Is 1 = 0.999... ? Really?

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Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
10
33%
No, 1 ≠ 0.999...
15
50%
Other
5
17%
 
Total votes : 30

Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Fri Jan 10, 2020 7:04 pm

Gib,

I’m going to kind of ignore your post because you’re not going to win this debate.

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).

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.
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Re: Is 1 = 0.999... ? Really?

Postby Silhouette » Fri Jan 10, 2020 9:16 pm

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!

What?

Unlike some here, I have no problem saying when I'm ignorant about something. It's how you learn, and therefore legitimately win in the long term, as opposed to deny, and therefore superifically win in only your own mind and only in the short term (which is stupid).

I don't know what Convergence theory is - I even looked it up and a cursory glance only finds references to Sociology, and nothing to do with "bounding infinities to make them quantities".
I have no idea what that means, and I strongly suspect it's just something you came up with and named yourself, perhaps you explained it elsewhere on the forum but certainly not here or anywhere that I read or remember reading.

I'm offering you an opportunity to detail your terminology and all the steps of its application so I can have a look at it.
It's not shooting myself in the foot to not be able to read your mind, even if I have the intelligence to work out more of what you're talking about than others seem to.

If you're just talking about convergent series, then that's fine - but your explanatory power will be increased if you use the standard terminology; in the same way that people will know what you mean better if you use the standard definitions of words.

It's true that the construction of \(0.\dot9\) converges to the quantity \(1\) (and I just explained the difference between representative constructions and quantities myself), but I dunno what the terminology of "binding infinities" has to do with this.

I already explained that \(1\) is also a representation of the actual quantity \(1\), which doesn't necessitate the quality of endlessness (e.g. \(1.0, 1.00, 1.\dot0\)) etc. makes no difference to the quantity represented.
And by contrast, \(0.\dot9\) is also a representation of the actual quantity \(1\), but necessitating the quality of endlessness (e.g. \(0.9, 0.99, 0.\dot9\)) etc. makes all the difference to the quantity represented.

Ecmandu wrote:Convergence theory states that an infinite sum (non convergent) equals a whole number (convergent)

So feel free to show how this works in practice without skipping steps or using unexplained non-standard terminology, such that you can successfully communicate all the essential details to others.

I don't know the answer yet, but you've asked me to have a look at it and I'm curious anyway - you have the opportunity to explain something I don't know to me here. Would make a first for this entire thread, so the change would be thoroughly welcome, especially if you explain it very well.

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.

Hi gib, I wasn't actually criticising what I quoted by you in case you were wondering, I was offering my own explanation around the concept because other people were too and I had improvements to offer.

I actually thought I covered what you refer to as "inventing a representation of the representation" through "the dot in \(0.\dot9\)" in what I parenthesised as "(or "ends" with indefinite continuation implied)". The dot above the 9 to denote recursion terminates the appearance of the representation with an instruction to imply indefinite continuation. Infinite sums do this as well with the infinity above the Sigma - it looks all defined with the finite terms all laid out with finite instructions about what to do with them, except that 1 term infinity that says "do the finitely defined thing an infinite number of times".

This is the whole trick that lures the non-mathematicians into a false sense of security in treating infinites as finites - they get fooled by the superficials of the representation and think that infinites can be defined, when it's only the finites that are defined, accompanied by an instruction to do the defined indefinitely. Indefinitely defined does not mean defined.

But in short, I've not read everything you've said because the thread's moving fast and there's so much wrongness being predictably and repeatedly peddled by the less qualified for me to try and handle - but what I have read of you mostly seems to make perfect sense.

obsrvr524 wrote:Who ever said that the lines have to be added end to end?

If it's the infinity that's twice as long, you have to add along the dimension that is going on infinitely. That would be consistency, but obviously it's impossible by definition and derivation.

For example, the real numbers represented along a number line go on infinitely along one dimension only - that's the sole dimension along which infinity applies.
Here, the number line doesn't go infinitely up or down, toward or away from the viewer - those dimensions and all others are finitely contained.
This is what I was explaining when I clarified that any mistaken identification of different sized infinities is actually only a product of the the finite constraints around the infinity - the size refers to these finites, not the infinity itself.
If you were to add another infinite number line side by side with another, you aren't changing the dimension along which the number lines are infinites, you're changing a finite constraint along the dimensions that the "infinite series" are finitely bound.

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 \(\infty - 1\).

This is really dense of you.

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.

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 \(s \div 10\).

Urgh.
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.

The \(0.9\) in \(s\) is NOT paired with \(0.09\) in \(s \div 10\). Let me explain in yet another way for you to not listen to...

The first term in \(s\) is \(\frac9{10^0}\). This looks like "9".
The first term in \(\frac{s}{10}\) is \(\frac9{10^1}\). This looks like "0.9"
But this doesn't mean we match the unit column with the tenths column - that would be to be fooled by superficial appearance.

