I think you’re misunderstanding the math a bit here. Let me give an example.
If you took a list of all the natural numbers, and a list off all multiples of 100, then you’ll find they have a 1 to 1 correspondence.
Now you might think “Ok, that means if we add up all the multiples of 100, we’ll have a bigger infinity than if we add up all the natural numbers. See, because when we add 1 for natural numbers, we add 100 in the list of multiples of 100. The same goes for 2 and 200, 3 and 300, and so on.”
But then you’ll notice a problem. The list of natural numbers already contains every multiple of 100 within it. Therefore, the list of natural numbers should be bigger because you’re adding more numbers. So now paradoxically, both sets seem like they should be bigger than the other.
The only resolution to this paradox is that both sets are exactly equal. I’m not smart enough to give a full mathematical proof of that, but hopefully that at least clears it up a bit.
Adding up 100 dollar bills infinitely and adding up 1 dollar bills infinitely is functionally exactly the same as adding up the natural numbers and all the multiples of 100.
The only way to have a larger infinity that I know of us to be uncountably infinite, because it is impossible to have a 1 to 1 correspondence of a countably infinite set, and an uncountably infinite set.
Infinity is a concept that can’t be reached so it can’t be counted up fully. Its not a hard number so you can’t get a full value from it since there is always another number to reach. Therefore you only peak at ∞ in any individual moment. You can never actually count it.
If you’re responding to the part about countable infinity and uncountable infinity, it’s a bit of a misnomer, but it is the proper term.
Countably infinite is when you can pick any number in the set and know what comes next.
Uncountably infinite is when it’s physically impossible to do that, such as with a set of all irrational numbers. You can pick any number you want, but it’s impossible to count what came before or after it because you could just make the decimal even more precise, infinitely.
The bizarre thing about this property is that even if you paired every number in a uncountably infinite set (such as a set of all irrational numbers) with a countably infinite set (such as a set of all natural numbers) then no matter how you paired them, you would always find a number from the uncountably infinite set you forgot. Infinitely many in fact.
It’s often demonstrated by drawing up a chart of all rational numbers, and pairing each with an irrational number. Even if you did it perfectly, you could change the first digit of the irrational number paired with one, change the second digit of the irrational number paired with two, and so on. Once you were done, you’d put all the new digits together in order, and now you have a new number that appears nowhere on your infinite list.
It’ll be at least one digit off from every single number you have, because you just went through and changed those digits.
Because of that property, uncountably infinite sets are often said to be larger than countably infinite sets. I suppose depending on your definition that’s true, but I think of it as just a different type of infinity.
Sure if you’re talking about a concept like money, but we’re talking about dollar bills and 100 dollar bills, physical objects. And if you’re talking about physical objects you have to consider material reality, if you’re choosing one or the other the 100 dollar bills are more convinient. Therefore they have more utility, which makes them have a higher value.
I was actually discussing this with my wife earlier and her position is that the 1 dollar bills are better because it’s tough to find somebody who’ll split a 100, and 100s don’t work in vending machines.
I thought the hundreds would be better because you could just deposit them in the bank and use your card, and banks often have limits on how many individual bills you can deposit at once, so hundreds are way better for that.
~~I was thinking that, bill for bill, the $100 bill will always be greater value. But I can see the plausibility in your argument that, when we’re counting both the value of the members of each set, the value of the $100 bill pile can always be found somewhere in the series of $1 bills. The latter will always “catch up” so to speak. But, if this line of reasoning is true, it should apply to other countably infinite sets as well. Consider the following two examples.
First, the number of rational numbers between 0 and 1 is countably infinite. That is, we can establish a 1-1 correspondence between the infinite set of fractions between 0-1 and the infinite set of positive integers. So the number of numbers is the same. But clearly, if we add up all the infinite fractions between 0 and 1, they would add up to 1. Whereas, adding up the set of positive integers will get us infinity.
Second, there are equally many positive integers as there are negative integers. There is a 1-1 correspondence such that the number of numbers is the same. However, if we add up the positive integers we get positive infinity and if we add up all the negative integers we get negative infinity. Clearly, the positive is quantitatively greater than the negative.
In these two cases, we see that a distinction needs to be made between the infinite number of members in the set and the value of each member. The same arguably applies in the case of the dollar bills.~~
Ah but you see if you take into consideration it’s talking about bills and not money in an account you have to take into consideration the material reality. A person who receives more 100s would have an easier time depositing and spending the money therefore they have higher utility, therefore they are worth more than the 1s.
Sure they may have the same amount of numbers (and 1, 2,3… May even be larger because you’ll eventually repeat in the 1:1 examile) but in reality the one with the 100s will have an easier time using their objects (100 dollar bills) than the ones who pick the 1 dollar bills
I think you’re misunderstanding the math a bit here. Let me give an example.
If you took a list of all the natural numbers, and a list off all multiples of 100, then you’ll find they have a 1 to 1 correspondence.
Now you might think “Ok, that means if we add up all the multiples of 100, we’ll have a bigger infinity than if we add up all the natural numbers. See, because when we add 1 for natural numbers, we add 100 in the list of multiples of 100. The same goes for 2 and 200, 3 and 300, and so on.”
