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March 17th, 2018, 08:16 PM   #11
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Quote:
 Originally Posted by AplanisTophet Because I am choosing to question the Archimedean property with this ...
Hey man I apologize if I was a little crunchy earlier. I have no idea how that got so out of hand. I should follow my own advice and either respond to the math or just go click on something else.

I do find your stuff interesting. We should just agree to disagree about certain other members and the overall parameters and limits of free expression on moderated forums.

If I get a chance I'll try to work through your OP.

Peace Brother.

Last edited by Maschke; March 17th, 2018 at 08:18 PM. March 17th, 2018, 08:41 PM   #12
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 Originally Posted by AplanisTophet Because I am choosing to question the Archimedean property with this, I might ask what the explosion principle would imply if it were flawed.
The Archimedean property has been proved beyond reasonable doubt, so if we were to assume it is false then explosion happens.
Quote:
 Originally Posted by AplanisTophet My hand waivy comparison was to assert that the paper will cross the point where the rationals align.
Just as mine was to assert that no such point exists.
Quote:
 Originally Posted by AplanisTophet Zeno shows how the math suggests Achilles cannot pass the tortoise, but when viewed properly we know he does.
This is a rather different matter and there are multiple different theories as to why Zeno's Paradox occurs. A mathematical justification must be perfect in order for it to be correct.

Last edited by Azzajazz; March 17th, 2018 at 08:43 PM. March 17th, 2018, 09:01 PM #13 Senior Member   Joined: Sep 2016 From: USA Posts: 680 Thanks: 454 Math Focus: Dynamical systems, analytic function theory, numerics The fact that you are choosing your irrational numbers to be algebraically independent is the reason the rationals won't "line up" after your shift. If you want some intuition about it, I suggest looking into diophantine conditions: https://en.wikipedia.org/wiki/Diophantine_approximation In a nutshell, there is a notion for how "close" an irrational number is to being rational and numbers which are "far" from rational are said to satisfy a diophantine condition. Moreover, irrationals which are algebraically independent will be different distances away from the rationals (in the diophantine sense). This turns out to have a lot of applications for a few reasons. One example which has a similar flavor to your intuition is the classification theorem for homeomorphisms of $S^1$. This is closely related to KAM theory which studies the near resonances and the breakdown of integrable Hamiltonian systems (differential equations which conserve energy) under small perturbations. Diophantine approximations played a large role in understanding this culminating with the KAM theorem in the 1950s which solved a large class of unsolved problems in celestial mechanics. See the following for details/cool examples: https://en.wikipedia.org/wiki/Kolmog...3Moser_theorem http://www.dam.brown.edu/people/meno...210/circle.pdf https://en.wikipedia.org/wiki/Orbital_resonance Thanks from Maschke and AplanisTophet March 18th, 2018, 06:37 AM   #14
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 Originally Posted by SDK The fact that you are choosing your irrational numbers to be algebraically independent is the reason the rationals won't "line up" after your shift.
That is very interesting and I thank you for posting it. My one question for you is whether you can offer proof of the above statement or if it is merely hand-waivy? March 18th, 2018, 06:43 AM   #15
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 Originally Posted by Azzajazz Just as mine was to assert that no such point exists.
Regarding the “spacing” of the rationals you mentioned, I see no reason why the game couldn’t be played with only dyadic rationals to achieve the “evenly spaced” set up that you sought. Again, it shouldn’t have to be though because the spacing between any two rationals is always rational and therefore uniform across the set. March 18th, 2018, 08:12 AM #16 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,696 Thanks: 2681 Math Focus: Mainly analysis and algebra The spacing doesn't matter does it? The reason it doesn't work is that you can't line up $r$ with $r'$ and rationals with rationals. The reason: there is no rational that is irrational. I confess that I don't see what the Archimedean property has to do with that. March 18th, 2018, 03:47 PM #17 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 552 This is a sort of meta-comment and very hand-wavy. If you accept that dealing with the infinite and irrational numbers takes us way beyond anything that any human has physically experienced, then it should be no surprise that the results of reasoning with such concepts may lead to counter-intuitive results. If you recognize that there are no known physical correlates to infinity and irrationals, you can also recognize that the counter-intuitive results do not affect our understanding of the actual universe that we live in. If we try the thought experiment with actual pieces of paper and dots, the dots will line up as far as any measurement device can validate. I for one do not find it difficult to accept that certain results must logically obtain in the ideal realm that mathematicians enjoy exploring without offending my intuition about the non-Platonic, physical universe that we seem to inhabit. They are two completely different universes. What does sometimes strike me as amazingly odd is how useful in the non-Platonic universe are the results of pondering the Platonic universe. (It might persuade some to become Platonists, but not me. I was probably about 17 when I read the parable of the cave, and it has not grown on me since then.) Thanks from v8archie and AplanisTophet March 18th, 2018, 04:05 PM   #18
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 Originally Posted by v8archie The reason it doesn't work is that you can't line up $r$ with $r'$ and rationals with rationals.
If $r$ and $r’$ are aligned, then we know the game is not won because the rationals are not aligned. That is correct. We are not trying to align $r$ and $r’$ though. Instead, we are trying to align the rationals in $\, (r, r+1\, )$ with those of $\, (r’, r’+1\, )$. It is not necessary for $r$ and $r’$ to align, but it is fair to then ask what happens if they do not. We then question the Archimedean property:

