Infinite sums vs infinite unions

Question

Why is it that

For every set $S$, there exists a set $\bigcup S$.

is something we take for granted (even though $S$ could be infinite), while

For every sequence $a_1,a_2,\dots$ of numbers, there exists a number $\sum a_i$.

is not something we take for granted? Addition is viewed as "something to be performed" and we generalize that idea to make meaning of an infinite sum. We don't assume right off the bat that we can perform arbitrary additions. But we don't hold a similar attitude with sets? Isn't unioning "something that has to be performed" as well?

In both cases, there is a "procedure" that I think is implicit.

Answer 1

It's a question of size. The real numbers are of finite magnitude. There is no reason to expect a bunch of finite things to sum up to a finite thing. The correct analogy would be to compare addition of numbers with union of finite sets. Then you also would not expect the union of a bunch of finite sets to be finite. It requires more care. Similarly, you could extend the meaning of number somehow to that it would become obvious that sums always exist (this will have to be done carefully).

Another important difference is that we do not define sets, while we do define numbers. So, you can set axiomatic properties on sets any way you like (as long as you don't create a contradiction), but you can't just assert that numbers exist, since they are defined rigorously. You must prove they exit.

Answer 2

I somewhat disagree with Ittay. This is not a question of size. It's a question of purpose.

Numbers, and here I'm referring specifically to real numbers,¹ have a particular purpose. They model the notion of length. When we think about something being long, we think about an interval of time, or distance or otherwise a notion of size, which is a real number. (You could argue that negative numbers make no sense here, but they do from a formal point of view; and we can ignore them for this discussion anyway.)

We know that the infinite sum of finite lengths does not have to be a finite length. It might, under some circumstances, but it does not have to be. We know that, because if we assume that it is, then we can derive all sort of contradictory things like $0=1$.

Sets, on the other hand, model the notion of a collection of mathematical objects. When we have two collections, we can talk about their union. But this goes even further. If you already know that $\{X_i\mid i\in I\}$ are all subsets of some $X$, then you can explicitly define the collection which is their union: $$\bigcup_{i\in I}X_i = \{x\in X\mid\exists i\in I: x\in X_i\}$$

The ability to unify a collection of collections into a single collection is useful. And as far as we know, it does not add any contradictions.

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But here's where things become very different between sets and numbers. If we compare sets to numbers, then arbitrary unions are only referred to unions indexed by other sets. This is similar to saying that a sum is indexed by another number. So it's a finite sum. So it exists.

On the other hand, "infinite sums" would translate to "union over a proper class of sets", which is rarely ever a set (it is exactly when there are "set many" different sets in our family). So in some sense, summation of sets is stricter and more limiting than summation of numbers.

But the entire thing is about purpose and usability. We have no use for an axiom stating that every series of numbers is again a number; but we have a use for an axiom stating that given a set of sets, we can take its union.

 

_Notes:

_{1. There is no formal definition for the concept of number. True, there is no formal definition for a set either; but as far as context goes, the notion of a set is far more robust than the notion of a number. When I say "two numbers", do I mean natural numbers, integers, rationals, real, complex, ordinal, cardinal, surreal, hyperreal? What do I mean when I say two numbers? This is very context dependent. However, when I say "set" it is almost always clear what I mean by that.}

Answer 3

In response to comments about numbers defined as sets:

Numbers can be sets but not all sets are numbers.

0 = $\emptyset $

1=$\{\emptyset\} = \{0\} $

2=$\{0,1\}=\{\emptyset,\{\emptyset\}\} $

n = {0,1,2,3,.....,n-1}

etc.

Note, all numbers are finite.

$\mathbb N =\{0,1,2,3,4,....\} $ is not finite and is not a number.

We can define n + 1 as $n \cup \{n\} $ and $n+m$ as $n + 1+pred (m)$. So example $2+3=(2+1)+2=((2+1)+1)+1=\{0,1\}\cup\{2\}+1+1=\{0,1,2\}\cup\{3\}+1=\{0,1,2,3\}\cup\{4\}=\{0,1,2,3,4\}=5$.

So sums are unions. So we can have infinite sums. But the result may be an infinite set. Which is not a number.

So we can assume infinite sums/unions exist but we can not assume the result is finite/a number.i.e. that the sum converges.

If it diverges it is not a finite set; not a number.