Uncountable Summation of Zeros

Question

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

Answer 1

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.

Answer 2

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 $0$.