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Sum of Uncountably Many Positive Numbers

Date: 09/15/2004 at 01:31:33
From: Mark
Subject: sum of uncountably many positive numbers

Let S be a set of uncountably many positive numbers.  I would like to
show that the sum of all the elements in S is infinite.  That is, 
there is no convergent sum of uncountably many positive terms.  I am
not seeing how to apply uncountability as the key to the proof.

For a proof by contradiction, we can assume that the sum of the 
elements of any sequence (countable subset) in S is finite.  
Furthermore, we can construct a countable sequence of disjoint
uncountable subsets S_i of S.  The sequence of sums of the elements of
S_i must have each element converge to zero.  This direction seems to
lead nowhere, though, because it fails to use the uncountability of
the S_i.

Date: 09/17/2004 at 02:31:18
From: Doctor Jacques
Subject: Re: sum of uncountably many positive numbers

Hi Mark,

First of all, I'm not quite sure how it is possible to define 
unambiguously the sum of the elements of an uncountable set.  We can 
define a finite sum using associativity and induction, and the sum of 
an infinite (countable and ordered) sequence as the limit of the 
partial sums (in general, this sum may depend on the summation order).

For an uncountable set, we would probably need to assume that there 
is a well-order defined on the set, and use some kind of transfinite 

In this case, however, we have only positive numbers, and we can make 
some reasonable assumptions.  If the sum of the elements of S (in 
whatever way it is defined) is finite, say equal to some real number 
M, then the sum of the elements of any subset of S is at most equal 
to M.

In particular, consider the subsets:

  T(a) = {x in S | x > a}

where a is a positive real number.  Notice that T(a) can only contain 
at most [M/a] terms, and is therefore finite.

Notice also that S is the union of all the T(1/n) for all positive 
integers n.  This means that S is a countable union of finite sets, 
and is therefore at most countable.

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jacques, The Math Forum 

Date: 09/18/2004 at 23:18:36
From: Mark
Subject: Thank you (sum of uncountably many positive numbers)

Thank you very much for answering this question!  I was fortunate 
enough to get a very similar answer from an old high school classmate:
Define T(a) = {x in S | x > a}, and observe that the whole set S is 
the union of T(1/2^n) (countably many finite sets, if the sum is 
finite).  Another classmate suggested looking at binary expansions--
which was not the most accurate notion but pointed towards the
T(1/2^n) idea.

Thank you again for your help. I hope this answer is useful to other
folks who are learning about countability.
Associated Topics:
College Logic

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