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March 16th, 2018, 10:02 PM   #1
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A Rational Game

This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]. Each piece of paper is then stuck to a wall, one just above the other, so that the line segments on each piece of paper are perfectly aligned. This means the line segments on each piece of paper run parallel to each other and drawing a vertical line between each 0 and each 2 would form a rectangle.

On the upper line segment we mark two numbers, $r$ and $r+1$, where:

$$r \in \, (0,1\, ) \text{ and } r \notin \mathbb{Q}$$

On the lower line segment we mark two more numbers, $r’$ and $r’+1$, where:

$$r’ > r,$$
$$r’ \in \, (0,1\, ),$$
$$r’ \notin \mathbb{Q}, \text{ and}$$
$$r’ - r \notin \mathbb{Q}$$

We are now ready to start our game. The object of the game is to slide the top piece of paper horizontally to the right so as to make all of the rational numbers in $\, (r,r+1\, )$ align vertically with all of the rational numbers in $\, (r’,r’+1\, )$. Fortunately, for purposes of this game, we are given the ability to slide ‘perfectly’ given that this game takes place only in a theoretical sense.

Despite our ability to slide the uppermost piece of paper with God-like precision, it is seemingly still impossible to win the game. The length that we would have to slide the upper piece of paper to the right must be a rational number because sliding an irrational length to the right would imply that none of the rational numbers align (ie, a rational plus an irrational will always be irrational). However, if we slide a rational length to the right, then $r$ and $r’$ could not be vertically aligned. In that case, the Archimedean property of the real numbers would ensure that an infinite number of rational numbers on each number line could also not be vertically aligned.

Therein lies what I feel is paradoxical. Seemingly, we should be able to slide the upper piece of paper so as to make all of the rational numbers on each line segment vertically aligned. If we choose to believe that we can in fact do so, then we contradict the standard view that the reals adhere to the Archimedean property which ensures there are an infinite number of rational numbers between any two irrational numbers.

Is there anyone who shares in my view that we ought to be able to get each rational number vertically aligned so as to win the game? Assuming that my view is correct (and in noting that I am not asserting it is), what would this imply?
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March 17th, 2018, 06:30 AM   #2
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This seems a lot like the sliding thing we went around on a couple of years ago. Is it basically another variant on the same idea?

ps -- Here it is. Some Fun and Simple Questions About the Rationals

Is this the same question in different form?

Last edited by Maschke; March 17th, 2018 at 06:34 AM.
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March 17th, 2018, 07:37 AM   #3
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Quote:
Originally Posted by Maschke View Post
This seems a lot like the sliding thing we went around on a couple of years ago. Is it basically another variant on the same idea?

ps -- Here it is. Some Fun and Simple Questions About the Rationals

Is this the same question in different form?
Yes, only back then I had zero knowledge of real analysis so you went through the painful yet amusing process of trying to get me to understand it. Since then I've understood the math behind why we can't 'align' the rationals in the above game, but this is a much clearer way to "demonstrate something that I find to be intriguing about the real numbers." I am still perplexed by it.

Although I understand the standard view as a rigorously proven mathematical concept, a part of me doesn't agree with it on a common sense level. I think we may assert too much based on the notion of what the infinite leads us to believe when perhaps we shouldn't be incorporating the infinite into rigorous mathematical arguments in the first place. There is no irrational that the rationals cannot approach down to an infinitesimal level and Physics could care less, for example, so why the need to distinguish between the two kinds of numbers? My common sense view is that the rationals must be alignable just as Achilles must catch the tortoise (see Zeno's paradoxes), but the math leads us to believe otherwise. I question the Archimedean property and with it many other things.

Last edited by AplanisTophet; March 17th, 2018 at 07:44 AM.
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March 17th, 2018, 09:05 AM   #4
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Quote:
Originally Posted by AplanisTophet View Post
Yes
Thanks, I thought I was having a flashback.

Can you please write a short and clear summary of your core issue? Your second post in this thread is handwavy and doesn't say anything. Frankly it's cranky. You know I've suggested before that the reason you come down so hard on Zylo is that you have a bit of the crank in you yourself. You denied it.

