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According to the last step in proof of the unmeasurability of Vitali_set, it said that summing infinitely many copies of the constant $\lambda(V)$ yields either zero or infinity, according to whether the constant is zero or positive. It sounds pleasing to my ear, but I still have a bit doubt in the reason of the sum of infinitely many copies of a positive real constant would definitely yield infinite.

Actually, $< \sum_{i=0}^n c>_{n=0}^{\infty}$ is indeed an strict increasing series when $c>0$. However, this fact seems cannot guarantee the inevitability of $\sum_{i=0}^\infty c=\infty$.

Take an example in infinite product. $1,2,4,8,16...$ is actually a strict increasing series too, but $\prod_{i=0}^{\infty}2$ can yield $0$ in some cases.

Moreover $9,99,999,\ldots$ is also a strict increasing series, but in some theory $...999$ is not a infinite but $-1$.

So my question on what basis can the conclusion that $\sum_{i=0}^\infty \lambda(V)$ is necessarily not between $1$ and $3$ be concluded?

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    @AsafKaragila Yes, additionally I also felt the purpose which physicists hold is to seek a proper theory for a certain kind of models(especially crystals), compared with mathematicians, to them it seems theories go first.2012-11-11

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What is $\sum_{n=0}^\infty x_n$? It is $\lim_{k\to\infty}\sum_{n=0}^k x_n$. This is a limit of real numbers.

Suppose that $x_n=1$ for all $n$, then what is this limit? The partial sum $\sum_{n=0}^k 1 = k+1$, and therefore this is the limit $\lim_{k\to\infty}(k+1)$.

Replacing $1$ by any other positive constant has the same effect.


It seems that the questions stems from mixing up contexts. One should never do that in mathematics. Real numbers are real numbers, they are not $10$-adic, they are not ordinals and they are not cardinals.

True, the natural numbers can be represented as cardinals, ordinals, real, $10$-adic numbers, and more. However each system carries out its own rules. In particular in the behavior of infinitary operations such as infinite sums and multiplications.

Even cardinals and ordinals, which are often thought as the same, behave differently with respect to infinitary multiplications. Let alone real numbers and cardinals, or real numbers and ordinals.

In measure theory we work with real numbers which means that the sums taken are sums of real numbers, and when taking infinitary sums of real numbers one apply the definitions for sums of real numbers.

For example, in the real numbers I am allowed to do this: $\frac12\sum_{i=0}^\infty 1=\sum_{i=0}^\infty\frac12$ Where as summation of ordinals or cardinals cannot be done because the object $\frac12$ is neither an ordinal nor a cardinal number.

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    Okay, I'm quite clear now, many thanks.2012-11-12