Summation of S(N): Infinite/Finite?

[moving finger]

The set S(N) of all natural numbers is generally believed to have infinite cardinality (ie S(N) has an infinite number of members) and yet every member of the set is believed to be finite. Infinite natural numbers are by convention "not allowed".

This leads to a contradiction, as follows :
The set S(N) of natural numbers is closed under the operation of addition, which basically means that the sum of any two (or more) natural numbers is also a natural number, and must be included in the set S(N).
Therefore, the arithmetic sum of ALL members of the set S(N) must also be a natural number, included in S(N).
However, if S(N) has infinite cardinality (as is generally supposed), then it follows that the process of summation of all of the members of S(N) (summation of an infinite number of finite natural numbers, all but one of which is greater than or equal to 1) must produce an infinite result. This would mean that the summation of all the members of S(N) is an infinite natural number - a contradiction.
On the other hand, if we insist that the result of summation of all of the members of S(N) must produce a finite natural number, then this can only come about if there are a finite number of members of S(N), ie S(N) has finite (not infinite) cardinality.

 

[matt grime]

No, you see, the rules of algebra do not allow you to conclude that the naturals are closed under infinite sums. They aren't. They are closed under adding two summands, and hence by induction on adding any finite number of summands. Infinite summands are not mentioned. We have repeatedly told you that you do not know what you're talking about, please try and accept that and learn. Had you been more open to new ideas rather than simply spouting your own beliefs and dismissing the knowledge of those more informed than you, you might not have got me quite so dismissive of your posts, and I may have replied in a friendlier tone.

 

[Gokul43201]

Here's your mistake (emphasis added)

Therefore, the arithmetic sum of ALL members of the set S(N) must also be a natural number, included in S(N).

 

[moving finger]

No, you see, the rules of algebra do not allow you to conclude that the naturals are closed under infinite sums. They aren't. They are closed under adding two summands, and hence by induction on adding any finite number of summands. Infinite summands are not mentioned. We have repeatedly told you that you do not know what you're talking about, please try and accept that and learn. Had you been more open to new ideas rather than simply spouting your own beliefs and dismissing the knowledge of those more informed than you, you might not have got me quite so dismissive of your posts, and I may have replied in a friendlier tone.

Thank you, Matt for your explanation.
I readily agree that I do not understand some of the concepts involved, but that is precisely why I am here to learn, and I appreciate you taking the time to discuss these things. The textbooks I have consulted gloss over these issues and do not discuss them. I apologise for the tone of my earlier posts.

Could you please elaborate further?

Does the above mean that
a) it is simply "not allowed" in algebra to do the summation over the entire S(N), the set of all naturals?
or does it mean
b) that this summation is "allowed", but the result of this summation over the entire S(N) is not a natural number?
or does it mean
c) something else entirely?

If (a), why is the summation not allowed?
If (b), what "is" the resulting sum, if it is not a natural?

 

[Euclid]

it is simply "not allowed" in algebra to do the summation over the entire S(N), the set of all naturals?
___________
The summation is not defined. You could, I suppose, define it however you like, but that it not conventional.
I have seen one book define infinity as a 'number' that is greater than every other, but I believe that was just for notational purposes. I don't remember which book this was (in this case, it was for writing the interval ($$a,\infty)$$)). You don't need this definition though (you can define ($$a,\infty$$)$$=\{x|a<x\}$$ without calling infinity a number), so I'm not sure why this particular book did this. Even in this case, though, the sum is still not defined, but you could say that any sum of numbers that is not bounded above is equal to infinity. My impression is that mathematicians like to avoid this talk, and instead this simply say the sum diverges.
In order to talk about infinite sums, you will have to use the notion of limit. See a calculus book for that.

 

[matt grime]

It is "not allowed" as it is not an operation defined on the naturals, as Euclid says. In fact one can show that the sum of an infinite number of naturals is a natural exactly when only a *finite* number of the summands are non-zero (assuming zero to be in our naturals). Infinity isn't a number, but we can abuse it to talk about finite things that grow without bound.

Let A and B be 3x2 and 2x4 matrices respectively. Then AB makes sense, BA does not even though I can write the expression BA. Just because you can talk about something doesn't mean it actually has meaning.

So, the natural numbers is the smallest set containing 1 and closed under adding finitely many summands. It contains no "infinite elements"; the set is infinite; all representations of any element in base 10, say, contain only a finite number of non-zero digits. If you write, as you have done in another thread, 11111... then you are not talking about a natural number.

Why should the resulting sum be anything? If it is "anything" then what ever it is is not a natural.

You can define extensions, larger objects, where that sum is meaningful but they aren't the naturals, ok, not the naturals. And there are infinitely many ways to define strictly different objects where that sum does make sense, and they are called the p-adic integers.

 

[Euclid]

One reason why you might not want to call infinity a number is that this would take away the familiar structure of the real numbers. This is because if infinity were a number, you would have to define addition and multiplication on it. What would would be the additive inverse of $$\infty$$? What about its multiplicative inverse? I don't think you could come up with anything that would save the structure.

 

[HallsofIvy]

We DEFINE the sum of TWO numbers as part of the basic structure of any number system. We then define "a+ b+ c" as (a+ b)+ c doing two sums of two numbers, define "a+ b+ c+ d" as (a+b+c)+ d, etc. That allows us to define the sum of any FINITE sequence of numbers, not an infinite sequence of numbers.

We can, as part of calculus, define an infinite sum in terms of limits IF the limit exists which turns out to be true only if the terms go to 0. It you try to do that with an infinite sequence of integers, the limit does not exist because a sequence of integers can't go to 0.