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Ecmandu wrote:Silhouette,
I like you, but damn are you arrogant sometimes! By saying this; I mean, man, can you shoot yourself in the foot sometimes!
Ecmandu wrote:Convergence theory states that an infinite sum (non convergent) equals a whole number (convergent)
gib wrote:Silhouette wrote:Building the representation of \(0.\dot9\) needs endlessness
Even then, we get around that by inventing a representation of the representation--the dot in \(0.\dot9\)--which in turns just ends up being a representation of the quantity.
obsrvr524 wrote:Who ever said that the lines have to be added end to end?
Magnus Anderson wrote:Here's a simple visual representation of why \(10 \times 0.\dot9 \neq 9 + 0.\dot9\).
The red line indicates something interesting. It represents \(0.\dot9\) but is it the same quantity as the one indicated by the green line? Obviously not. You can see that the red rounded rectangle is narrower than the green rounded rectangle. In other words, the number of terms is not \(\infty\) but \(\infty - 1\).
Magnus Anderson wrote:That's correct.
\((9 + \underline{0.\dot9}) \div 10 \neq \underline{0.\dot9}\)
The two underlined numbers don't have the same number of 9's.
In other words, the two infinite sums don't have the same infinite number of non-zero terms.
Therefore, they aren't the same number.
That's my claim. It's up to other people (which includes you) to point out flaws within my argument if they wish to do so.
That's precisely the point of our disagreement. The \(0.9\) in \(s\) is paired with \(0.09\) in \(s \div 10\).
Magnus Anderson wrote:This isn't to say that I'm not sure about my claims regarding Wikipedia proofs. I'm pretty confident about them.
Magnus Anderson wrote:A hotel that is both full and not full. Not a logical contradiction at all.
Ecmandu wrote:Like I stated about 10 posts ago in this thread...
Convergence theory is the theory that infinite sums converge.
Ecmandu wrote:The “convergence” is ROUNDING (second grade math)
Rounding is not under any circumstance an equality.
Ecmandu wrote:Gib,
I’m going to kind of ignore your post because you’re not going to win this debate.
Ecmandu wrote:I will say that on the ground, before we abstract the infinity, we use temporal logic FIRST!!! We execute the decimal readout in TIME and then LATER, we realize that it’s an infinite expansion (or not).
Ecmandu wrote:Like I stated before, if someone makes a logical error about infinite expansions, the only way to clean up that error is to return to the original temporal abstraction.
gib wrote:Ecmandu wrote:Gib,
I’m going to kind of ignore your post because you’re not going to win this debate.
Wow! Can I try that? Can I win a debate by ignoring someone on the ground that they're "not going to win"?Ecmandu wrote:I will say that on the ground, before we abstract the infinity, we use temporal logic FIRST!!! We execute the decimal readout in TIME and then LATER, we realize that it’s an infinite expansion (or not).
Isn't that what I said?Ecmandu wrote:Like I stated before, if someone makes a logical error about infinite expansions, the only way to clean up that error is to return to the original temporal abstraction.
Sure, you retrace your steps to find out where you went wrong. But I don't think anybody ever added up a bunch of 9s to find out it sums to 1. You couldn't prove anything either way with that approach. Does it end in 1, or does it end in 1 minus an infinitesimal? Adding 9s 'til the end of time will only give you a headache.
I think the conclusion that 0.999... = 1 was reached by something like this: https://www.relativelyinteresting.com/d ... y-equal-1/
If you want to go back and see where they went wrong, the full proof is right there for you to pick apart.
I'm not sure where "temporal abstraction" comes in, or even what that means. My guess is that you mean, at some point, we even have to go back to imagining adding a bunch of 9s together and rediscovering that it goes on forever. Not sure what that accomplishes since no one's denying that in the first place. Like I said, we discover it once, and we remember. We can also define. Either one works.
Silhouette wrote:Ecmandu wrote:Like I stated about 10 posts ago in this thread...
Convergence theory is the theory that infinite sums converge.
Right, so it's just a theory that I just accept?
You've said "what" it is.
You posted what you called "notes" a few posts ago in this thread - I read them over and over. I saw your mentions of convergence theory and I wasn't clear of the exact steps of how it applies.
Can you not elaborate on "the theory that infinite sums converge" in a clearer way than before, with clearer explanations of exactly what's changing between each step?Ecmandu wrote:The “convergence” is ROUNDING (second grade math)
Rounding is not under any circumstance an equality.
Yes.
Rounding is basic math and isn't an equality...
I'm asking you to show more exact steps of how basic concepts such as rounding fit in with the potentially less basic operations that you're applying to these equations, which certainly don't so far equate in the standard way.
This is why they need further, clearer elaboration.
You don't need to be defensive, I'm just asking an optional favour. You could just refuse, that's fine.
