I think “is” may be ambiguous in this context. I would say that natural numbers can be expressed as the sum of an infinite series. I think that is synonymous with the mathematical ‘is’. (Aside: my understanding is that category theory is an attempt to rebuild math without using the ‘=’ operator (or without using it the way it’s traditionally used. Unfortunately, I don’t understand it any further than that, and I don’t know its implications for questions like these.)
You need a contradiction to create a proof by contradiction. You’ve assumed that a finite number can’t be expressed as the sum of an infinite series, and I agree that if we take that as a given, we could use it to create a proof by contradiction. But we don’t have to take that as a given: there is no contradiction between something being both a finite number and the sum of an infinite series.
This assumes something that isn’t true for uncountably infinite sets. You can’t list them, even in theory. There is no next largest real number.
Before you answer my last post. Consider that I pointed out that the best English word for you in argument 1 was “represents”
It’s the same in argument 2. I misspoke and said “substitutes”. However, if I use the word “represents” as well, it changes the meaning and ability to refute it substantially
If (i) must start with (1), end with (n) and increase by (1), then (2.5) is not a valid value for (n).
On the other hand, a sum of natural numbers (1 + 2 + 3 + \cdots + n) is equal to (\frac{n(n+1)}{2}). If we plug (2.5) into that, we get (4.375) as a result. We might also be able to find a sum with an index that starts with certain number, ends with (2.5) and increases by certain value.
Have you considered limiting (n) to numbers that have no fractional component but wihtout limiting it to integers?
But you want to put (\infty) in there. So let’s talk about (\infty) as a fuzzy blob of impossibly big numbers, not as limitlessness, as the spectrum of numbers you would find if you transcended the limits of the limitless.
Why would you break the pattern? (Assuming you know what I mean.) Obviously, if you break the pattern, you might not get a number that is less than (1).
In the case of (\sum_{i=1}^{n}), (i) starts with (1), increases by (1) and ends with (n). The number of terms is (n), so the sum stops (is complete) after (n) number of terms.
In the case of (\sum_{i=1}^{\infty}), (i) starts with (1), increases by (1) and does not end. The fact that (i) does not end tells us that the number of terms is (\infty). This means the sum stops (is complete) after an infinite number of terms. (I assume you’re one of the people in this thread who have no problem with the concept of “actual or completed infinity”.)
(\sum_{i=1}^{\infty}) does not mean that (i) ends with (\infty). In other words, (\sum_{i=1}^{\infty} i) is not equal to something like (1 + 2 + 3 + \cdots + \infty). (Note that such a sum would have more than (\infty) terms.)
The value of (i) is always a natural number. So if we are asking a question such as “What’s the value of the sum of terms whose index is (i, 1 \leq i \leq x)?” then (x) cannot be anything other than a natural number since the range of (x), in such a case, must be the range of (i) – and this means that (x) can’t be (\infty). But if we’re asking a question such as “What’s the value of the sum after (x) number of terms?” then the range of (x) goes from (1) to the number of terms of the sum. If the number of terms is infinite, then (x) can be (\infty). And it is precisely this question that we’re asking.
What is the value of the sum (\sum_{i=1}^{\infty} \frac{9}{10^i}) after an infinite number of terms?
You’re saying it’s (1), I am saying it’s less than (1).
My argument (which is basically James’s argument) is that the pattern of this sum prohibits its value after (x) number of terms to be equal to (1) for any (x > 0). (You can limit the value of (x) to numbers that have no fractional component, if you want.) Since (\infty) is greater than (0), it applies to (\infty) as well.
By what logic does the value of this sum become (1) after an infinite number of terms?
This is what I was trying to explain to you before.
If you halve 1, it’s 1/2 + 1/2
If you halve THAT!
It’s 1/4+1/4+1/4+1/4. Etc…
If you keeps doing this:
At infinity (convergence)
EVERY!!! Whole number solves as zero.
I still think you are confusing all your arguments. You switch back and forth when it suits you without understanding (or caring) about the implications.
You quoted me. I guess it’s somehow relevant to what I said in that quote.
(\frac{1}{\infty}) is a number greater than (0) but smaller than every number of the form (\frac{1}{n}, n \in N). That’s what it means (so it’s not really meaningless.)
My argument proves that when numbers converge at infinity (and in saying this, infinity is NOT A NUMBER!)
That 1=0.
Thus, infinities do not converge.
All you did was change infinity to “lowest possible ‘number’ that’s not equal to zero, which by my argument, makes every whole number equal to “the lowest possible number not equal to zero” which is still a contradiction.