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.
What is worth? Would you rather have 500 1s or 5 100s? You already said you’d take the 100s, why? I would take the 100s because I personally value the convenience of 100s more than the 1s, so to me, a single 100 is worth more than 100 1s. Worth doesn’t need to imply the monetary value of the money.
The convenience/utility makes the 100s worth more even if they’re both valued the same
Don’t care. They both get deposited at my bank the same.
You already said you’d take the 100s, why?
An acknowledgement to the point you were making that was a digression from the discussion at hand. My mistake apparently.
I would take the 100s because I personally value the convenience of 100s more than the 1s, so to me, a single 100 is worth more than 100 1s.
I personally would not accept a Genie wish for either as the mass would create a black hole that would destroy the universe. There is no “practical consideration” when dealing with infinities. Knowing how to work with infinities is useful for complex mathematics, but there is no real world application until you simplify away the infinities.
Worth doesn’t need to imply the monetary value of the money.
It very much does imply the monetary value of the money. It can mean other things if you want to define it as such, but you need to before hand in such a case. It was not defined differently by OP.
I see I made a mistake by acknowledging your argument, and trying to indicate my understanding there of, before trying to get back to the point at hand that an infinite number of $1 bills is worth the same amount as an infinite number of $100 bills.
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.”
What is worth? Would you rather have 500 1s or 5 100s? You already said you’d take the 100s, why? I would take the 100s because I personally value the convenience of 100s more than the 1s, so to me, a single 100 is worth more than 100 1s. Worth doesn’t need to imply the monetary value of the money.
The convenience/utility makes the 100s worth more even if they’re both valued the same
Dollar value.
Don’t care. They both get deposited at my bank the same.
An acknowledgement to the point you were making that was a digression from the discussion at hand. My mistake apparently.
I personally would not accept a Genie wish for either as the mass would create a black hole that would destroy the universe. There is no “practical consideration” when dealing with infinities. Knowing how to work with infinities is useful for complex mathematics, but there is no real world application until you simplify away the infinities.
It very much does imply the monetary value of the money. It can mean other things if you want to define it as such, but you need to before hand in such a case. It was not defined differently by OP.
Also you never said why you’d pick the 100s.
I see I made a mistake by acknowledging your argument, and trying to indicate my understanding there of, before trying to get back to the point at hand that an infinite number of $1 bills is worth the same amount as an infinite number of $100 bills.
I won’t make the same mistake again.
Nah. We’re taking the singles and hitting up the strip club boiii.
Only valid retort