Quote:
 Originally Posted by v8archie I confess that I don't see what the Archimedean property has to do with that.
Ok, good question. The proof that there is a rational between any two irrationals stems from the Archimedean property. See, e.g., Theorems 1.1.4 and 1.1.6 on pages 5 and 6 of the following: Trench, W. (2013). Introduction to Real Analysis. Available at http://ramanujan.math.trinity.edu/wt...L_ANALYSIS.PDF

We now return to the game. We know that we can’t move the upper piece of paper a distance $r’-r$ if we expect the rationals to align. Rather, we must move the upper piece of paper a rational distance, in which case all of the rationals in the portion of $\, (r, r+1\, )$ that lies over $\, (r’, r’+1\, )$ will also be aligned.

So, let's just calculate what rational number to shift by, right? No. We can't because we are left with an infinite series of steps to traverse and at best place our hopes on a convergent infinite series as an approximation. This is why I incorporate motion into my demonstration of the problem.

When moving the upper piece of paper from left to right, we cross all possible distances up to and including the point where the rationals would align. These distances do not have to be real numbers (though it isn’t necessary for the game, as an example, we could move a hyperreal or infinitesimal distance to the right as well, noting the real numbers are just approximations and contain no infinitesimals). Motion, just like with Zeno's paradox involving Achilles and the tortoise (we know Achilles must catch the tortoise), is not plagued by our inability to calculate at an infinitesimal level.

I assert therefore that by moving the upper piece of paper a distance $r'-r$ to the right, we may have gone too far or we may have not gone far enough.

If we ‘nudge’ the paper a little to the right or left so as to make the rationals in $\, (r, r+1\, )$ align with the rationals in $\, (r’, r’+1\, )$, we are left asking what real numbers on the number line lie in the space between $r$ and $r’$ and between $r+1$ and $r’+1$. If all the rationals are aligned, no rational number can lie in either space.

Quote:
 Originally Posted by v8archie The spacing doesn't matter does it?
In the sense that Azzajazz was referring to, no it didn’t, as again I could run the game with the object being the alignment of just the dyadic rationals and it would alleviate his concerns (though again this isn’t necessary, because if all the dyadic rationals were aligned, then all of the other rationals would be too).