So prove me wrong and write something clear and precise and sensible. You can't just say, "Ooh my intuition this and that so the Archimedean property is wrong and besides, physics!" and then bust some OTHER crank's chops. You're just projecting.

You know that I'm not gratuitously attacking you. When you write math, I respond with math. When you get cranky, I call you cranky. That's fair.

How about perhaps a numerical example to show your shifting issue. I'm sure you know that the difference of irrationals is usually irrational, so you can't hope to shift by an irrational and make the rationals line up or make a single point "fall off the end" which is what you were trying to do last time.

Last edited by Maschke; March 17th, 2018 at 09:18 AM.
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March 17th, 2018, 11:17 AM   #5
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Quote:
Originally Posted by Maschke View Post
Thanks, I thought I was having a flashback.
You're not.

Quote:
Originally Posted by Maschke View Post
Can you please write a short and clear summary of your core issue?
I did. It's the OP. I have no "issue," just describing what I find intriguing.

Quote:
Originally Posted by Maschke View Post
Your second post in this thread is handwavy and doesn't say anything.
Correct. It wasn't meant to, save for in a philosophical sense.

Quote:
Originally Posted by Maschke View Post
Frankly it's cranky.
That's impossible, seeing as how I didn't assert anything. Describing one's intuition ("common sense") without making an assertion is, by definition, not crankery. Nut job.

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Originally Posted by Maschke View Post
You know I've suggested before that the reason you come down so hard on Zylo is that you have a bit of the crank in you yourself. You denied it.
I was honest with zylo because I think he suffers from a potential mental illness and possible drug use. I asked you to quit encouraging zylo because you're here for your own amusement more than anything and it's doing zylo a disservice to sit here and lead him on. I don't recall you calling me a crank over zylo's stuff, but cranks don't admit when they're wrong by definition. I did a couple years ago, so again by definition, you can't call me a crank. I was wrong when it came to understanding real analysis. The above aspects of it didn't (and still doesn't) sit well with my intuition.

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Originally Posted by Maschke View Post
So prove me wrong and write something clear and precise and sensible.
Prove you wrong about what?

Quote:
Originally Posted by Maschke View Post
You can't just say, "Ooh my intuition this and that so the Archimedean property is wrong and besides, physics!" and then bust some OTHER crank's chops. You're just projecting.
I don't recall asserting that the Archimedean property is wrong. I'm just questioning it. I can talk about my intuition all I want too. Now you're just being an ass, IMHO.

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Originally Posted by Maschke View Post
You know that I'm not gratuitously attacking you.
Ditto.

Quote:
Originally Posted by Maschke View Post
When you get cranky, I call you cranky.
If I start declaring that Math is broken, you can call me a crank. That would be fair.

Quote:
Originally Posted by Maschke View Post
How about perhaps a numerical example to show your shifting issue.
Read the OP. Again, I have no "issue," just describing why the math doesn't jive with my intuition. I didn't assert anything other than we seemingly can't get the rationals to align, which is exactly what you would say, mathematically speaking.

Quote:
Originally Posted by Maschke View Post
I'm sure you know that the difference of irrationals is usually irrational, so you can't hope to shift by an irrational and make the rationals line up or make a single point "fall off the end" which is what you were trying to do last time.
No, I wasn't trying to do that last time, which is part of the reason last time lasted so long. The moment you mentioned that I should gain a basic understanding of real analysis I read an introductory text on it and boom, all was understood. I'm cringing reading this last comment of yours. Seriously Maschke?

Last edited by AplanisTophet; March 17th, 2018 at 11:19 AM.
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March 17th, 2018, 12:32 PM   #6
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Just post some math. I'll respond with some math. A small numeric example would be helpful. Like, $r_1 = \sqrt 2$ and $r_2 = \pi$ and we shift this by that and now the question is ...

Just like that.

And I think you're the one on drugs. If you insist on making personal attacks I can do that as well as anyone. WTF is wrong with you to say something like that about a member who isn't even here? And I live in the US where the vast majority of the population is whacked out on one substance or another, legal and otherwise. What of it?

