I will assume you actually mean a series of rationals, as series of naturals of course cannot converge unless every term beyoud a certain point is zero.Originally Posted by Nylo
If it does not converge it is not a number.
The Gregory-Leibniz series:
is an infinite series of rationals that converges to an irrational number. Every finite truncation is rational but the sum is not.
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No, I meant what I said. Can you provide any reference regarding your assertion that in a sum of naturals, if it is infinite and the terms are not zero beyond a certain point (i.e. it doesn't converge), the result cannot be claimed to belong to the set of the naturals as well? Thanks.
What are you using as a definition of the natural numbers? I have never seen a definition of the natural numbers (or integers or rationals or real numbers, for that matter) that includes "infinity". What you are asking for, essentially, is a reference to a definition of "natural numbers". Here's one:Peano axioms - Wikipedia, the free encyclopedia
All natural numbers are finite, therefore an infinite sum of non-zero naturals being greater than every natural (i.e. infinite) is not a natural number.
Which is my last post in this thread, I do not have the time to teach you elementary number theory, or maths in general.
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I was asking for a citation for this, which you insist on not providing.
I agree that a number claimed to be greater than every natural would not be a natural number, because naturals are endless by definition, so you can always find a bigger one. But I disagree that an infinite sum of non-zero naturals is greater than any natural. It is just greater than any computable one, than any finite natural. The same infinite sum plus three, for example, is also infinite, and is a greater one. You can substract them, indeed, and the result is greater than zero. It is three, as the infinite terms of the sum cancel each other in the substraction.therefore an infinite sum of non-zero naturals being greater than every natural (i.e. infinite) is not a natural number.
HallsofIvy, I've gone through the Peano axioms, as they appear in the wikipedia, three times already, and I couldn't find a single one of them that stops me from claiming that an infinite sum of naturals is a natural. Perhaps you can cite the specific paragraph that escapes me?
In fact, if we assume that an uncalculably high natural is not really a Natural, if we say like you both suggest that Natural numbers can only be finite, I have now found another strategy to be able to COUNT the INFINITE subsets of N. Which would be an even greater heresy than being able to count the reals. And it does not even use infinite sums, so it would not have the drawback that zzephod seems to have found in my recent sequence of all reals. It will take me a while to find the way to express it in a mathematical formula, but just give me a few hours.
I got it! It was easier than I thought. The following expression is a valid bijection between the set of subsets of N and N. It is only valid if we consider the premise that you have stated before: that any natural number has to be finite.
Given A, a subset of N, its position in the sequence of all subsets of N is calculated as follows:
With your premise, if any natural is finite, then the highest element in A is a finite natural, and if it is, the maximum number of naturals that your A can have is also finite (and equal to its highest natural plus 1), so the sum would be finite, and a finite sum of finite naturals is a finite natural.
So now you choose. Either all naturals are finite, and then I have found a bijection between N and the set of subsets of N, or in fact infinite naturals are allowed into the naturals set, in which case I have found a bijection between N and the Reals.
WHAT DO YOU CHOOSE?
OK let me choose for you. Infinite, at least if calculated from adding other Naturals, is included in the Naturals. It is easy to prove.
For any A non-empty subset of N with c elements, A necesarily contains at least 1 element bigger than c-2.
N is a subset of N and has infinite elements, according to Cantor himself. Therefore it has at least 1 element which is bigger than infinite - 2. Which means that such an element is also infinite and, notwithstanding, it is included in the Naturals. Therefore Naturals need to include the element infinite for it to be an infinite set.
So this disprooves what you said before, zzephod. No surprise that you didn't offer a link supporting your assertion and defended yourself as "not having to teach basic mathematics to me". Basic mathematics say that infinite is indeed included in the naturals. And arrogance is, if I don't remember wrong, one of the capital sins.
Because infinity is not a number (so it is true by definition if you like). Alternatively it can be proven using the induction axiom of the Peano axioms.
