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This might be a bit of a basic question but my current understanding is that we cannot. Still it makes me wonder if we can propose a mapping between two countable sets why not?

For example why is this expression incorrect?

$$\sum_{i\in \mathbb{N}} i =\sum_{i \in \mathbb{Q}_+}i$$ Since our LHS and RHS are both positive infinity.

Any help is appreciated

Blue
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EconJohn
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    What does "$=$" means in this case ? This is all the question. If this means they both diverge, that is true. – Lelouch Jan 26 '22 at 21:44
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    Infinity is not a number. Infinity should be used with extreme care in statements. It is true that both expressions diverge to infinity. Before that however, it is worth noting that both expressions are identical in their meaning. – JMoravitz Jan 26 '22 at 21:45
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    Depending on what you mean by $\mathbb{N}$ and $\mathbb{Z}^+$ it would be like saying that $1+1=1$ because two water drops might merge and become one. Anyway, if the two sets are the same, you're saying nothing really meaningful. – egreg Jan 26 '22 at 21:46
  • FWIW your question reminds me why Cauchy et al introduced limits to put The Calculus on a rigorous footing. – pshmath0 Jan 26 '22 at 21:47
  • Why do you presume that that identity is incorrect? Even if we include $0$ as a natural number, it doesn’t change the value of the LHS. – Radial Arm Saw Jan 26 '22 at 21:48
  • The symbol '$\infty$' means different things in different contexts. You have to define what you mean by it, then you can see if two things are equal. Also, you've got sums, not sets. There are mappings between countable sets. – J126 Jan 26 '22 at 21:48
  • @JMoravitz Thanks. I didn't see that. I'll delete the comment. – J126 Jan 26 '22 at 21:49
  • My current understanding is that in general $$\infty-\infty \neq 0$$. Is this true? – EconJohn Jan 26 '22 at 21:51
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    I will reiterate... infinity is not a number. You ought not perform arithmetic with infinity. You ought to be extremely careful with phrasing statements correctly when talking about infinity. There is no reasonable meaning one can give to the expression $\infty-\infty=0$. Can you insist on working in a setting where you can perform arithmetic with infinity? Yes... but in such a setting many things break and certain expressions become undefined, $\infty-\infty, \frac{\infty}{\infty},\frac{0}{0}$ among others. For that reason and others you are encouraged to stay far away until more mature – JMoravitz Jan 26 '22 at 21:52
  • @JMoravitz very helpful comment. Thank you. If you make this comment an answer i will accept it – EconJohn Jan 26 '22 at 22:00
  • Whoops i made a mistake in writing Z instead of Q for the rational numbers – EconJohn Jan 26 '22 at 22:15
  • @EconJohn you mention in the same question infinity as a cardinality (words mapping and countable) and as the sum (value) of a series ($+\infty$). This hints that you may be making a confusion here. – Arnaud Jan 26 '22 at 22:22

2 Answers2

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I am assuming that, for you $\Bbb Z^+$ is $\{0,1,2,3,\ldots\}$. If that so, no, you don't have$$\sum_{i\in\Bbb N}i=\sum_{i\in\Bbb Z^+}i.\tag1$$That's so because the LHS of $(1)$ is the series$$0+1+2+3+4+\cdots,$$whereas the RHS of $(1)$ is the series$$1+2+3+4+5+\cdots$$These are two distinct series.

On the other hand, you do have$$\sum_{i=0}^\infty 2^{-i}=\sum_{i=1}^\infty i2^{-i},\tag2$$although, again you have two distinct series. But this time you have two convergent series and the sum of both sides of $(2)$ is $2$. It happens that the notation $\sum_{i\geqslant k}a_i$ is ambiguous; it is both a series and its sum (when the series converges). But there is not ambiguity in the case of $(1)$, since non of the series converges.

Note that it is not correct that both sides of $(1)$ are equal to $\infty$. What happens is that both series diverge to $\infty$.