By the way I never encouraged Zylo. I do support free speech and a tolerant moderation policy that allows for alternative views. That includes your stuff too. Nobody here has spent more time working with your ideas than I have. "No good deed goes unpunished" as they say.

Get back to the math. If you can explain your idea clearly I'll respond the best I can.

Last edited by Maschke; March 17th, 2018 at 12:40 PM.
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March 17th, 2018, 01:24 PM   #7
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Quote:
Originally Posted by Maschke View Post
Get back to the math. If you can explain your idea clearly
Find the first line of text in the OP that isn't clear to you and I'll clarify. The OP is all math.

Quote:
Originally Posted by Maschke View Post
I'll respond the best I can.
It seems you already have.

When it comes to Zylo, there is a slew of research regarding immersing one's self in something like mathematical research and its connection to mental illness. Cantor himself suffered, as did many other great mathematicians. It's not a laughing matter. I am concerned for Zylo and perhaps you should be too. It's not an attack on him.
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March 17th, 2018, 02:11 PM   #8
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Quote:
Originally Posted by AplanisTophet View Post
Find the first line of text in the OP that isn't clear to you and I'll clarify. The OP is all math.



It seems you already have.

When it comes to Zylo, there is a slew of research regarding immersing one's self in something like mathematical research and its connection to mental illness. Cantor himself suffered, as did many other great mathematicians. It's not a laughing matter. I am concerned for Zylo and perhaps you should be too. It's not an attack on him.
I see there won't be math today. Maybe someone else will respond more to your liking. All the best.
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March 17th, 2018, 04:49 PM   #9
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Here's my thoughts on the issue, in slightly hand-wavy terms:

Imagine for a second we're not trying to align the rational numbers, but a sequence of randomly spaced dots. Obviously there's no guarantee that we can line up two sets of dots spread over different sections of the line.
Having said that, this situation with the rationals is more complicated. Talking about the spacing of the rationals and whether they are "evenly spaced" makes little to no sense because of their density over the reals. In such cases where we are talking about arbitrarily small (or large) values, intuition is your enemy. Unfortunately, sometimes we have to convince ourselves that something is true because the maths says it is.

As for assuming whether your intuition is correct and what the consequences would be, look up the explosion principle.
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March 17th, 2018, 06:02 PM   #10
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Quote:
Originally Posted by Azzajazz View Post
Here's my thoughts on the issue, in slightly hand-wavy terms:

Imagine for a second we're not trying to align the rational numbers, but a sequence of randomly spaced dots. Obviously there's no guarantee that we can line up two sets of dots spread over different sections of the line.
Having said that, this situation with the rationals is more complicated. Talking about the spacing of the rationals and whether they are "evenly spaced" makes little to no sense because of their density over the reals. In such cases where we are talking about arbitrarily small (or large) values, intuition is your enemy. Unfortunately, sometimes we have to convince ourselves that something is true because the maths says it is.

As for assuming whether your intuition is correct and what the consequences would be, look up the explosion principle.
Any rational plus another rational will be rational. The “spacing” as you put it (agree it’s tough to use that term with a dense set like the rationals but I get what you’re picking at) must always consist of rational lengths and is not random. It’s computing the amount of the shift that appears impossible because there isn’t a first rational in either segment to align with each other. If there was, we could shift easily just as we can shift from [0,1] onto [0.33, 1.33]. The same would be true if the difference between $r$ and $r’$ was rational, in which case shifting would be easy. Since we can’t compute the rational by which to shift by in order to win the game, I chose to represent the problem by sliding a paper horizontally so as to force it to reach (and pass if the motion continued) the point where the rationals would align.

Because I am choosing to question the Archimedean property with this, I might ask what the explosion principle would imply if it were flawed. My hand waivy comparison was to assert that the paper will cross the point where the rationals align just as Achilles will pass the tortoise in Zeno’s paradoxes linked above. Zeno shows how the math suggests Achilles cannot pass the tortoise, but when viewed properly we know he does. Here, we also know the paper will pass the point where the rationals align even though the math implies it isn’t possible because we cannot compute the rational number by which to shift by.
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