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).
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.
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.
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).
Those are all aleph 0 infinities. There’s is a mathematical proof that shows the square of infinity is still infinity. The same as “there is the same number of fractions as there is integers” (same size infinities).
It might give use different growth rate but Infinity is infinite, it’s like the elementary school playground argument saying “infinity + 1” there is no “infinity + 1”, it’s just infinity. Infinity is the range of all the numbers ever, you can’t increase that set of numbers that is already infinite.
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.
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.
If you want to argue which you would rather have, that could be another argument. However, we’re weighing the monetary value of both arguments of both arguments. Neither of them is greater than the other if they are both the set of all numbers already.
Why are people upvoting this post? It’s completely wrong. Infinity * something can’t grow faster than infinity * something else.
Afaik it can, buy not this way.
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.
for $1 bills: lim(x->inf) 1*x
for $100 bills: lim(x->inf) 100*x
Using L’Hôpital’s rule, we take the derivative of each to get their ratio, ie: 100/1, so the $100 bill infinity is bigger (since the value of the money grows faster as the number of bills approaches infinity, or said another way: the ratio of two infinities is the same as the ratio of their rates of change).
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.
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.
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.
Something to do with the…Greek? Hebrew? Klingon? Letter Aleph
it’s actually Vulcan
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*xbut the area scales likex^2(which gets larger faster asxapproaches infinity).Those are all aleph 0 infinities. There’s is a mathematical proof that shows the square of infinity is still infinity. The same as “there is the same number of fractions as there is integers” (same size infinities).
It might give use different growth rate but Infinity is infinite, it’s like the elementary school playground argument saying “infinity + 1” there is no “infinity + 1”, it’s just infinity. Infinity is the range of all the numbers ever, you can’t increase that set of numbers that is already infinite.
oh, you mean like taking the ratio of the derivatives of two functions?
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
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.
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If you want to argue which you would rather have, that could be another argument. However, we’re weighing the monetary value of both arguments of both arguments. Neither of them is greater than the other if they are both the set of all numbers already.