If you want me to "man up and consider (your) argument is true", I need to know very exactly and very clearly what it is. Are you able to communicate that?
gib wrote:Yeah... you realize notation doesn't stand for a process, right? It stands for a quantity, a static quantity, something we are to presume is already there. There is only a process in the attempt to visualize the notation. Our minds find that they need to keep adding more 9s to fully visualize the entire series. But the notation they're trying to build already represents the quantity whether it's complete or not. The quantity itself doesn't need to "build up", it's just there.
\(0.\dot9\) approaches but never attains \(1\). This means two things: 1) not a single one of the partial sums of \(0.9 + 0.09 + 0.009 + \cdots\) (which is equivalent to \(0.\dot9\)) is equal to \(1\), and 2) the greater the number of terms that constitute a partial sum of \(0.9 + 0.09 + 0.009 + \cdots\), the closer it is to \(1\).
Come to think of it, I'm not sure two scenarios are any different: two identical lines or one line with an infinite gap an infinite distance away. To say the gap is an infinite distance away is equivalent to saying it's at the end of the line. But then what's at the end of the other line? More line? For all intents and purposes, if we're talking about "the end of the line", I'd say it's fair to say that's where the lines end. So the gap has effectively been push out of the line and the two are once again identical.
And where do the gaps disappear to in this scenario:
This problem arises when you imagine each point taking its turn to fill the gaps. If each point takes its turn, you'd need an eternity to complete the thought experiment and answer the question above. But what about each point moving at the same time? This is how we are to imagine Hilbert's Hotel. Each guest moves to the next room simultaneously, not one after the other. Of course, in the case of the gaps in the line, each point would have to move a different amount. The first point moves one position, the second point moves two positions, the third point moves three positions, etc..
Max wrote:You had an infinite number of points. Then you removed every odd point. To say that the resulting line is the same line is to say that you removed no point from it, which is a logical contradiction.
gib wrote:Woaw, woaw, woaw! What happened to the step of moving the remaining points to fill the gaps?
See, this is why it's you who is ignoring the counter-examples I bring up. You're response is to repeat the same original logic. You keep saying: it's just logical that if you remove stuff from other stuff you have less stuff. Then you make a leap from finite examples (which I agree is trivially true) to infinite examples. My counter-examples hold in the latter case, in the case of infinite things. I'm not refuting your original logic--of course if you subtract a number of things from a set of things, you get less things--but I'm refuting your right to carry that logic over to infinite things. Your response to that is just to reiterate the original logic and repeat the generalization--as if doing so a number of times will eventually invalidate my counter-examples. You need to address my counter-examples (which granted, you are), otherwise you're not arguing anything new.
gib wrote:I think the conclusion that 0.999... = 1 was reached by something like this: https://www.relativelyinteresting.com/d ... y-equal-1/
Silhouette wrote:If you want to say that the greens are "1 longer" than the reds, why not also say "but the reds are 9 broader than the greens"?
Again, we go back to Hilbert's Hotel and how infinities are undefined because you can easily end up with answers that are both bigger and smaller.
Do you then try to say something like the reds are therefore 9 times bigger than the green, or perhaps the "green minus 1", or maybe even 9-1 times bigger?
But then the quantities within the reds add up to the 1 green quantity that's not in the reds so...
Basically your argument here successfully says and shows absolutely nothing.
I have been pointing out the flaws in your argument because I wish to help you learn.
Yet even the non-mathematician expressing mathematical expertise over professional mathematicians doesn't want to learn, because what? You're an adult? You want to *feel* competent, or at least not incompetent? You see how I responded to Ecmandu at the start of this post? We're all students, and the less you're ruled by your insecurities, the better you'll learn if you simply admit you're NOT an expert and also not ACT like you're an expert nonetheless.
But this doesn't mean we match the unit column with the tenths column - that would be to be fooled by superficial appearance.
You actually do need mathematical expertise to deeply understand "2 + 2 = 4" - have you not seen the size of the proof that 1+1=2?
Not everything is complicated in life, indeed. Mathematics is, even if you have an uncomplicated understanding of it.
The whole point of the hotel is to show the contradictions presented by the intrinsically indefinite nature of infinity
Ecmandu wrote: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!!
Magnus Anderson wrote:Ecmandu wrote: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!!
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"?
Magnus Anderson wrote: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.
Magnus Anderson wrote:We didn't double anything because we weren't multiplying by 2 but dividing by 2.
What's your point?
Ecmandu wrote:Magnus Anderson wrote: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
Ecmandu wrote:Gib, I’m not always the nicest person.
Ecmandu wrote:When I saw that you were in a perpetual trap with Magnus and didn’t make the argument that an infinity divided by two doesn’t equal two infinities but rather 1/2 infinity and that no quantity has been added or subtracted, D'Oh! Really dropped the ball on that one. I assumed (probably correctly) that you’re not going to take the “lead” on this.