This should be within your ability to prove, so try it.But I disagree that an infinite sum of non-zero naturals is greater than any natural.
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That is a load of bollocks, and you know what our answers are likely to be since you have been through all of this before elsewhere.I got it! It was easier than I thought. The following expression is a valid bijection between the set of subsets of N and N. It is only valid if we consider the premise that you have stated before: that any natural number has to be finite.
Given A, a subset of N, its position in the sequence of all subsets of N is calculated as follows:
With your premise, if any natural is finite, then the highest element in A is a finite natural, and if it is, the maximum number of naturals that your A can have is also finite (and equal to its highest natural plus 1), so the sum would be finite, and a finite sum of finite naturals is a finite natural.
So now you choose. Either all naturals are finite, and then I have found a bijection between N and the set of subsets of N, or in fact infinite naturals are allowed into the naturals set, in which case I have found a bijection between N and the Reals.
WHAT DO YOU CHOOSE?
And that is my final final word on this thread.
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From the Peano's axioms, which define N, all Naturals have a follower. Done. You still didn't provide a citation.
Infinite is not a number, it is an idea that means "no upper limit". If you exclude infinite, you exclude this idea. You are putting a limit to the naturals. When we say that an infinite sum's result is infinite, we don't mean that infinite is its value, as infinite is not a value. We mean that it is not possible to find a finite number which is bigger than the result you get.Because infinity is not a number (so it is true by definition if you like).
OK guys, I rest my case you are right. I've been talking all the time about naturals and reals, but it seems that I was actually dealing with hypernaturals and hyperreals. Probably because they're the only thing that make sense to me. And from wikipedia, "The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets", which is precisely what we are talking about.
As I don't understand the concept of having infinite sets of naturals in which all of them are finite naturals, which I find completely flawed, but it's what the standard arithmetic relies upon, I will not try to understand Cantor. I'll just consider it a curiosity of the past.
I have no problem with that. There is nothing in set theory that says that the property of a set must be transferred or pushed onto the property of its elements. I understand intuitively why all natural numbers are individually finite whilst the entire set is infinite. But ONLY if infinity it recognized to simply be an endless process.
As soon as it's treated as a "Completed Cardinal Quantity" (which is how Cantor treats it), then it makes no sense at all. According to Cantor the infinite set of real numbers is some how 'more endless' than an already 'endless' collection of natural numbers which is an absurd concept, IMHO.
We talk about "Proof by Contradiction" or "reductio ad absurdum" (i.e. reduction to absurdity) as being a valid means of proving something in mathematics.
However, the very idea that something could be more "endless" than "endlessness itself", is already an absurd concept. Yet this is what Cantor is asking us to accept.
I totally reject this idea for many reason.
The actual concept is irrelevant in any case when we simply ask whether Cantor's diagonal "proof" using a list of decimal numerals is valid. It's not a valid proof for the reasons I had given.
Completed lists of numbers are innately rectangular by the very nature of the numeral systems being used to express them. They cannot be made to be square in any case. Yet Cantor's diagonal method of constructing a list of decimal numerals requires that his list necessarily be square. This is forced by his very method of creating the list using a diagonal line that necessarily moves to the next column with every new digit he creates.
It's impossible for Cantor to create a "completed" list of all possible numbers using that method of construction.
So any other arguments that might be given for idea that there exist "more" real numbers than natural numbers cannot depend upon Cantor's diagonal proof.
His proof is necessarily flawed.
As I've stated before, even if his conclusion could be "proved" using some other method, that still doesn't validate his "proof". All it validates is that he luckily came to the correct conclusion using a totally invalid method of reasoning.
So for me it's not even about the conclusion at all. Cantor's method of proof is simple wrong. Period.
In order for his proof to be meaningful complete numerical lists of numbers would need to be innately square. But they aren't. So his proof is no proof at all. It's based on a totally invalid assumption that he could construct a completed list of numbers using his diagonal method of construction, which simply isn't possible to do.
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