Is 1 = 0.999... ? Really?

Magnus already debunked those methods in this thread.

Magnus,

To claim victory, you have to actually address the actual argument.

Honestly, I know this thread has moved fast lately, but if you read the last two pages, and had any intellectual integrity, you’d notice that my arguments in particular are crushing, and you would have addressed them head on.

I consider this you conceding the debate; throwing the towel in

I suppose what you want to say is that the red rectangle is 9 rows higher than the green one. That’s true. Why didn’t I mention that? Because it was unnecessary to do so. You’re implying that it was but without bothering to explain why.

Here’s an updated image:

Compare the purple rectangle with the green one. They are equal in height but they differ in width. The purple rectangle has one term less than the green rectangle.

Perhaps you want to argue that the red rectangle and the purple rectangle do not represent the same value? If so, explain why.

But honestly, I don’t really think you found a flaw. You’re merely posturing.

What do you mean when you say “infinities are undefined”? What does that mean? As far as I know, the concept of infinity is well defined. It means “endless”.

The red rectangle is (10) times the green rectangle without the first term (which is (0.9)).

They don’t. That’s the point.

That would be you.

Remember, you need to show me the flaw. Where’s the flaw? Show it.

Sure, you’re a good-natured person. You have no flaws. Other people do, even though you can’t point them out (:

Maybe you should listen to your own advice.

That’s exactly what you have to do. And it’s not merely a supeficial appearance. Doesn’t matter how many times you say it.

The point is that if (0.\dot9) in (9.\dot9) and (0.\dot9) on its own have the same infinite number of non-zero terms (or non-zero digits, in plain terms) then (9.\dot9 \div 10) doesn’t give you (0.\dot9). Why is this so? Because (9.\dot9) has all of the non-zero digits (or terms) that (0.\dot9) has + one more (which is the (9) that comes before the decimal point.) So when you divide it by (10), and shift all of the digits to the right, you don’t get (0.\dot9), because the result has one non-zero term (or digit) more than (0.\dot9) does.

Don’t be silly.

Not all of it.

And that means exactly what? What does “contradictions presented by the intrinsically indefinite nature of infinity” mean?

All in all, you have no arguments. Around 80% of your post has nothing to do with pointing out flaws in other people’s arguments and everything to do with your frustration and vanity.

Magnus,

I’m actually getting pissed at you.

I’m not even going to ask you if an infinite orange exists (and how absurd that is - even though you say all infinities can be quantified! - let alone 2 infinite oranges!!)

I’m going to approach your fucking “dot argument”

If you take an infinite set and you remove every other quantity… you state that this makes 2 infinities!!

ACTUALLY!!! It makes (2) 1/2 infinities!!! Nothing was added or subtracted!!!

Answer me this!!

Because in not replying to messages that address your points, and then repeating the same shit, I’m starting to think you’re a troll to your own sentences!

If we take an infinite sequence of oranges (which you seem to be fan of) and remove every other, we get two smaller infinite sets of oranges. The resulting infinite sets are half the size the original infinite set.

Basically, (\infty = 2 \times \frac{\infty}{2}) where (\infty) represents the same infinite quantity wherever it occurs.

What exactly is your argument?

What do you mean “nothing was added or subtracted”?

Finally, you addressed me on this, thank you!

The issue is this:

Let’s say I have “1”

Let’s say I divide “1” in half.

As far as “1” is concerned, nothing was added or subtracted!!! 1/2+1/2 still equals 1!

You can’t then say “well now there are (2) “1’s”, which is exactly what you’re doing!

No!, even using your logic that infinity CAN be quantified, I can disprove you, by merely asserting that nothing can be added or subtracted to the initial infinity.

Let’s look at this with subtraction:

You say that infinity minus one has changed the nature of the infinity: not true.

Just like 1/2 infinity doesn’t make 2 infinities, 1 minus infinity doesn’t make 2 quantities.

Man, I hope you understood that!

You have an orange. You use a knife and cut it into two halves. What do you have now? You don’t have two oranges, you have two halves.

The same applies to infinite sets. You have an infinite set e.g. an infinitely divisible orange. You use a knife and cut it into two halves. What do you have? You don’t have two infinite sets that are the same size as the original set, you have two infinite sets of smaller size.

Exactly, you left a part out though…

Yes, you have (2) HALVES!!

But! You also STILL have 1 orange!

You haven’t actually DOUBLED anything!

You’ve HALVED something that’s still there.

We didn’t double anything because we weren’t multiplying by 2 but dividing by 2.

What’s your point?