José Carlos Santos
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  • $\mathbb{Z^+}$ refers only to positive integers. But the identity (1) is true regardless of whether we include $0$ as an element of $\mathbb{N}$ because if we add $0$ to a quantity, that does not change the value of the quantity. – Radial Arm Saw Jan 26 '22 at 22:00
  • @RadialArmSaw That is just wrong. Again: the series $0+1+2+3+4+\cdots$ is different from the series $1+2+3+4+5+\cdots$. A series and its sum are different things. If we are dealing with a convergent series $a_1+a_2+a_3+a_3+\cdots$, then yes, its sum is equal to the sum of $0+a_1+a_2+a_3+a_3+\cdots$. – José Carlos Santos Jan 26 '22 at 22:02
  • we have a sum. We are adding together the elements of two distinct sunsets. Those two sums happen to be identical. – Radial Arm Saw Jan 26 '22 at 22:05
  • @RadialArmSaw No, we don't have a sum. We have a series. It's not the same thing. – José Carlos Santos Jan 26 '22 at 22:10
  • Whoops I realized i put down Z instead of Q for rational numbers! – EconJohn Jan 26 '22 at 22:11
  • @JoséCarlosSantos ok. I’m sorry for misunderstanding. – Radial Arm Saw Jan 26 '22 at 22:13
  • @JoséCarlosSantos what is the distinction between saying $\sum^n_{i=0} i$ is a sum vs its a series? Is a series the set of intermediate sums - in this case ${0, 0+1, 0+1+2, \cdots, 0+1+2 \cdots + n-1 + n}$? – joseville Jan 26 '22 at 22:18
  • @joseville The series is a sequence of intermediate sums, while the sum is the limit of the sequence. In the case of OP, $(0,1,3,...)$ is different from $(1,3,6,...)$ even if both have the same sum – Wastaken Jan 26 '22 at 22:22
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    It is also useful to remark that it only makes sense to talk about a sum when the sequence of intermediate sums converges (in this case one can think about divergence as converging to infinity with the metric in $\mathbb{R}\cup{\infty,-\infty}$). For example it makes no sense to talk about a sum when the limit does not exist – Wastaken Jan 26 '22 at 22:34
  • @SamueleMonitto I think my confusion is the symbolism. I always thought the symbol $\sum$ always refers to a sum, but that is not the case, is it? How do we differentiate the difference symbolically if a symbol can refer to different things? – Radial Arm Saw Jan 26 '22 at 22:44
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    @RadialArmSaw The symbol refers to the sequence and not the sum, the meticulous way to write the sum when the sequence converges would be to write $lim \sum^n_{i=0} a_n$ to indicate the sum, however an abuse of notation is made and we identify the series with its sum(when it converges). In this case there is added ambiguity in a sense that it can be considered convergent or not whether you consider it a sequence in $\mathbb{R}$ or $\mathbb{R}\cup{\pm\infty}$ . Usually context is what matters in differentiating the symbols, the author usually would specify what he means in this situation. – Wastaken Jan 26 '22 at 23:25
  • In this case, to avoid the confusion, one would probably write the limit notation if he means to specify that those limits are the same. – Wastaken Jan 26 '22 at 23:35
  • @SamueleMonitto thanks for the clarification! So we can say that my confusion is the fault of the OP. Haha – Radial Arm Saw Jan 27 '22 at 00:46
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First of all, let's get the stuff about the mapping between countable sets out. The fact that we can construct a bijective mapping between countable sets means exactly that, we can send every element of one set to one and only one in the other. Note that this doesn't say that the mapping has anything to do with how it does it, it just means it's possible to do. For this reason, knowing that $\mathbb{Q}$ and $\mathbb{N}$ are in bijection tells us nothing about those series. In fact, any series is indexed by a countable set, but not all series diverge to $\infty$ !

Now to the meat of the question. I assume that you mean that both those series diverge to infinity. First of all, you should be very careful on saying what you mean by $$\sum_{i \in \mathbb{Q}_+}i$$

But rather than trying to make sense of it, which is not actually terribly important to the arguement, substitute it with any divergent series to $+\infty$ , let's say $$\sum_{n=0}^{\infty}a_n$$

Now you ask, is this expression true? $$\sum_{n=0}^{\infty}a_n=\sum_{i\in\Bbb N}i$$

The answer would be in fact yes, but not in the way you imagine it. We can consider the extended real line $\mathbb{R}\cup\{\pm\infty\}$ and in fact both of those values are the same, i.e. $+\infty$ . Great! But wait a minute, now we should be able to subtract $-\infty$ from both sides and get $+\infty-\infty=0$, right? No. The reason why this can't be done is that, any new set you define is just a set, and then you can start to define operations on this new set. For subsets of the real numbers, you can just use the operations on $\mathbb{R}$ , but this set is bigger, so we need to define a new operation so that we can say what it means to sum a number and $+\infty$, or multiply and add $+\infty$ and $-\infty$ . The sad truth, though, is that we can't really do the operations in a way that perfectly extends our operations on the reals, makes it so that every element has a multiplicative and additive inverse AND makes the infinities behave like you would expect them to. What all this means is that, yes, you can construct a system where divergent sequences(series) are convergent to $+\infty$ (using an appropriate metric space structure), but you can't also have a well defined operation that behaves like you want, so while we can say that yes, in fact, $+\infty=+\infty$ and $-\infty=-\infty$, we can't say that $\infty-\infty=0$ and $-{\infty}+{\infty}=0$ . The first expression simply means that they are equal as elements of the set, while the second would imply an operation that we haven't defined.

Wastaken
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