I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
this post was submitted on 06 Jan 2024
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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.
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.
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.