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Graham Oppy writes in »Philosophical Perspectives on Infinity«:

Without countable additivity, it seems – for example – that we must lose the result that an arithmetic sum of an infinite series is the limit of the partial sums.

Why is that the case? Can anybody explain? The value of an infinite series usually just is defined as the limit of its partial sums.

  • What is the context here, what "countable additivity" property is it referring to? Is it discussing measure theory or probability theory? – 6005 Dec 24 '14 at 5:21
  • It's just measure theory. – Ystar Dec 27 '14 at 4:44
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I'd strongly suggest moving this to the Math.SE forum, see this post, but seeing as how you are keeping it here, I'll offer some thoughts:

Oppy is discussing what properties a measure should possess, from a foundational perspective. Note that a measure is defined on a measurable space, which consists of a set, a sigma-field on that set, and a set-function that obeys the measure axioms:

  1. The measure of the empty set is 0
  2. If set A is a subset of set B, the measure of set B should be at least as large as the measure of set A
  3. The measure of the (finite or countable) union of mutually disjoint sets is the sum (finite or countable) of the measure of each such set.

Note that (3) is an assumption, and one that Oppy justifies by appeal to a very basic mathematical property that we take for granted...the limit of infinite sums. I have seen other authors (e.g., Robert Ash) also take pains to justify the need for countable additivity by appealing to the richness of the mathematics allowed by assuming/requiring countable additivity, not on any physical grounds.

In fact, there are set functions that are finitely additive (all partial sums exist) but that don't converge to the value of the set function at the limit itself (think jump discontinuity) when a countable sum is taken.

Specifically for your question, Oppy was addressing Zeno's paradox about distance. The distance to be traveled is being divided into an infinite sequence of intervals, each half as long as the previous one. Now, under the axioms of measure, we'd expect that no matter how we divide up the distance to be covered, as long as we specify a countable set of divisions, the sum of the length of the divisions should give the total length (i.e., a meter is a meter, no matter what scale we measure it in or how we break it up).

Thus, if we reject countable additivity, then we can have measures (e.g., length) with counterintuitive properties, such as the total distance covered by the runner being dependent on the way we break up and add the partial distances. By requiring countable additivity, we are forcing a common-sense definition of length (and measure in general) and excluding definitions that lead to weird behavior.

There are several interesting counterexamples that show what can happen if we try to work with non-measurable sets (Banach-Tarsky paradox, Vitali Sets) or if you try to work with set functions that are only finitely additive

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Graham Oppy's statement is correct. Without "countable additivity," the result of an arithmatic sum of an infinite series would not be the limit(the total) of the partial sums. The reason is that we would not be able to add the partial sums!

  • But we don't add partial sums. We take the limit of the sequence of partial sums. And limits of sequences are perfectly well defined without any reference to countable additivity. – user4894 Dec 31 '14 at 18:58
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The statement

Without countable additivity, it seems – for example – that we must lose the result that an arithmetic sum of an infinite series is the limit of the partial sums.

is surely false, for the simple reason that it's a definition and not a result.

Given an infinite sequence x_1, x_2, ... we define repeat define its sum as the limit, if it exists, of the corresponding sequence of partial sums.

And the limit of a sequence only depends on the formal definition. The sequence x_1, x_2, ... converges to L if for all epsilon > 0 there exists N such that if n > N, then |x_n - L| < epsilon.

[Why doesn't philosophy.stackexchange support LaTeX markup?]

That definition in turn depends only on the construction of the real numbers (as equivalence classes of Cauchy sequences, or of Dedekind cuts) carried out within ZF set theory. Any competent undergrad math major could outline the steps involved.

There is no need for countable additivity to define the sum of a convergent infinite series. Perhaps the OP can supply more context for this obviously incorrect quotation.

Countable additivity is useful in measure theory so that we can say that the union of disjoint intervals of lengths 1/2, 1/4, 1/8, ... respectively has measure 1. But you don't need an assumption of countable additivity to define the limit of a sum of real numbers. That result only depends on the proper definition of the real numbers and the development of the theory of convergent sequences and series.

In fact mathematicians can even define the sum of an uncountable collection of real numbers (although if the sum is finite, we can prove that all but countably many of the summands must be zero); yet, uncountable additivity is not required of a measure.

Oppy is either confused, is ignorant of basic real analysis; or is being quoted out of context by the OP.

  • That's the context, last paragraph of page 106: books.google.com/… – Ystar Jan 1 '15 at 12:14
  • Perhaps Oppy means something deeper than the basic mathematical definition? I don't know, if we view the infinite sum of positive numbers x₁, x₂, ... simply as the set [0, x₁) ∪ [x₁, x₁+x₂) ∪ [x₁+x₂, x₁+x₂+x₃) ∪ ⋯ = [0, lim n→∞ S(n)) with S(n) = x₁+x₂+x₃+ ⋯ +x_n we come pretty close to an explanation why the basic calculus definition makes sense... – Ystar Jan 1 '15 at 12:15
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The value of an infinite series usually just is defined as the limit of its partial sums.

True; but it may make no sense ie 1+1+1... adds up to what? or 1+(-1)+1+(-1)... or 1+2+3+4...

One can observe that [1+(-1)]+[1+(-1)]+...= 0+0+0...=0 And that 1+[(-1)+1]+[(-1)+1]+... = 1+0+0+... = 1

This shows that infinite sums are dependent on how you sum them, this is against the associative law in finite sums; a property that one wants to conserve.

Furthermore, certain infinite sums, (I think to Euler who first showed this) sum to different values if they were re-arranged; this is against the commutat8ive law for finite sums; again a property that one wants to conserve.

'Countable additivity' is a term usually used in measure theory; it is a fact, that using Dirac measures that the theory of 'infinite sums' as usually understood is subsumed measure theory; countable additivity is a general feature of measure theory (there is a theory also using finite additivity); and in the context of 'infinite sums' it ends up as the limits of a sequence of partial sums. (I don't view this as being of particularly philosophical import - its an important but technical result in the theory).

  • That's true about series in general, but if you stick to series with nonnegative terms, that anomaly goes away. And your answer doesn't seem to bear on the original question at all. – user4894 Dec 31 '14 at 18:47
  • I don't know about you; but for me, its of philosophical interest that (generalised) associativity & commutivity is lost in an infinite context; this is how I answered it. The original question is more mathematical than philosophical, and its better answered at Math.SE; though I put in my pennys-worth here. – Mozibur Ullah Jan 2 '15 at 13:35
  • 1) We're just talking about convergent series of positive numbers. 2) Yes, ∑ -(-1)ⁿ⋅(1/n) = log(2), but summing the terms with odd n gives +∞, summing the terms with even n gives -∞, OF COURSE we can't change the order, nothing that mysterious! 3) I believe strictly mathematically, Oppy's statement is false. I believe that he meant something different (deeper, more philosophical). After all infinite sums should work in real life, it's not just a matter of definition... – Ystar Jan 4 '15 at 6:56

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