Why infinite sums of positive real constants definitely yield infinite?

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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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Updated on June 18, 2022

Comments

  • Popopo
    Popopo 1 day

    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?

    • Michael Greinecker
      Michael Greinecker over 9 years
      Where is the limit of $9, 99, 999,\dots$ equal to $-1$?
    • Popopo
      Popopo over 9 years
      @MichaelGreinecker $...999$ sometimes be dealt as $-1$ (in 10-adic), not the limit of $9,99,999...$
    • Asaf Karagila
      Asaf Karagila over 9 years
      @Popopo: I think that the problem is that you are mixing contexts. The real numbers are not $10$-adic numbers, and they are not cardinal numbers, and they are not anything except the real numbers. If you agree that measure theory is done in the context of real numbers you cannot give "counterexamples" from a different context. This is like saying "Oh, religious Jews don't eat pork, but Christians do. So Judaism is inconsistent", within the real numbers the products and sum you mentioned are infinite because they are limits of strictly increasing sequences and therefore larger than any number
    • Andreas Blass
      Andreas Blass over 9 years
      @AsafKaragila: Just in case Popopo remembers the minor bits of your comments while ignoring (as (s)he seems to have done) the main points, perhaps clarify that not every limit of a strictly increasing sequence of reals is larger than any number, i.e., some increasing sequences converge.
    • Popopo
      Popopo over 9 years
      @AsafKaragila Ummm...your opinion indeed make a sense. However I still have a bit confusion. Different religious indeed should not be confused, but are numbers merely mythical characters? Moreover if they actually are, what is the relation between characters shared the same name from different myths? Are they identical? Or totally dictinct with each other?
    • Popopo
      Popopo over 9 years
      @AsafKaragila e.g. there is a man son of Virgin Mary named Jesus in Gospel and has an attribute 'is a son of God' whereas there is a man son of Virgin Mary also named Jesus in Quran which is a prophet but does not has such an attribute. The question is are they identical? If NO is the answer then it sounds strange to me because there is unique such Jesus in history.
    • Asaf Karagila
      Asaf Karagila over 9 years
      @Popopo: Yes. Numbers are mythical creatures if you prefer to think about them that way. But this makes even more sense. Real numbers are pixies and cardinals are trolls. The fact that there are pixies and trolls which behave similarly to unicorns (natural numbers) does not mean that pixies are trolls, nor that unicorns are either.
    • Asaf Karagila
      Asaf Karagila over 9 years
      My point is that there is a cardinal number $2$ and there is a natural number $2$ and there is a real number $2$, and despite the fact that those share common properties, when applying infinitary operations to these objects the cardinal number reveal its nature in a different way than the real number does.
    • Popopo
      Popopo over 9 years
      @AsafKaragila Okay, I'm much enlightened now. In addition, are the collection of all unicorns(named $\mathbb N$) and an infinite large troll called $\omega$ identical? It seems in one myth(ZFC) they are, whereas in another(surreal number theory) they aren't.
    • Asaf Karagila
      Asaf Karagila over 9 years
      This is why I usually make the distinction between $\mathbb N$ and $\omega$ (and I have made it and clarified it in several past questions dealing with this topic exactly). Much like $2$ has no additive inverse in $\mathbb N$; no multiplicative inverse in $\mathbb Z$; no square root in $\mathbb Q$; and only one third root in $\mathbb R$; when we specify a framework we treat the objects as objects in that framework. They don't switch around just because we know they could be thought as objects in a different land.
    • Popopo
      Popopo over 9 years
      @AsafKaragila Actually, but it seems things go different in Physics. Photon in quantum field theory behaves different from electromagnetic waves in classical field theory, but it seems physicist always think they are in the same land and both denote the same object light.
    • Asaf Karagila
      Asaf Karagila over 9 years
      @Popopo: Well whoop-dee-doo... another thing physicists don't pay attention to. Luckily for them they can confirm or refute their hypotheses in experimentations. For mathematicians it's a bit different, we cannot observe infinite sums or products in nature. One of the things which make mathematics a deductive science and not an inductive one.
    • Popopo
      Popopo over 9 years
      @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.
  • Popopo
    Popopo over 9 years
    What about $\prod_{n=0}^{\infty}x_n$ when $x_n=2$ for all $n \in \omega$? $\lim_{k \to \infty}\prod_{n=0}^{k}x_n=\omega$ but $\prod_{n=0}^{\infty}x_n=0$ in some cases...
  • Asaf Karagila
    Asaf Karagila over 9 years
    @Popopo: No. The product of real numbers is not the product of cardinals. Furthermore $\infty$ is not $\omega$. Also the product of countably many copies of $2$ is never countable (if not empty).
  • Popopo
    Popopo over 9 years
    But does $\lim_{k \to \omega}\prod_{n=0}^{k}x_n=\sup\{\prod_{n=0}^{k}x_n|k < \omega\}=\sup\{2,4,8,16,...\}=\omega$ hold?
  • Asaf Karagila
    Asaf Karagila over 9 years
    @Popopo: No!! You are taking a product of cardinals, not a product of real numbers! There can never be any real number corresponding to $\omega$. Not even a hyperreal number can correspond to $\omega$!! Read my comment to your question. Mixing up contexts is bad. This is like saying that a regular graph and a regular cardinal have something in common because both are called regular.
  • Asaf Karagila
    Asaf Karagila over 9 years
    @Popopo: And as cardinals go, the product is not continuous. This means that the limit of finite products is not the product over the infinite index. On the other hand, in ordinal arithmetic this is true, and there $2^\omega=\omega$. But those are ordinals and cardinals and neither are real numbers.
  • Popopo
    Popopo over 9 years
    Okay, I'm quite clear now, many thanks.