Is 1 = 0.999... ? Really?

Yeah!

Oh, you guys are going to love this!

So, I was using my notes in the last two posts, and I realized a contradiction in convergence theory:

Get this!

So you have a number like 0.9… that converges to the number 1!

Here’s the deal. 0.09… is converging to 1 as well.
Meaning: 0.199… 0.009… is converging to 1 as well…

Meaning 0.01999…

What this means is that you end up with the “infinitesimal” before the 9’s necessarily have to start.

This means that as you count through the convergence, you either have infinitesimal 1 before the 9’s or you have the 9’s equaling 1.111…

Dammit! Ok, this is really important !!!

For 0.9… to equal 1, the solution is 3.1…!!!

Like I said before…

0.9… =1

That means that 0.01999… equals 0.2999…

0.2999… equals 3.1111…

I’m getting there! Let me work through this process!

If 1=0.9…

Then 0.009… = 0.01

Which means that 0.9… equals 1.1…

When you add those up,

You get 2.000…

That’s it!

That’s the disproof!

So what I’m trying to explain here, is that in the proof that 0.999… equals 1; ALL nines repeating always equal 1, which is why I say that the solution to 0.999 actually equals, not just 1, but 1.111… (unless you have an infinitesimal which is impossible (the one never gets expressed)

As I stated earlier, when you carry from the right to left, that makes 0.999… equal 2, with no infinitesimal to flip the 9’s, you’re still left with 1.999… which equals 2.

You want to get at the truth?

Work with me here.

I know I’m close.

Because an infinitesimal is impossible, you have to carry from left to right.

This causes all the 9’s to eventually be 1’s… which not equal to 1 but is equal to 1.1…

1.1… + .999… = 2

Some clarifications:

If you have an infinite line of people standing in front of you, and you remove the person standing at the end of the line (or at the beginning of the line, it’s the same thing), you’d have a shorter line of people standing in front of you. So even though the queue is a ray, and not a line segment, you can still talk about its length.

It does not. That’s because the symbol already has a meaning, one that has nothing to do with quantities. Even if, instead of using a known symbol, you used an unknown symbol to which you assigned no meaning, we’d still be talking about a symbol that does not represent a quantity.

The symbol (\infty) is by definition a quantity albeit not the kind of quantity we’re used to. By definition, (\infty) is a number greater than every real number which is why we consider expressions such as (\infty > 1) as being true. You can’t do something like that with your :male_sign: symbol. Indeed, the symbol has no defined relation to other numbers. Does (:male_sign: > 1)? We can’t know, the result is undefined.

The problem occurs with equations such as (\infty + 1 = \infty). True or false?

If what that equation means is “If you take some infinite quantity and add one to it, you’d get an infinite quantity as a result” then the equation is TRUE.

However, if what that equation means is “If you take some infinite quantity and add one to it, you’d get the same infinite quantity as a result”, then the equation is FALSE.

In the first case, the symbol of infinity represents ANY INFINITE QUANTITY IN GENERAL. In the second case, the symbol of infinity represents THE SAME INFINITE QUANTITY WHEREVER IT OCCURS.

The meaning of the symbol of infinity is what has to be well-defined in order to avoid making pseudo-proofs such as this one:

$$
s = 9 + \underline{0.\dot9}\
s \div 10 = \underline{0.\dot9}\
s - s \div 10 = 9 + 0.\dot9 - 0.\dot9 = 9\
s \times (1 - 1 \div 10) = 9\
s \times 0.9 = 9\
s = 9 \div 0.9\
s = 10\
$$

((9 + \underline{0.\dot9}) \div 10 \neq \underline{0.\dot9})

The two underlined numbers aren’t the same: the two infinite sums don’t have the same infinite number of non-zero terms (or in plain terms, they don’t have the same infinite number of 9’s.)

That’s why (0.\dot9 - 0.\dot9) in the third equation of the proof does not equal to (0).

The infinitesimal quantity has conveniently disappeared.

Well, geez, man, not sure how else to put it. “Fewer” implies finitude. It implies an actual quantity relative to a larger quantity. I shouldn’t need proof of this, that’s just the common definition. If you need proof, what you’re really asking is: why should I accept your definition when I have my customized one that fits with my theories on infinity. And to that I say: fine, we can go with either definition, but then you gotta help me understand your definition.

