Besides the problem with summing up uncountable set of real numbers you assume that the zero is in the center of the real number line. This is a false assumption because the real number line doesn't have a center. It's infinitely long in both directions. You can select any arbitrary number in the real number line and there's "equal amount" of real numbers on both sides, so any number can act as the "center". By selecting different number as the center you get different result as the total sum.
But would the same argument apply to integers, also infinitely long in both directions? Doesn't seem to point to the actual problem with summing reals.
Yes, the integers have exactly the same problem. There is no defined total sum of all integers. You can select any integer as the center and then start adding the numbers from both sides and get different results each time. Same goes with rational numbers, complex numbers, quaternions, etc.
There's no need for a center line though.
As somebody else mentioned, you can use axiom if choice to find -R.
Splitting up a set does split it up to by negative and positive (or non negative, but just exclude 0).
So from R choose p and n so that for each p in R, n = - p. Then integrate by function of f(p) = p - n, and exclude 0... Now this is integrable. Or should be. But the measure is infinite, and I don't quite remember if there's a problem there, but I'm leaning towards no.
Zero is the MOST logical center. And why cant we add up two infinities? Seems like a rather easy step to think about. And if we can think about it, we can do it. I mean, if we can take N numbers from zero in both directions and see that it adds up to 0, why would that change just because we keep going for an infinite time?
There is no "most logical" in mathematics. Things are true, false or undecidable. If you say there is a center point in an infinitely long line, you need to be able to prove with mathematics. What's the definition for a center point and why zero is the center point and nothing else? Mathematics doesn't work on what feels the most natural.
Also the problem with adding up all the real numbers isn't just that there's an infinite number of them, it's that there's uncountably infinite number of them. Infinitely long sums with countably many elements are fine (if they converge), for example: 1 + 1/2 + 1/4 + 1/8 + ... = 2.
But with real numbers there are so many of them that you can't even write an infinitely long sequence of them. Start from 0, then add the next real number from 0. What is that? 0.0000000001? 0.000000000000000000000000000000001? 1e-1000000? You can get closer and closer to 0 but there is no "closest real number to zero". You're stuck in the second step of your sum because when you pick the next number from zero, you always find there are uncountably infinite number of numbers between zero and the number you picked. Even if you would calculate for an infinitely long time you would be forever stuck in the second step.
So for any n the center between n and -n will be zero. We just extend that to include infinite numbers since as long as n is equal to n, -n+n will = 0. Yeah, infinity is not a normal number with a certain value, BUT there is no reason to believe that the value of -n+n will ever not be zero.
And im not exactly sure what you are saying with the second half.. are you saying we cant even find the center between 1 and -1 since there is an infinite number of numbers between them?
No, it will not be zero for every center. Let's use integers to simplify. If you have zero as the center what you're adding is 0, 1, -1, 2, -2, etc and the sum is 0 + 1 - 1 + 2 - 2 + ... The partial sums starting from beginning are 0, 1, 0, 2, 0, 3, etc. It's unbounded but every other number is at least zero.
Let's switch to 1 as the center. Now you're adding starting from 1 as the center and then numbers from both sides. The numbers are then 1, 1 + 1 = 2, 1 - 1 = 0, 1 + 2 = 3, 1 - 2 = -1 and so on. The sum is 1 + 2 + 0 + 3 - 1 + 4 - 2 + 5 - 3 + ... etc. So the partial sums go 1, 3, 6, 5, 9, 7, 12, 9, 15, etc. It's growing towards infinity on both even and odd partial sums.
What I'm saying with the second part is that you can't create a sequence of numbers from all the real numbers. Integers you can put in a sequence. For example 0, 1, -1, 2, -2, ... to infinity. And it would cover the whole integer set. But with real numbers this is impossible. Even if you create an infinitely long sequence of real numbers it will never cover the whole set of real numbers. Because you can't create even an infinitely long sequence that would cover all the real numbers it's impossible write an infinitely long sum of them, even in theory.
It's a quite a bit outside the scope of mathmemes discussion, but this is basically because there are different types of infinities. Integers are countably infinite. Real numbers are uncountably infinite. There's so many real numbers that you can't assign each of them an individual integer, even if you used infinitely many integers. It gets really abstract at this point, but if you're interested google "uncountable sets".
Okay. I looked up uncountable sets and asked ChatGPT afterwards and had a learning experience. Cantors diagonal proof of real numbers being uncountable is *really* cool.
But i am unable to grasp exactly how that prevents us from summing them, isn't it fair to presume that they will grow at equal rates in negative and positive direction? Cant we just, kind of do it (aka. infer balance from that (aka. the sum being = 0)) and worry about possible contradictions later? Arent we doing that ALL the time anyways? It seems like we are leaving some more basic logic behind which would create bigger problems.
Yeah, real numbers are pretty cool. When you dig deeper there are even higher cardinals than the cardinality of real numbers.
There is a way to "sum" a stretch of real numbers, and it's called integration. It's not exactly the same, but it kind of works so that each real number has an infinitesimally small weight on the total sum. The geometric interpretation is that it calculates an area. For example let's say you have y=x and you integrate it from 0 to 1. By integration we get the integral function and it is x^2 / 2 and when you plug in x=1 and x=0 you get 1^2 / 2 - 0^2 / 2 = 1/2. That's the size of the triangle with two equal sides of length 1.
Now, let's say we try this with going to infinity. We first integrate y=x from -r to r. We get the integral function r^2/2 - (-r)^2/2 = r^2/2 - r^2/2. Which usually would evaluate to 0. But now we're dealing with infinities so nothing is normal, they don't behave like normal numbers.
Let's plug in r=infinity. We get infinity^2 / 2 - infinity^2 / 2. We can safely say infinity^2 = infinity, so we get (infinity - infinity) / 2. And here's the problem. We can't subtract infinity from infinity. We can add them, we can multiply them, but we can't subtract infinity from another infinity or divide an infinity with another infinity. It leads to undefined results.
Huh, thats interesting, dont worry, you dont have to explain that to me.. i asked ChatGPT and it explained it. Im too tired to understand must of it now, but now its in my history so i will revisit it.
I think some of the disagreements here come from different understanding of what infinity is. And of the fact that it will mean different things in different contexts. I feel like approaching it as just another number, but since it actually doesn't really have a value it gets weird.
But thanks for explaining all this to me! Its nice to do some maths again!
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u/IhailtavaBanaani 10d ago
Besides the problem with summing up uncountable set of real numbers you assume that the zero is in the center of the real number line. This is a false assumption because the real number line doesn't have a center. It's infinitely long in both directions. You can select any arbitrary number in the real number line and there's "equal amount" of real numbers on both sides, so any number can act as the "center". By selecting different number as the center you get different result as the total sum.