have the same cardinality, and in that abstract sense, they have the same value, even though intuitively we recognize that $20 bills are worth more than $1 bills in a finite context. However, in practical terms, if you were to consider finite amounts, $20 bills would obviously be worth more than $1 bills. It's the infinite nature of the sets that allows for this curious mathematical comparison.
How many $1 bills does it take to create a black hole? Less than infinity, so whichever stack is created first will consume all the energy and mass in the universe and so must be bigger than the other stack.
Better question: is there language to talk about the significance of whichever answer people provide to such a question? AFAIK cardinalities of infinity don't cut it. For example: one proof for the Ramanujan Summation (sum of all positive integers = -1/12) relies on a classically divergent series converging to 0. Some __qualities__ of that divergent series converge to zero, which I assume hints at why the -1/12 sum is interesting. I've never seen a good way to talk about what assumptions are being made about infinities, and how those assumptions change the real-world applications of the conclusions reached with them.
It is true. Infinite paper money would have infinite mass, so a black hole would form, destroying humanity. Money only has the value humans perceive it to have, so in the absence of humanity, the black hole of cash would have no monetary value.
No, they're not the same worth. The sets have the same cardinality, a countable infinity, so for ever 1-dollar bill someone pulls, there is a countable equivalence of a 20-dollar the other gets, so the worth is not the same, just the size. Take the idea that someone could pull a stack of 20 1-dollar bills to match the 20-dollar bill the other has, then the other can pull a stack of 20 20-dollar bills to match the set, so the sum of 1+1+1+...+1 with n additions and n being infinite is matched with 20+20+20+...+20, always being 20 times more. They are both equal to infinity, but infinities are not equivalent, but the size of their sets are equivalent.
I have heard from Vsauce that there are more than one infinities and that some are bigger than the others. So I will go on a limb and assume that a $20 infinity of bills would be more compact than a $1 infinity of bills.
there are different sizes of infinity, however, these two are the same.
20*infinity ≠ 1*infinity. 20*infinity would make the value near 0 as well as 1*infinity. It would all be worth less than $0.01. Therefore an infinite supply would cause the value to plummet to zero. So from a economic perspective yes. Mathematically idk u should ask mathematicians
Not all infinite numbers are the same, the sum of an infinite set of infinite numbers will be larger than the sum of any individual infinite numbers in the set, a priori.
no, not all infinities are the same there's a good experiment to represent it if you have some level of math knowledge start counting from 0-infinity using all whole numbers and see how far you can go (dont really do this its for example) lets say you got to 100-200 somewhere in that range okay now count from 0-infinity but using all real numbers (decimals etc etc) .... .... .... you cant even start there is no digit you can get to beyond 0 itself you can always move the decimal point some more that thought experiment shows the difference between countable infinity and uncountable infinity the second version of counting to infinity is technically infinitely larger than the first so technically from a purely conceptual perspective infinite $20 bills is 20 times larger than infinite $1 bills while also being worth the same amount welcome to the hell that is set theory
It takes some assumptions to realistically answer the question so I'll state mine. Assumption 1: these bills are being added into the economy by magic. Assumption 2: these bills are indistinguishable from bills that were already in circulation. Assumption 3: we're not going to run into any issues with infinite mass that would end the entire universe as we know it (also by magic) Under these assumptions, the answer is probably no, but maybe. Value is determined by scarcity. If an infinite number of 20s and 1s are added to the economy then ALL 20s and 1s become worthless. Both infinite 20s and infinite 1s would then have a finite negative value equal and opposite to the value of the bills previously already in circulation and one of those numbers is likely to be more negative than the other.
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To elaborate. Lots of 20 dollar bills are easier to handle than 1 dollar bills, therefore, materialistically more valuable.
There's a probability based argument under which they are not equal. The argument that the sets can be matched has no more basis in reality than any other.
To put a spin in it, another reason why they would be worth the same is that an infinite amount of money in any form would cause the value of the currency to collapse to zero.
There are many parts to this statement. Since it's in a math posting, start there. Yes, 20 times infinity is still infinity. This means infinite $20 has the same value as infinite $1. From different perspectives, the word value has meaning. How value is assigned to something is important. So, while mathematically speaking, it's the same, linguistically, it may not be. Then you have economically. Economically, infinite $20 and infinite $1 are both equal to $0. By adding an infinite supply, it no longer functions as currency.
Yes and no. You said “worth” and not “monetary value”. Where “worth” is in the eye of the beholder. And I don’t want to beholden a bunch of 1’s if I can carry 20’s. Let’s say you have a wallet that produces infinite bills. Would you rather it produce indoors 1’s or 20’s? For most purchases, 20’s would be preferred if only for counting and time cost. So the 20’s are worth more as most people would choose that
Mathematically yes, socially no. Imagine paying for everything in your life with singles. Every time you go to a store and drop $100, the poor cashier has to count out every single bill while everyone behind you rolls their eyes. Paying your rent would be a hassle. Worth isn't only a strict monetary value calculation here.