My point is that if there is a highest order of infinity, everything is a fraction and doubling or even adding is impossible

Since there seems to be a lull right now, I’m going to clarify this:

The highest order of infinity is my cheat:

1.) rational number
2.) uncounted number
3.) different rational number
4.) different uncounted number

Etc…

If you “ divide” it in half, you’ve divided NOTHING from the TOTAL system!

If you subtract one element from it, you’ve subtracted NOTHING from the TOTAL system!

You cannot add to it (my cheat), because my cheat is the highest order infinity.

No matter how many times you divide an infinite orange, the infinite orange still exists in the form of all its pieces, you’ve divided nothing! It’s still all there.

Ecmandu

Nonsense! You’re building zombie universes to save us all!

Hey, I still gotta few tricks up my sleeve.

Am I entrapped by Magnus… or is he entrapped by me? BWAHAHAHA!!!

Uh, er… oh, I get it! :smiley:

Magnus,

Idunno man, the argument they’re trying to make kinda hinges on their not understanding it… but I am admittedly speculating.

With the exception of your statement that any of the partial sums is equivalent to (0.\dot9), this makes good sense. But that statement is the catch. I assume that by “partial sum” you mean a finite number of terms, which–by definition–cannot be equivalent to (0.\dot9). And no, you don’t get to carry a fact about the partial sums over to the full sum.

Glad you understand that, but most equalist deniers’ argument seems to hinge on temprality. So what did you mean by “approaches” (remember, you said it was the same as “builds up to”)?

Well, we come back to the crux of my issue. I need to understand what you mean by “size”.

Honestly, I had to read that several times over to get what you’re saying. But I think I get the gist. I think your argument revolves around the answer to this question: “But then what’s at the end of the other line? More line?” You seem to be saying, with respect to the first line, yes, there’s more line at the end of the line (that’s this: (\bullet \bullet \bullet \cdots + \bullet \bullet \bullet \cdots)). And with respect to the second line, no, there’s just gaps (because they got pushed here).

But I wonder how you arrive at that answer: yes, there’s more line. I think we can both agree that, initially, both lines are infinite, that they don’t have an end. But if we are gonna talk about the infinite distance to which each line extends as the line’s “end”, we have to say: that’s where the lines end. The line’s “end” is, after all, just a metaphor for the infinity that results from the line not having an end. So as far as the metaphor goes, the line’s “end” is where the line terminates–at infinity, so to speak–but as far as the literal facts go, there is no end. In either case, your diagram ((\bullet \bullet \bullet \cdots + \bullet \bullet \bullet)) is a poor representation.

A better argument might have been that it would take an eternity for the gaps to reach the line’s end, so the operation of filling all the gaps can’t actually be done. That’s why I brought up the point about taking turns vs. simultaneous movement, to which you said:

In what sense are they pushed out in this scenario? We could talk about the points being pushed out in the previous scenario because there was the illusion of an ever growing gap moving along the line towards infinity (at infinity, it gets “pushed out”). But in this scenario, the illusion is different. If we imagine that the second point moves twice as fast as the first, then they both start moving and end moving at the same time. Same if we imagine the third point moving three times as fast as the first, and the fourth point four times as fast, and so on. The illusion that arises from this is that of each gap, begining as 1 point wide, all closing at the same rate, and all moving towards the front of the line (not the end way off at infinity). The first gap doesn’t move (because it’s at the front of the line already), the second gap moves as fast as the first point (because it’s bound at one end by it), the third gap as fast as the second point, the fourth gap as fast as the third point, and so on. They all converge to the front of the line and vanish before they get there. ← So in what sense do the gaps get “pushed out”?

All this notwithstanding, I still don’t understand your definition of “shorter” and “longer”. I said that your answer to my question about what the difference between the lines is would explain your definitions, so if we put a pin in the above disagreements for a sec, I guess I can conclude that the difference, you’re saying, is that the first line has “more line” at the end whereas the second just has “gap”. So what it would mean for the second line to be “shorter” is something much like what it means in the conventional sense–that the line terminates before the other line (which you denied, btw)–but this termination is an infinite distance away. The other line, on the other hand, continues on at infinity… to… what?.. second infinity?

I’d have to presume that this was true of the second line before we removed the points, and when we moved the remaining points to fill the gaps, they were shifted from second infinity to the first, thereby leaving nothing but gap. But what about third infinity? Would the points in third infinity come rushing in to fill the gap in second infinity? Or is second infinity where the lines really end? :smiley:

In any case, the lines definitely look the same in first infinity.

Um… Wow! You literally just reiterated your point and completely ignored my question. Yeah, the lines would be different if you just removed every odd point and left it at that. D’Uh! You’d have gaps all through it. It would be like swiss cheese. But I’ll ask again very slowly… what… happened… to… the… step… of… moving… the… remaining… points… to… fill… the… gaps…[question mark]. ← I love how you construe this (necessary) step as “going further” and saying there’s no need for it. Obviously, if you think this results in a contradiction, then maaaybe there was a need for it after all. Think about it, Mags, think about it.