Ecmandu wrote:What I saw in the post that I replied to, was you still being perpetually entrapped by Magnus ...
Ecmandu wrote:Arguments about infinity, if you’re not careful, can go in forever!
Magnus Anderson wrote:Why do you think that people don't understand this?
Magnus Anderson wrote:I already explained what it means to say that an infinite sum approaches but never attains certain value.\(0.\dot9\) approaches but never attains \(1\). This means two things: 1) not a single one of the partial sums of \(0.9 + 0.09 + 0.009 + \cdots\) (which is equivalent to \(0.\dot9\) <-- Not true) is equal to \(1\) <-- True, and 2) the greater the number of terms that constitute a partial sum of \(0.9 + 0.09 + 0.009 + \cdots\), the closer it is to \(1\). <-- True
Magnus Anderson wrote:We speak of time with regard to infinite sums out of convenience, not because infinite sums have temporal dimension (they don't.)
Magnus Anderson wrote:Come to think of it, I'm not sure two scenarios are any different: two identical lines or one line with an infinite gap an infinite distance away. To say the gap is an infinite distance away is equivalent to saying it's at the end of the line. But then what's at the end of the other line? More line? For all intents and purposes, if we're talking about "the end of the line", I'd say it's fair to say that's where the lines end. So the gap has effectively been push out of the line and the two are once again identical.
The two lines are identical in some ways but not all. They are identical in the sense that 1) they are both infinite/endless and 2) they both contain no gaps. But they are not equal in terms of size.
Magnus Anderson wrote:If you take \(\bullet \bullet \bullet \bullet \cdots\) and take every odd inch out, you get \(\circ \bullet \circ \bullet \cdots\). In order to make the resulting line equal to the original line in terms of size, you have to fill every gap. Not merely push it out, but fill it. The above doesn't do this. The above merely splits the line into two halves, \(\bullet \bullet \bullet \cdots\) and \(\circ \circ \circ \cdots\).
Take the original line \(\bullet \bullet \bullet \cdots\) and split it into two equally-sized lines by removing every second inch from it. The result is \(\bullet \bullet \bullet \cdots\ + \bullet \bullet \bullet \cdots\). Compare that to the result that you get when you take \(\circ \bullet \circ \bullet \cdots\) and split it into two halves. What do you get? You get the following:
\(\bullet \bullet \bullet \cdots + \bullet \bullet \bullet \cdots \neq \bullet \bullet \bullet \cdots + \circ \circ \circ \cdots\)
Magnus Anderson wrote:It's the same thing. The gaps aren't filled, they are merely pushed out.
Magnus Anderson wrote:gib wrote:Woaw, woaw, woaw! What happened to the step of moving the remaining points to fill the gaps?
If you have an infinite number of points and remove every odd point from it, it is a logical contradiction to say that the resulting line is the same line. That tells you that, unless you add new points to the resulting line, the two lines can't be equal. You don't have to go any further than this. You don't just ignore logic. But that's precisely what you're doing.
Magnus Anderson wrote:You didn't fill the gaps. You merely pushed them out.
Magnus Anderson wrote:I do respond to your counter-arguments...
Magnus Anderson wrote:On the other hand, you never pointed a flaw in my argument. Oh, please You never said "This is the flaw of your argument!" <-- Do you literally need me to say those words? And if you did something similar, you never explained why it's a flaw.
Magnus Anderson wrote:You want all of the attention for yourself. And when you don't get the amount of attention that you want, you accuse the other of ignoring you.
obsrver524 wrote:Magnus already debunked those methods in this thread.
gib wrote:Come to think of it, I'm not sure two scenarios are any different: two identical lines or one line with an infinite gap an infinite distance away. To say the gap is an infinite distance away is equivalent to saying it's at the end of the line. But then what's at the end of the other line? More line? For all intents and purposes, if we're talking about "the end of the line", I'd say it's fair to say that's where the lines end. So the gap has effectively been push out of the line and the two are once again identical.
The two lines are identical in some ways but not all. They are identical in the sense that 1) they are both infinite/endless and 2) they both contain no gaps. But they are not equal in terms of size.
Magnus Anderson wrote:Silhouette wrote:If you want to say that the greens are "1 longer" than the reds, why not also say "but the reds are 9 broader than the greens"?
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.
Magnus Anderson wrote: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 \(B\).
Ecmandu wrote:Ahh... gib, zombie universes are so last year!
I’m doing hyper dimensional mirror realities now.
Ecmandu wrote: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!
Magnus Anderson wrote: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.
Obviously! That would take an eternity.
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. Now I'm being thrown for a loop. 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 \(B\).
Isn't that mapping kind of arbitrary? Couldn't you do the same mapping B to A?
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.
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