I’m going to repeat this!!!
Better yet, “Keep going indefinitely” is the same as “Stop at the largest number”.
Let (L) denote the largest number. Since (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for each (n > 0), and since (L > 0), it follows that (\sum_{i=1}^{L} \frac{9}{10^i} < 1).
No Magnus, doesn’t.
With infinity, there is no “largest number”
Then it is not a number, but a … motion of sorts. It has a dynamism to it.
Numbers neither do or don’t ‘end’, they never even begin.
Even though the range of their decimals might extend infinitely, the number doesn’t move.
But “infinity” does seem to be required to be on the move for it to be comprehensible.
Infinity ((\infty)) = a number that is greater than every integer
Largest number ((L)) = a number that is greater than every other number that one can think of
(0 < 1 < 2 < 3 < \dotso < \infty < 2\infty < 3\infty < \dotso < \infty^2 < \infty^3 < \dotso < \infty^\infty < \dotso < L)
Whatever “repeats indefinitely” cannot have more than (L) repetitions.
“Keep adding 9s and then stop at n number of 9s” simply means “Let the number of 9s be an integer”.
“Keep adding 9s and never stop” simply means “Let the number of 9s be a number greater than every integer”.
Fixed Cross
As far as I know, “indefinite” means something like “we don’t know when or if it will ever end.” Infinite means “we definitely know it won’t end.” Infinite is the property of being endless, it’s not a quantity. Quantities are the things you find on the number line. Infinity is a property of the number line itself. It is where the number line extends to (or more accurately, the property of its extension being unlimited).
It’s LaTeX.
You can see how we do it by quoting our posts and looking at in the text editor.
This board doesn’t seem to have all LaTeX features enabled though. I know the mars symbol 
can’t be posted in LaTeX.
Magnus seems to be the real LaTeX guru. He uses it even to say (n > 0). I’m not that hardcore. I’d rather just type out n > 0.
That’s a bit ambiguous, turning on what exactly is meant by “indefinitely”.
What (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for any n means is: pick any integer from the number line. You are completely unlimited in which number you pick. But it does not mean: pick infinity. And not just because infinity isn’t a number on the number line, but because substituting (\infty) for n actually means: don’t pick a value for n. Just keep adding forever.
Magnus,
Only you could be driven to say something like this. I honestly wonder if you would have said something like this before this conversation.
Persuasive.
Magnus, Magnus, Magnus… you should know my response to this by now. You might have read somewhere that, maybe, possibly, infinity is not a quantity? Ring a bell? I’m honestly more astonished at this point that you aren’t able to predict this than the fact that you actually believe your own arguments.
Well, this goes way back to a point I made much earlier in this thread (in fact, I think it was the point I jumped into this thread with). You don’t have to calculate the sum of anything. You already have the answer: (\sum_{i=1}^{\infty} \frac{9}{10^i} = 0.\dot9). All the 9s, as infinite as they are, are already there (you just can’t write them out). The only question at this point is: does (0.\dot9) = 1? And there happens to be a nice proof online that it does indeed equal 1:
X = (0.\dot9)
10X = (9.\dot9)
10X = 9 + (0.\dot9)
10X = 9 + X
9X = 9
X = 1
As you’re so fond of saying, Magnus: where’s the flaw?
Let’s not confuse our two arguments. You, like me, do not believe that 0.999… = 1.
My argument with you Magnus is about your belief in orders of infinity.
For people who belief in convergence, every whole number equals an infinite amount of convergent series (sequences) therefor, every whole number is infinite (contradiction)
That’s not my issue with you Magnus
Stop trying to conflate the two debates and make this conflation an issue between us.
You also conflate infinity as a process and completed infinities.
Basically, you are confused; inarticulate.
Just a question I’ve been dying to ask: when we say that a set consists of infinity members, how do we know it’s exactly (\infty) as opposed to (\infty) + 1 or (\infty) - 1? Or 2(\infty) or (\infty)/2?
If you agree that (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for every (n > 0), and if you agree that (\infty > 0), then it necessarily follows that (\sum_{i=1}^{\infty} \frac{9}{10^i} < 1). Everything else is irrelevant. The only condition is that (n) is greater than (0). No need to satisfy Gib’s definition of the word “quantity”.
(\sum_{i=1}^{n} \frac{9}{10^i} < 1) holds true for every (n > 0). It does not only apply to integers. It literally applies to anything greater than zero.
We declare how big it is.
When we say that a set consists of a finite number of members, how do we know it consists of exactly (5) members as opposed to (100) or (200)? Well, we don’t, unless we specify it further.
When we say that a set consists of infinite members all we can say is how relative it is to other infinite sets
So for example the infinite set of primes is a smaller set than the infinite set of integers because there are fewer primes / more integers
Also infinity plus one / infinity minus one are both still infinity for only infinity minus infinity will result in infinity actually being negated
[b]
(\infty) + I = (\infty)
(\infty) - I = (\infty)
(\infty) + (\infty) = (\infty)
(\infty) - (\infty) = 0
[/b]
(\infty) + ANY NUMBER = (\infty)
(\infty) - ANY NUMBER = (\infty)
Again, that’s not what I said.
What I said is that you need to show me the statement (S) and its negation (\neg S) that are implied by ((1, 2, 3, \dotso, 0)).
And you have to take into account this:
What exactly do you think you can prove by this?
The first expression is true if what it represents is “A number greater than every integer + 1 = a number greater than every integer”. Strictly speaking, however, this is not a mathematical expression since “A number greater than every integer” is not a specific number, but rather, a class of numbers (i.e. the number of numbers greater than every integer is greater than (1).)
But what if by (\infty) we refer to some specific infinite quantity such as “The smallest number greater than every integer”? (Note that there is only (1) such number.) In such a case, the first expression is not true because “The smallest number greater than every integer + 1 > The smallest number greater than every integer”.
And what about the second expression?
The second expression is true only if (\infty) refers to a specific number. Otherwise, it is not true. “A number greater than every integer - a number greater than every integer” is not necessarily equal to (0) in the same way that “A number greater than (3) - a number greater than (3)” is not necessarily equal to (0) (e.g. (5 - 4 = 1 \neq 0).)
Infinity cannot be defined as the smallest number greater than any integer because it is not a number as such
And adding one onto the largest integer makes that number the largest integer and it carries on ad infinitum
Where two infinities are identical the answer is 0 when they are subtracted
And so for example the infinite set of primes - the infinite set of primes = 0
Actually (\infty) - (\infty) = (\infty)
All the results can be derived from the starting equation (\infty) + I = (\infty)
when (\infty) = (\infty) then (\infty) - (\infty) = 0
when (\infty) < (\infty) then (\infty) - (\infty) = < 0
when (\infty) > (\infty) then (\infty) - (\infty) = > 0
If you accept that (\infty)+1= (\infty) then none of those three equations makes sense. That’s because they all depend on some sort of definite value/quantity for “each” infinity.
The size of an infinity is relative to how many members it has
For example the infinite set of integers is smaller than the infinite set of irrationals
So even though they are both infinite one is demonstrably larger than the other one