You can map 1000 one dollar bills to every single 100 dollar bill and never run out, even in the limit, and therefore conclude (equally incorrectly) that the infinite $1 bills are worth more, because infinity isn’t a number.
Yes, the mapping that you’re describing isn’t useful; but that doesn’t invalidate the simple test that we can do to prove that two infinite sets are the same. And the test is pretty intuitive in some cases. If for every fork there is a spoon, then they must be an equal number. And if for every positive whole number (e.g “2”) there is a negative whole number (e.g. “-2”), then the set of positive whole numbers and the set of negative whole numbers must be the same.
Another thing that seems unintuitive but might make the concept in general make more sense is that you cannot add or do any other arithmetic on infinity.
I agree with you that adding infinity to itself just gets you the same number. But now we’re starting to ask: just what is infinity? We could think of infinity as a collection of things or we could think of infinity like we do numbers (yes, I know infinity is not technically among the real numbers). In the latter sense we could have a negative infinity {-1…-2…-3…etc.} that is quantitatively “worth less” than a countable positive infinity. On the other hand, if we think of infinity as a collection of stuff, then you couldn’t have a negative infinity because you can’t have less than zero members in the set.
I’ll admit that I’m getting out of my depth here since I know this stuff from philosophical study rather than maths proper. As with anything, I’m happy to be proven wrong, but I’m quite sure about the 1-1 correspondence bit.
Your 1-1 relationship makes sense intuitively with a finite set but it breaks down with the mathematical concept of infinity. Here’s a good article explaining it, but DreamButt’s point of every set of countable infinite sets are equal holds true because you can map them. Take a set of all positive integers and a set of all positive, even integers. At first glance it seems like the second set is half as big right? But you can map them like this:
Set 1 | Set 2
1|2
2|4
3|6
4|8
5|10
6|12
If you added the numbers up on the two sets you would get 21 and 42 respectively. Set 2 isn’t bigger, the numbers just increased twice as fast because we had half as many to count. When you continue the series infinitely they’re the same size. The same applies for $1 vs $100 bills.
$1|$100
$2|$200
$3|$300
In this case the $1 bills are every integer while the $100 bills is the set of all 100’s instead of all even integers, but the same rule applies. Set two is increasing 100x faster but that’s because they’re skipping all the numbers in between.
I understand that both sets are equally infinite (same cardinality). And I do see the plausibility in your argument that, in each case, since there’s an infinite number of bills their value should be equally infinite. If your argument is correct, then I should revise my understanding of infinity. So maybe you can help me make sense of 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. 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 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.
But clearly, if we add up all the infinite fractions between 0 and 1, they would add up to 1.
0.9 and 0.8 are in that set, so they would add up to at least 1.7. In fact if you give me any positive number I can give you a (finite!) set of (distinct) fractions less than 1 which sum to more than that number. In other words, the sum is infinite.
I meant to say that we can infinitely divide the numbers between 0 and 1 and then match each with an integer. But I realized that the former wouldn’t be rational numbers, they would be real numbers.
That aside, I see now that the original idea behind the meme was mistaken.
I meant to say that we can infinitely divide the numbers between 0 and 1 and then match each with an integer. But I realized that the former wouldn’t be rational numbers, they would be real numbers.
That aside, I see now that the original idea behind the meme was mistaken.
Yes, the mapping that you’re describing isn’t useful; but that doesn’t invalidate the simple test that we can do to prove that two infinite sets are the same. And the test is pretty intuitive in some cases. If for every fork there is a spoon, then they must be an equal number. And if for every positive whole number (e.g “2”) there is a negative whole number (e.g. “-2”), then the set of positive whole numbers and the set of negative whole numbers must be the same.
I agree with you that adding infinity to itself just gets you the same number. But now we’re starting to ask: just what is infinity? We could think of infinity as a collection of things or we could think of infinity like we do numbers (yes, I know infinity is not technically among the real numbers). In the latter sense we could have a negative infinity {-1…-2…-3…etc.} that is quantitatively “worth less” than a countable positive infinity. On the other hand, if we think of infinity as a collection of stuff, then you couldn’t have a negative infinity because you can’t have less than zero members in the set.
I’ll admit that I’m getting out of my depth here since I know this stuff from philosophical study rather than maths proper. As with anything, I’m happy to be proven wrong, but I’m quite sure about the 1-1 correspondence bit.
Your 1-1 relationship makes sense intuitively with a finite set but it breaks down with the mathematical concept of infinity. Here’s a good article explaining it, but DreamButt’s point of every set of countable infinite sets are equal holds true because you can map them. Take a set of all positive integers and a set of all positive, even integers. At first glance it seems like the second set is half as big right? But you can map them like this:
Set 1 | Set 2
1|2
2|4
3|6
4|8
5|10
6|12
If you added the numbers up on the two sets you would get 21 and 42 respectively. Set 2 isn’t bigger, the numbers just increased twice as fast because we had half as many to count. When you continue the series infinitely they’re the same size. The same applies for $1 vs $100 bills.
$1|$100
$2|$200
$3|$300
In this case the $1 bills are every integer while the $100 bills is the set of all 100’s instead of all even integers, but the same rule applies. Set two is increasing 100x faster but that’s because they’re skipping all the numbers in between.
I understand that both sets are equally infinite (same cardinality). And I do see the plausibility in your argument that, in each case, since there’s an infinite number of bills their value should be equally infinite. If your argument is correct, then I should revise my understanding of infinity. So maybe you can help me make sense of 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. 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 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.
0.9 and 0.8 are in that set, so they would add up to at least 1.7. In fact if you give me any positive number I can give you a (finite!) set of (distinct) fractions less than 1 which sum to more than that number. In other words, the sum is infinite.
I meant to say that we can infinitely divide the numbers between 0 and 1 and then match each with an integer. But I realized that the former wouldn’t be rational numbers, they would be real numbers.
That aside, I see now that the original idea behind the meme was mistaken.
Um no? 3/4 + 5/6 > 1
If you mean the series 1/2 + 1/3 + 1/4 + … that also tends to infinity
I meant to say that we can infinitely divide the numbers between 0 and 1 and then match each with an integer. But I realized that the former wouldn’t be rational numbers, they would be real numbers.
That aside, I see now that the original idea behind the meme was mistaken.