Let’s try this: imagine the scenario I described earlier, the one with two infinite parallel lines. For all intents and purposes, the same length. Now remove every odd point from one of the lines. Then move all remaining point into the spots left behind by the points you removed. According to you, the line with the points removed is now “shorter”. But since we moved all points into the spots left empty from the points we removed, the lines are perfectly identical. It’s as if we didn’t remove any points at all. We’re back to the initial state of the scenario. So here’s your chance to shine. Help me understand how the lines are different now. Help me understand what it means that the line we removed points from is now “shorter”.

You are, once again, dismissing counter-examples. Nothing wrong with following a string of logic to arrive at certain conclusions, but when someone shows you how counter-examples can be drawn in special case (like when the number is infinite), you can do two things: 1) dismiss it and just reiterate your original logic, or 2) consider the counter-example and rethink your original logic. You can then show how your original logic still holds and give an explanation for why, or you can concede that your original logic doesn’t actually hold in the special case of the counter-example. Seems to me like you’re choosing 1).

Unless we’re talking about Anderson’s magical fantasy world of shorter and longer infinities, this thread is about reality. The only way confusion would arise is if we insist on having contradictions between the concepts and the empirical evidence while still maintaining the concepts represent reality. If I say the concept of a flat world implies that you’d fall off the edge of the world if you walked far enough, and empirical evidence shows that you don’t, you’d only be confused if you insisted your flat-earth concept is still reality. The alternative is to accept that the concept is wrong. You can still say that the concept of a flat-earth implies that you’d fall off the edge if you walked far enough, but what good does holding onto that concept do if the empirical evidence shows that it’s wrong?

Ugh Ok, let’s try this. Infinity means no end. That means surpassing all numbers. As soon as you reach a particular number, you know you’re not at infinity because otherwise you could say you’ve reached the end of all numbers, and you know that just doesn’t make sense (right?). An infinite quantity just means beyond quantity. ← That’s why it’s not a quantity. Give me any quantity, any quantity at all, and I will show you it’s not infinity.

So then tell me, how many gwackometers is 1cm?

This should be good.

I am often amazed at what people need proof of.

Glad we got that cleared up.

So (\infty) stands for: an infinite number of Xs (organisms, train carts, points in line, whatever) where X is the unit, correct? Then I forbid you to use it in arithmetic. If (\infty) was the unit (as in 2(\infty) meaning 2 infinities), then I’d say run wild, have fun, but you’re not saying that. You’re saying you want to take the quantitative value of (\infty) and multiply it by 2, leaving whatever there is an infinity of as the unit, and to that I say: STOP IT! slaps hand

No, no, no, you’re just making the unit implicit. I mean, really get rid of it, as in you just have the number 13, not 13 centimeters, just 13. But this is a non-issue as I see above you don’t mean to get rid of units entirely but simply use whatever object there is an infinity of as the unit. Again, I forbid you!!! ominus voice echoing

You’re insisting that because symbols serve to represent something, we know what those are. But I’m saying that the whole point of variables (which are symbols) in algebra is to have a way of doing arithmetic with unknowns. You can know them, but you don’t have to. I was saying that if you want to use (\infty) as a variable, it has to stand for an unknown because the actual value of infinity is off limits.

I know I sound like an incomrehensible crazy person…

I’m working again on the 0.999… /= 1 argument, and if as Magnus states, silhouette is the “ecmandu whisperer”…

It’s a lot to ask silhouette, but can you help me disprove your proof that 0.999… does equal 1!!!

My argument is this:

The number 1 never occurs with the infinitesimal argument … thus, the 0.999… never falls like dominoes from right to left to create the whole number from the infinite sequence. Are we good so far??

So next, this means the only other option is to carry from left to right.

This means that you have to add 0.999… to 0.111…

The first time you do this you get 1.0999… and 1.0111…

The second time you do this, you get 2.10999…

See where I’m going with this sil???

Like I stated before, if you have no infinitesimal and the only other option is to carry from left to right, you have no equality for 1=0.999…

So, Silhouette! I’m asking you to “man up” here, and consider my argument is true.