In the sense that we are questioning the space between $r$ and $r’$ when the rationals are aligned, however, spacing is everything. What fills that space? The implication is that only irrationals can fill that space, if any real numbers at all fill that space. The proof that there is a rational between any two irrationals may not even be applicable here, and so maybe I'm not even running afoul of the Archimedean property. That proof requires two real numbers that are firmly positioned on a number line for the purpose of demonstrating that a rational exists between them, but my game fails to even assert that $r$ will hold an official real number's position on the number line when the rationals are aligned.

In the very least, I think this game of mine sheds some interesting perspective on the real numbers. At the most, we might assume that we can align the rationals and then ask what the consequences of such an alignment are. I hope you are enjoying it.

At this point I turn to some common sense, or philosophical, points that I understand do not amount to much in a rigorous mathematical sense. I note the following simply to point out what prompts me to analyze further the above mathematics of it all:

1) The game suggests there is space between the unaligned rationals when $r$ and $r’$ are aligned, so what fills that space? Where our number line appears to be ‘flooded’ with irrationals as compared to rationals (the measure of the irrationals on the relative line segments is 1 while the measure of the rationals is 0), why would we expect there to be rationals between EVERY pair of irrationals absent the Archimedean property?

2) The cardinality of the irrationals is greater than that of the rationals, so again, why would we expect there to be rationals between EVERY pair of irrationals absent the Archimedean property?

3) As SDK pointed out, Diophantine approximations suggest that rationals can approximate certain irrational numbers better than others, so why would we expect that they can always get ‘closer’ in this case to $r’$ than $r$ would be if the rationals were aligned per the game’s instruction?

4) Just as Zeno showed that math can be misleading in Achilles and the Tortoise, the Archimedean property may be leading us astray when it comes to assessing the reals on what would essentially be an infinitesimal level (no, I’m not asserting the reals themselves contain infinitesimals). Modern math’s explanations rest with the notion of a convergent infinite series, but Zeno’s problem is not only of finding the sum, “but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event" … Zeno’s arguments…, because of their simplicity and universality, will always serve as a kind of Rorschach image onto which people can project their most fundamental phenomenological concerns (if they have any).” Frankly, I have such concerns and I’m projecting.

Last edited by AplanisTophet; March 18th, 2018 at 04:18 PM. March 18th, 2018, 05:22 PM   #19
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Too much to parse in one go, but I'll comment a bit now and return later.
Quote:
 Originally Posted by AplanisTophet We are not trying to align $r$ and $r’$ though. Instead, we are trying to align the rationals in $\, (r, r+1\, )$ with those of $\, (r’, r’+1\, )$. It is not necessary for $r$ and $r’$ to align,
As you pointed out in the last thread, the moment they don't align, there are an infinitude of rationals that aren't matched with anything in the other interval. So the game as I understood it is lost.

Quote:
 Originally Posted by AplanisTophet So, let's just calculate what rational number to shift by, right? No. We can't...
This should be "we can't - full stop". We can't because there is no first rational. "It's turtles (or tortoises) all the way down." Of course, in a physical context this is problematic as JeffM1 points out. But then the real world doesn't exist on a continuum. Everyday models break down at the quantum level which is one significant problem with Zeno. He assumes that the infinitesimal level means something, when in reality the Archimedean property just keeps going. March 18th, 2018, 06:09 PM   #20
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 Originally Posted by v8archie This should be "we can't - full stop". We can't because there is no first rational.
If I had instead stated that $r’-r \in \mathbb{Q}$, then we could despite there being no “first rational,” so the lack of a first rational is not dispositive. I assume you are still looking to add to this as indicated though, so maybe you would have deleted that if I’d given you more time. If so, my apologies. Tags game, rational Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post brhum Algebra 2 June 22nd, 2014 07:05 AM musicgold Economics 0 March 14th, 2012 09:04 PM stuart clark Algebra 1 May 19th, 2011 01:14 AM lkzavr Economics 1 April 22nd, 2009 12:35 PM lkzavr Applied Math 1 December 31st, 1969 04:00 PM

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