But then you’ll notice a problem. The list of natural numbers already contains every multiple of 100 within it. Therefore, the list of natural numbers should be bigger because you’re adding more numbers. So now paradoxically, both sets seem like they should be bigger than the other.
The only resolution to this paradox is that both sets are exactly equal. I’m not smart enough to give a full mathematical proof of that, but hopefully that at least clears it up a bit.
Adding up 100 dollar bills infinitely and adding up 1 dollar bills infinitely is functionally exactly the same as adding up the natural numbers and all the multiples of 100.
The only way to have a larger infinity that I know of us to be uncountably infinite, because it is impossible to have a 1 to 1 correspondence of a countably infinite set, and an uncountably infinite set.
Infinity is a concept that can’t be reached so it can’t be counted up fully. Its not a hard number so you can’t get a full value from it since there is always another number to reach. Therefore you only peak at ∞ in any individual moment. You can never actually count it.
If you’re responding to the part about countable infinity and uncountable infinity, it’s a bit of a misnomer, but it is the proper term.
Countably infinite is when you can pick any number in the set and know what comes next.
Uncountably infinite is when it’s physically impossible to do that, such as with a set of all irrational numbers. You can pick any number you want, but it’s impossible to count what came before or after it because you could just make the decimal even more precise, infinitely.
The bizarre thing about this property is that even if you paired every number in a uncountably infinite set (such as a set of all irrational numbers) with a countably infinite set (such as a set of all natural numbers) then no matter how you paired them, you would always find a number from the uncountably infinite set you forgot. Infinitely many in fact.
It’s often demonstrated by drawing up a chart of all rational numbers, and pairing each with an irrational number. Even if you did it perfectly, you could change the first digit of the irrational number paired with one, change the second digit of the irrational number paired with two, and so on. Once you were done, you’d put all the new digits together in order, and now you have a new number that appears nowhere on your infinite list.
It’ll be at least one digit off from every single number you have, because you just went through and changed those digits.
Because of that property, uncountably infinite sets are often said to be larger than countably infinite sets. I suppose depending on your definition that’s true, but I think of it as just a different type of infinity.
Sure if you’re talking about a concept like money, but we’re talking about dollar bills and 100 dollar bills, physical objects. And if you’re talking about physical objects you have to consider material reality, if you’re choosing one or the other the 100 dollar bills are more convinient. Therefore they have more utility, which makes them have a higher value.
I agree. I’m just being a math nerd.
I was actually discussing this with my wife earlier and her position is that the 1 dollar bills are better because it’s tough to find somebody who’ll split a 100, and 100s don’t work in vending machines.
I thought the hundreds would be better because you could just deposit them in the bank and use your card, and banks often have limits on how many individual bills you can deposit at once, so hundreds are way better for that.
If you’re considering material reality then you can’t have an infinite amount of it.
That’s a concession of the premise, you obviously can’t have infinite anything, but if you could then the 100s would bring more utility
But the utility is not the issue in the premise.
“Would you rather have an infinite number of $1 or $100 bills?” Obviously $100 bills, but they are worth the same amount.
If utility isn’t the reason why you’re picking 100s then why would you if they’re the same amount?
Utility is irrelevant to the statement “an infinite number of $1 bills is worth the same amount as an infinite number of $100 bills.”
Nah. We’re taking the singles and hitting up the strip club boiii.
Only valid retort
~~I was thinking that, bill for bill, the $100 bill will always be greater value. But I can see the plausibility in your argument that, when we’re counting both the value of the members of each set, the value of the $100 bill pile can always be found somewhere in the series of $1 bills. The latter will always “catch up” so to speak. But, if this line of reasoning is true, it should apply to other countably infinite sets as well. Consider the following two examples.
First, the number of rational numbers between 0 and 1 is countably infinite. That is, we can establish a 1-1 correspondence between the infinite set of fractions between 0-1 and the infinite set of positive integers. So the number of numbers is the same. But clearly, if we add up all the infinite fractions between 0 and 1, they would add up to 1. Whereas, adding up the set of positive integers will get us infinity.
Second, there are equally many positive integers as there are negative integers. There is a 1-1 correspondence such that the number of numbers is the same. However, if we add up the positive integers we get positive infinity and if we add up all the negative integers we get negative infinity. Clearly, the positive is quantitatively greater than the negative.
In these two cases, we see that a distinction needs to be made between the infinite number of members in the set and the value of each member. The same arguably applies in the case of the dollar bills.~~
EDIT: I see now that I was mistaken.
Ah but you see if you take into consideration it’s talking about bills and not money in an account you have to take into consideration the material reality. A person who receives more 100s would have an easier time depositing and spending the money therefore they have higher utility, therefore they are worth more than the 1s.
Sure they may have the same amount of numbers (and 1, 2,3… May even be larger because you’ll eventually repeat in the 1:1 examile) but in reality the one with the 100s will have an easier time using their objects (100 dollar bills) than the ones who pick the 1 dollar bills