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A peculiar line of philosophical inquiry drew me to this question: “Is the summation of an uncountable number of zeros, zero”? I am not very familiar (read as “basically know nothing”) with mathematics and only have a naive intuitive understanding of general concepts. With that being said, my thought process regarding the question went as follows:

Natural numbers seem to be the abstractions of things with numerical values in the real world, with zero being a numerical abstraction of “nothing” or privation. Given this, it seems to make perfect sense why 0 + x equals x in finite arithmetic, seeing that you are in essence adding nothing to x, resulting in a lack of increase or decrease. This also seems to hold with a countably infinite summation of zeros (if I am not mistaken). Yet in the case of an uncountable amount of zeros, the results seem to be blurred.

Following the initial logic (which I know is not rigorous in the least bit), it seems as if the summation of an uncountable number of zero should also be zero, as no matter how many times one iterates “nothing”, nothing should result. Yet looking through the internet, I saw many comments speaking about how one must define uncountable summation for it to have any meaning. The issue is that I do not see why one can’t simply extend the concept of summation to uncountables, it seems quite straightforward.

So, to reiterate my question: “Is the summation of an uncountable number of zeros, zero”?

Apologies if the question comes off as shockingly naive, but I had no other avenue to turn to. Thank you in advance, all answers are very much appreciated.

Edit: The post that was linked as the duplicate solution is quite difficult for me to understand, I was looking for an elementary, intuitive response. Thank you

Bill Dubuque
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AminGow
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  • If by $0$ you do mean also small quantities like the sequence $1/2, 1 /3, 1/4... $ then your distress is true.... The above sum is infinite. – dmtri Nov 05 '23 at 09:41
  • @dmtri No, I am asking strictly in terms of 0 – AminGow Nov 05 '23 at 09:43
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    @AminGow there are just 2 things you need to get clear while defining any computation mathematically ( rigorously ) - 1) you need to be clear what are the objects you are using to do the computation , in your case it seems to be just zero, which you already defined as a natural number ( or whole number ) thus any arithmetical operation can be done on it, 2) the operation/computation itself those objects need to follow, in this case - addition, So i for one don't understand the need to ask for defining a particular case - 0 + 0.... no matter how many times - is always 0... –  Nov 05 '23 at 10:52
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    cardinality of the set of number of times 0 is added has nothin to do with it - cuz the operation is defined as such, rigorously..... if you replace the zeroes with other natural numbers - then the result will just be uncomputable..... that's it.... some people may say about infinite series like - ramanujan summation , but then they are bein a fool, cuz the operation in the series is not 'addition' in the same sense as defined here, ... so i hope you got your answer - 2 basic points , you satisfied both of them –  Nov 05 '23 at 10:53
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    @AdityaMishra Thanks a thousand, your response is clear, concise, and non-disparaging, it’s much appreciated. – AminGow Nov 05 '23 at 11:36
  • @user1203803 your mistake is that doing 0+0+... , no matter how long your sequence is, you can only make a countable sum of zeros. But the question is about an uncountable sum of zeros. – FCardelle Jun 10 '24 at 07:19

2 Answers2

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This depends entirely on how you formalize the definition over an uncountable index set.

The definition I see most often is, provided $a_i \ge 0$ for all $i$, $$ \sum_{i \in I} a_i := \sup \left\{ \sum_{i \in F} a_i \, \middle| \, F \subseteq I \text{ is finite} \right\} $$ In this sense, if $a_i = 0$ for all $i \in I$, even $I$ uncountable, then $\sum_{i\in I} a_i = 0$ because every finite sum is $0$.

PrincessEev
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  • Thank you for the answer, it is much appreciated. Though I do have a follow up question. Wouldn’t this simply be pushing the problem back? The result of the summation in this case would be simply be informing us that all finite subsets of an uncountablly infinite set of 0s have a sum of 0s. Though what I am wondering is if you have an uncountable set of 0s, and you decide to find the sum of all the elements in the set, what would the result be? – AminGow Nov 05 '23 at 09:41
  • @princesseev I don't understand why the requirement $F\ne\emptyset$. Also, using $\sup$ only works if $\forall i, a_i\ge0$. For the general case you either do the limit over the directed set of the finite subsets of $I$, or you do $\sum_{i\in I} \max{a_i,0}-\sum_{i\in I} \max{-a_i,0}$ when at least one of the terms is finite. – Sassatelli Giulio Nov 05 '23 at 09:48
  • The stipulation about the empty set, ironically, arose from thinking about sums involving negatives a little too carelessly. My bad. – PrincessEev Nov 05 '23 at 10:13
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Whether the uncountable sum of zeros is zero or not simply depends on the definition of uncountable sum you're using. After all, concepts in mathematics require formal definitions to be rigorous, and there is no rule other than courtesy saying that these definitions conform to any sort of common sense or colloquial meaning. Though I would be rather surprised by any definition of uncountable sum that results in the uncountable sum of zeros being nonzero or undefined, and maybe use a different term than "uncountable sum" in that case.

Magma
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