So we have two infinite parallel lines that are of the same length. We pick one of them and remove every odd inch from it. Then we fill in the gaps that we created using remaining inches. By doing this, the gaps disappear leaving the two lines looking perfectly identical.

The problem is that there isn’t enough inches remaining to fill in the gaps without creating new gaps elsewhere. This illusion is created by moving the gaps out of our sight.

If you don’t see it, it’s not there.

And if you keep pushing things out of your sight, you can keep reassuring yourself they don’t exist.
Especially if this process is an infinite one (:

Here’s the line we started with:
( \bullet \bullet \bullet \bullet \bullet \bullet \cdots )

Now here’s the line with odd inches removed from it:
( \circ \bullet \circ \bullet \circ \bullet \cdots )

There’s an infinite number of inches out of our sight. We don’t see them, we merely see the ellipsis “…” which tells us there is more to this line than what we see. What we want to do now is take three inches from the remaining inches that we don’t see, so that we can fill in the gaps. We can do that, because there’s still an infinite number of inches remaining, so we do that and we get:

( \bullet \bullet \bullet \bullet \bullet \bullet \cdots )

Voila! The line looks like the original one! They now appear to be identical! But what happened to those gaps? Where did they go? Well, they went out of our sight. They didn’t magically vanish. We don’t know exactly where they went, but they are somewhere out of our sight.

So the lines aren’t really identical. They merely look like they are.

The gaps can’t magically vanish. The only thing we can do is push them out of our sight forever thereby creating an illusion that the two lines are identical.

They are not.

This “paradox” is known as Hilbert’s paradox of the Grand Hotel:
en.wikipedia.org/wiki/Hilbert%2 … rand_Hotel

I think Carleas mentioned it somewhere at the beginning of this thread (40-50 pages ago . . .)

Magnus, a paradox doesn’t mean that you’re right or wrong … have you disproved the paradox?

No.

So let’s look at the two gods argument again. A god is exactly half as omniscient as another god. Can that occur?

What does it mean to know half of everything if everything is infinite?

What about everything * 2?

Do you see why so many of us are debating you yet???

What’s everything * 2?

You mention contradictions frequently, but have an absolute blind spot to this very basic contradiction.

That’s why everyone else here is arguing against you.

That’s what you’re doing but you’re not recognizing it.

Allow me to repeat myself:

Whatever the quantity, if you subtract one from it, the quantity must change.

Suppose there is an infinite line of people somewhere in the universe and that YOU are one of the people waiting in it.

Suppose now that I take you by your hand, remove you from the line and place you somewhere outside of it.

The line is the same as it was before except that you’re no longer part of it. Noone joined the line, noone left it – except for you.

If you say that the number of people waiting in that line is the same as before, it either means that I didn’t really took you out of that line (that you’re still there) or that I did but that someone else joined it. Both are contradictions.

You never addressed this argument.

Magnus! If I subtracted you, you’d cease to exist. You would not be part of “another set”. You’d have no bearing on the infinity.

You aren’t following, Ecmandu.

I’ve addressed this argument before. Basically, the word “infinite” does not mean “the largest number possible” (let alone “everything”.) The word “infinite” means “endless”.

So “everything” by you! Doesn’t mean “endlesss”?

Keep going! I’m curious.

Not at all. And note that this isn’t merely “by me”.

To subtract does not mean to remove from existence. It means to remove from something where something can literally be anything.

Lol

So, you think that endless doesn’t imply everything, and that everything doesn’t imply endless.

Keep going.

“Everything” means “every element of some set” where “some set” can be literally any set.

E.g. “Everything you want” means “Every element of the set of things that you want”.

The set can be finite or infinite. It does not matter.

“Infinite” merely means “without an end”.

You can have a set made out of five infinite sets and the entire set would be more than any one of its five infinite sets.

You could ask “Which one of the sets do you want, Max?” And I could say “I want everything”. That would mean “Give me all five infinite sets, Sir”. In that particular context, “everything” would be representing five infinite sets. A finite number of sets each containing an infinite number of things.

Ok, that’s interesting.

So what if I say, “everything that exists throughout all existence, the cosmos”

And then a guy like you comes along and says, “every Tickle me Elmo that exists throughout existence an all the cosmos”

They are not equalities are they?

So why try to put them forward as such?