Honestly, I don’t see the difference.

Yes, that’s why I said “which granted, you are”. But when you don’t, you’re just reiterating your initial argument. You’re like this guy:

[youtube]http://www.youtube.com/watch?v=4xgx4k83zzc[/youtube]

Look, Magpie, we’ve both been addressing each other’s points, leveling counter-arguments, pointing out each other’s flaws, but of course, neither of us wants to agree with the other or concede that the other is making some valid points. But only one of us is confusing that for a lack of addressing points, leveling counter-arguments, and pointing out of flaws. If I point to a flaw in your argument, and you disagree with me, that doesn’t mean I haven’t pointed out any flaws, it just means it wasn’t a decisive blow to your argument (or to your persistence in the argument).

I wasn’t loved enough as child. :crying-green:

Oh, well then by all means, let’s all stand down and put this thread to rest. :smiley:

Ahh… gib, zombie universes are so last year!

I’m doing hyper dimensional mirror realities now.

You know, it’s funny, I put my whole soul on the line for this!

I have no choice (if you knew what I knew, you’d have no choice either) so it’s not nor should be seen as a sacrifice.

But man, the spirits you dredge up out of existence doing this line of work would scare the shit out of anyone!

Let me expand upon this.

The two lines look completely identical but they aren’t truly completely identical. They are identical in some ways. They are identical in the sense that they are both infinite/endless and in the sense that they have no gaps (they are made out of the same elements.) But they are not equal in size.

(A = {1, 2, 3, …}) and (B = {1, 2, 3, …}) are two identical sets in the sense that they are both infinite/endless and that they both have the same elements (every element present in (A) is also present in (B) and vice versa.) But they are not necessarily equal in size.

Given any two infinite sets, you cannot determine whether they are equal in size or not by looking at their elements.

Even though (A) and (B) have the same exact elements, you can say that (A) is twice the size of (B). There is nothing wrong with that. This can be easily represented using the following relation (A \mapsto B):

$$
1 \mapsto 1 \
3 \mapsto 2 \
5 \mapsto 3 \
\cdots
$$

Every member of (B) is associated with exactly one element from (A) but the reverse is not true – there are members of (A) that are not associated with any element from (B). So (A) has more elements than (B).

You can think of it as if (A) is made out of two infinite sets the same size as (B) with one containing even numbers and the other odd numbers.

Once you start performing operations on these sets, you must remain consistent. You can’t double the size of (B) and then say it’s the same size as before. That’s nonsense.

I think you have to stop using the word “size”. James spoke of 'degrees of infinity".

I don’t posture.
Let me show you how hypocritical this example still is:

Your objection to (\frac{9.\dot9}{10}=0.\dot9) is that at some “end” to the infinite recursion of 9s there’s a spare 9 for (0.\dot9) that doesn’t match to (9.\dot9)

Yet somehow for your visual display of (0.\dot9)s, starting one “1 decimal place” after the other makes it shorter. Here the ends are equal, but previously they weren’t.

Convenient how you can judge sizes arbitrarily to fit your point, no?
It’s almost as though indefinites can give you and Hilbert any answer you want… 8-[

You even prove this yourself through your example:

You can arbitrarily match one-to-one correspondence however you like so that A can be any “size” larger or smaller than B and vice versa.

It’s almost as if infinite means in-finite i.e. not bounded i.e. you can’t bound any specific size to it.

I mean Jesus, man. If you wanna do all my work for me in proving yourself wrong - go for it. I’m sick of you endlessly trying to pretend my reasoning is at fault.

For the bijection of (\frac{9.\dot9}{10}) and (0.\dot9), you insist on arbitrarily corresponding different numerical positions just to fit your point when you could do it another way because you naively think dividing by 10 literally does nothing more than shifting digits.

You can’t define the infinite because infinite means boundless and define means to give bounds - I’ve said this so many times and you’ve not once addressed the obviousness of this explanation, and you still insist I’m just “telling” you what to think as if the explanatory logic behind this obvious fact was merely some kind of subjective demand.

And you actually think I’m being silly about the extensive proof that 1+1=2
Yet again you prove that you have absolutely no idea about the domain you’re trespassing upon here.

It’s about bloody time!

Sounds like a bad acid trip.

Magnus,

What you just did seems really bizarre. You just took two identical sets and said: you know what… only the odd numbers from A map to B. And since every odd number in A maps to a number in B, it’s really the set of odd numbers in A which is the same size as the set of all numbers in B. Therefore, the full set of numbers in A is twice that of B. But why not do it the other way around? Why not all the odd numbers in B mapping to all numbers in A? Why not every third number so that B is only a third the size of A? Why not every fourth?