In both cases the first term is dealing with the quantity in the unit column.
It's only positional notation that puts the resultant decimal representation in the tenths column, but appearance doesn't transfer the first quantity to be matched with the second term.
To think it does is - as I explained - getting fooled by how things look rather than the essence of the quantity to which they're referring: which is the first term, which operates on the quantity in the units column.

Is this all coming together for you yet, or are you still distracted by all the pretty lights?
The second term in \(s\) also looks like "9/10" in fractional notation.
The second term in \(\frac{s}{10}\) also looks like "9/100" in fractional notation.
This is slightly less misleading, but if you still can't resist the urge to translate it back into decimal notation to make it look like the one-to-one correspondence is "misaligned" such that there's a "spare" unmatched digit at some fictitious "end" to infinity, then you're going to continue to miss the point.

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.

Have you considered that, given your position as a non-mathematician, maybe you shouldn't be pretty confident about them?
You're hedging here, with your lip-service "I might, in fact, be wrong about why Wikipedia proofs are flawed", and your confidence in your simplistic "proofs".

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?
It's for this reason that "a bunch of mathematicians" cannot get away with "2 + 2 = 5", like some kind of political mob taking power.
But this does not prevent non-mathematicians from simply accepting and memorising the equation on a superficial level, like even children do from a very young age.
If you think the mathematical world works like a direct democracy rather than an elitist meritocracy where proving yourself wrong actually ascends you up the ranks, then this is just another example of your superificial non-mathematical intuitions and assumptions.
Not everything is complicated in life, indeed. Mathematics is, even if you have an uncomplicated understanding of it.

I actually have an uncommonly high tolerance threshold in person, which I am proud of and I consider it a personal failure when it fails. I am willing to publically talk about such weaknesses as well as my strengths - the public element does not affect my authenticity whatsoever, so it certainly doesn't make me proud of my frustration. The fact that you consider the public element to be a factor is no doubt why you're having so much trouble accepting the degree to which your mathematical abilities can currently enable you to deal with topics like this one. If you actually listened to people who know what they're talking about, and your admission of being a non-mathematician ought rationally to be followed by this line of behaviour, you might actually stand a chance to learn and grow to become a mathematician - so you'll progressively empower yourself to competently and legitimately deal with topics like this one! If that prospect doesn't tempt you, then I don't know what to say - enjoy the limbo of self-denial, intellectual stasis and long debates with people getting angry at you "because that's their flaw".

Magnus Anderson wrote:A hotel that is both full and not full. Not a logical contradiction at all.

Again! The whole point completely passes you by. WHOOSH.

The whole point of the hotel is to show the contradictions presented by the intrinsically indefinite nature of infinity, and therefore to not treat them as finites that can have sizes. It's the finite constraints upon which infinity can operate that give the series any "size", it's not the infinity that therefore has size. The fault is not with the thought experiment, but with the subject of the thought experiment - the thought experiment is just a messenger. Again - you get fooled by the superficials!

Are you really sorry for disappointing me though?
I don't think you're hiding the fact that you're saying this rhetorically, but maybe you really should seriously consider the possibility that being sorry might actually help us all out?
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Fri Jan 10, 2020 10:01 pm

Silhouette,

Like I stated about 10 posts ago in this thread...

Convergence theory is the theory that infinite sums converge. The “convergence” is ROUNDING (second grade math)

Rounding is not under any circumstance an equality.
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Re: Is 1 = 0.999... ? Really?

Postby Silhouette » Fri Jan 10, 2020 10:33 pm

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?
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Re: Is 1 = 0.999... ? Really?

Postby gib » Fri Jan 10, 2020 10:36 pm

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.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Fri Jan 10, 2020 10:58 pm

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.


Gib, I’m not always the nicest person. I often just say what I need to say.

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, I assumed (probably correctly) that you’re not going to take the “lead” on this.

What I saw in the post that I replied to, was you still being perpetually entrapped by Magnus ...

Arguments about infinity, if you’re not careful, can go in forever!
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Fri Jan 10, 2020 11:03 pm

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?


Silhouette,

I’ve been yelled at my whole life, I’m always defensive. “You’re not good enough, you’re not smart enough, this piece of lint on the floor is worth more than you”. That’s not only how I grew up, that’s how I still live.


We agree that 0.999... is infinitely regressive.

We agree that 1 is not infinitely regressive.

“Convergence theory” ( and maybe that’s my own term - I forget) is self explanatory, it’s the theory that infinite sums converge to non infinite sums.

Show you an example ?? I just did with 0.999... and 1. The topic of the thread.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sat Jan 11, 2020 12:57 am

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.


Why do you think that people don't understand this?

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 \(0.\dot9\)) 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 \(1\).


We speak of time with regard to infinite sums out of convenience, not because infinite sums have temporal dimension (they don't.)

