Is 1 = 0.999... ? Really?

So every number that is not greater than every integer is a finite number?

Infinitesimals are finite numbers?

It holds true for every quantity greater than (0). And since (\infty) is a quantity greater than zero (I hope you agree on that one), it holds true for (\infty) just as well.

You know very well what I’m talking about (or at the very least, you are supposed to know, since all it takes is a little bit of attention.)

There is no injective non-surjective function between the two sets.

Do you know what logic is?

It doesn’t work the way you are explaining it.

I have to see a tree and not see the same tree at the same time in order to prove whether the tree is there or not? That’s not logic, that’s some sort of bizarre psychosis

Yes, boggles the mind doesn’t it?!

Yes.

We never came to an agreement that (\infty) is a quantity.

Even if we did, it brings into question whether we can still say (\sum_{i=1}^{\infty} \frac{9}{10^i} < 1) since the whole reason one would agree that (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for all n > 0 is because this says: if you keep adding 9s after the decimal point but you stop at some number n of 9s, then you will still be below 1. Setting n to (\infty) says something very different. It says: if you keep adding 9s and never stop, then you will get a value that [doesn’t] equal 1. That’s a good reason to question whether it holds for any value of n when we allow (\infty) as a value.

:laughing-rolling: I’m “supposed” to know. That’s rich. Just like I was “supposed” to know that (0.\dot9) is not infinite nor finite. The fact of the matter is, your statement was wrong. There is an injective function between A and B. If there were an injective function between any two sets, it would be A and B. But if you add “non-surjective” then sure. What does that prove?

It is indeed psychosis but your own (: That’s not what I said.

The first part of the disagreements comes down to whether infinite also means indefinite.
If it does, then it is not a quantity but rather something like a condition, of a set or whatever.

In as far as the logic itself goes, unless there is some change of the rules after a definite number of decimals, Magnus’ number will always be less than 1. I have no idea how you guys are using all this code, I cant even quote it.

So: “keep adding 9s after the decimal point but you stop at some number n of 9s, then you will still be below 1” is wrong. The formula doesn’t provide for a “stop at some number”, it rather says to keep going indefinitely.

But, not to keep going indiscriminately. You have to keep going with a specific task which by definition precludes any step from altering the result of the previous step. Which is what would have to happen for 1 to be reached.

That’s exactly what you said. It’s on board record.

You said that if I can’t prove that 0.123…0 is not true AND false, then I can’t disprove it.

Absolutely. It makes no sense.

Well, they actually aren’t.

Do you agree that (\infty) is greater than every integer?

Do you agree that an infinite number of apples is more than zero apples? and more than one apple? and two apples? and three apples? and so on?

In other words, do you agree that (0 < 1 < 2 < 3 < \cdots < \infty)?

If you do, then you have to accept that, since (\sum_{i=1}^{n} \frac{9}{10^i} < 1) for every (n > 0) and since (\infty > 0), that (\sum_{i=1}^{\infty} \frac{9}{10^i} < 1).

How do you calculate the result of an infinite sum that never stops?

Magnus!!! Honestly dude!!!

Infinity is not a number!!

It is not greater than or less than an integer!!!

You are so fucking confused on so many levels

“Keep going indefinitely” in the context of (0.\dot9) is the same as “Stop at infinity”.

No it doesn’t! It means “it is an infinity, that which never ends”.

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 :male_sign: 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.