Putting those questions aside, I can see how this carries over to my example of the two lines. Before removing the points, the points in each line are like the numbers in each set A and B. Then when I remove the odd points from the second line (call it line B) and move the remaining points into the gaps, that’s like mapping every odd number in A to every number in B. Have I got that right?

I don’t know if I want to go down the road of arguing whether the set of all odd numbers from A is half the size of A or B (I think I’d prefer your statement that one cannot determine the size of infinite sets… admittedly, whether or not infinite sets have a size isn’t clear to me). Suffice it to say, they’re both still infinite. What I’d rather return to is: how do you define “shorter”?

So let’s say we label the points in each line. Let’s label the points in line A a1, a2, a3, etc. The points in line B are b1, b2, b3, etc. Now, with labels attached to them, the lines would look different after removing every odd point and moving the remaining points to fill the gaps (they weren’t quite the same to begin with, what with line A having ‘a’ in the labels and line B have ‘b’, but let’s ignore that for now). You’d see a1 paired up with b2, a2 paired up with b4, a3 paired up with b6, and so on. I suppose your point is that even without the labels, it’s which point is paired up with which point that makes the difference. The first point in line A ends up paired with the second point in line B, the second point in line A paired up with the fourth point in line B, etc.

It’s as if each point has a special identity. If it’s the identity of each point that matters, then the lines were never identical in the first place. Point a1 is not point b1 (even without the labels).

But none of this addresses what it means for line B to be “shorter” than line A after removing every odd point and moving the remaining points to fill the gaps. By the conventional definition, both lines would still appear to be just as long (or if you’d like, their lengths remain just as undefined, no reason to say one is shorter than the other). I’m still looking for this unconventional way of defining “shorter”. Did you want to grab the concept I proposed earlier? That “shorter” means there is nothing but gap at the “end” of line B whereas there is “more line” at the end of line A? Or is it the history of the lines? What they went through? In that case, “shorter” means line B, in its past, had half its points removed. ← That could work to say line B is now shorter than its previous length, but it still raises some questions. Why did we decide to say the two lines were the “same” length in the beginning? And how could we tell that line B is now shorter than it used to be if we didn’t see every odd point being removed? Much remains to be fleshed out with this definition.

Emphasis is mine.

viewtopic.php?p=2614566#p2614566

viewtopic.php?p=2614807#p2614807

viewtopic.php?p=2615136#p2615136

viewtopic.php?p=2615491#p2615491

It won’t take eternity. You don’t need eternity in order to come to conclusion that every element in (A) is also present in (B) and vice versa. That wasn’t my point.

My point was that in order to determine whether any two infinite sets are equal in size or not, you have to know how each member of one set is represented in the other set. This can be as simple as “The way elements appear in one set is the way they appear in other sets”. If we said that this applies to (A = {1, 2, 3, …}) and (B = {1, 2, 3, …}) then we would be correct to say that the two sets are equal in size. But if a different rule applied (e.g. every number in (B) is represented by an odd number in (A) and every even number of (A) is represented by a non-numerical symbol in (B)) then a different conclusion would follow.

Note that (\bullet \bullet \bullet \bullet \cdots) is not a set, it is a sequence. Remember that unlike sets, sequences allow repetitions. Your confusion is created precisely by the fact that every element in the sequence (\bullet \bullet \bullet \bullet \cdots) is identical to every other. There is a number of ways to resolve this confusion. One way is to represent this sequence as a set. So instead of (\bullet \bullet \bullet \bullet \cdots), we’d have (L = {P_1, P_2, P_3, P_4, \dotso}) where “P” stands for point. Let’s say that the way elements appear in one set is the way they appear in all other sets. Take every odd point out and you get (L’ = {G_1, P_2, G_2, P_4, \dotso}) where “G” stands for gap. Remove the gaps and you get (L’’ = {P_2, P_4, P_6, \dotso}). Now, compare (L) to (L’‘). Do they have the same number of elements? Of course not. (L) has all of the elements that (L’') does plus some more.

Magnus,

in quoting James “that it’s to small to measure”

What’s being stated here is that it’s too small to explain. Explanations are measurements!

Yet, here he is, “explaining” it

I’m coming at this argument from your logic, not my logic.

Your logic is that infinity can be quantified.

So I used your logic to prove that everything except the highest order of infinity (in magnitude) (based on your logic), is actually fractional and has to be, but NEVER additive (2 infinite lines) - instead, you have (2) 1/2 infinite lines which still equals 1 infinite line.