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.

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:

\(\bullet \bullet \bullet \cdots + \bullet \bullet \bullet \cdots \neq \bullet \bullet \bullet \cdots + \circ \circ \circ \cdots\)

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..


It's the same thing. The gaps aren't filled, they are merely pushed out.

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?


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.

You didn't fill the gaps. You merely pushed them out.

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.


I do respond to your counter-arguments, even though I don't really have to.

On the other hand, you never pointed a flaw in my argument. You never said "This is the flaw of your argument!" And if you did something similar, you never explained why it's a flaw.

What you do is what everyone else (Ecmandu, Silhouette, Phyllo, Carleas, etc) does: you merely present your own independent arguments.

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.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Sat Jan 11, 2020 1:16 am

gib wrote:I think the conclusion that 0.999... = 1 was reached by something like this: https://www.relativelyinteresting.com/d ... y-equal-1/

Magnus already debunked those methods in this thread.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sat Jan 11, 2020 2:13 am

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
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sat Jan 11, 2020 2:35 am

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:

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.

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.


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".

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?


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

But then the quantities within the reds add up to the 1 green quantity that's not in the reds so...


They don't. That's the point.

Basically your argument here successfully says and shows absolutely nothing.


That would be you.

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

I have been pointing out the flaws in your argument because I wish to help you learn.


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

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.


Maybe you should listen to your own advice.

But this doesn't mean we match the unit column with the tenths column - that would be to be fooled by superficial appearance.


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.

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?


Don't be silly.

Not everything is complicated in life, indeed. Mathematics is, even if you have an uncomplicated understanding of it.


Not all of it.

The whole point of the hotel is to show the contradictions presented by the intrinsically indefinite nature of infinity


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.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sat Jan 11, 2020 2:47 am

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!
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sat Jan 11, 2020 3:16 am

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"?
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sat Jan 11, 2020 3:25 am

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"?


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!
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sat Jan 11, 2020 3:46 am

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.
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sat Jan 11, 2020 4:03 am

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.


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.
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sat Jan 11, 2020 4:15 am

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

What's your point?
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sat Jan 11, 2020 4:19 am

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
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sat Jan 11, 2020 5:09 am

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


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.
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Re: Is 1 = 0.999... ? Really?

Postby gib » Sat Jan 11, 2020 5:11 am

Ecmandu

Ecmandu wrote:Gib, I’m not always the nicest person.


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

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, #-o D'Oh! Really dropped the ball on that one. I assumed (probably correctly) that you’re not going to take the “lead” on this.


Hey, I still gotta few tricks up my sleeve.

Ecmandu wrote:What I saw in the post that I replied to, was you still being perpetually entrapped by Magnus ...


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

Ecmandu wrote:Arguments about infinity, if you’re not careful, can go in forever!


Uh, er... oh, I get it! :D

Magnus,

Magnus Anderson wrote:Why do you think that people don't understand this?


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

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


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.

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.)


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")?

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.


Well, we come back to the crux of my issue. I need to understand what you mean by "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:

\(\bullet \bullet \bullet \cdots + \bullet \bullet \bullet \cdots \neq \bullet \bullet \bullet \cdots + \circ \circ \circ \cdots\)


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:

Magnus Anderson wrote:It's the same thing. The gaps aren't filled, they are merely pushed out.


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? :D

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

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.


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.

Magnus Anderson wrote:You didn't fill the gaps. You merely pushed them out.


Honestly, I don't see the difference.

Magnus Anderson wrote:I do respond to your counter-arguments...


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:



Magnus Anderson wrote:On the other hand, you never pointed a flaw in my argument. Oh, please :icon-rolleyes: 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.


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).

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.


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

obsrver524 wrote:Magnus already debunked those methods in this thread.


Oh, well then by all means, let's all stand down and put this thread to rest. :D
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Re: Is 1 = 0.999... ? Really?

Postby Ecmandu » Sat Jan 11, 2020 5:27 am

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!
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Re: Is 1 = 0.999... ? Really?

Postby Magnus Anderson » Sat Jan 11, 2020 5:48 am

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.


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.
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Re: Is 1 = 0.999... ? Really?

Postby obsrvr524 » Sat Jan 11, 2020 7:46 am

I think you have to stop using the word "size". James spoke of 'degrees of infinity".
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Re: Is 1 = 0.999... ? Really?

Postby Silhouette » Sat Jan 11, 2020 7:54 am

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:

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.

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:

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 \(B\).

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.
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Re: Is 1 = 0.999... ? Really?

Postby gib » Sat Jan 11, 2020 8:42 am

Ecmandu wrote:Ahh... gib, zombie universes are so last year!

I’m doing hyper dimensional mirror realities now.


It's about bloody time!

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!


Sounds like a bad acid trip.

Magnus,

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.


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.
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