March 17th, 2018, 08:16 PM  #11  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Quote:
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  
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244  Quote:
Quote:
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 
March 18th, 2018, 06:37 AM  #14 
Senior Member Joined: Jun 2014 From: USA Posts: 644 Thanks: 55  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 handwaivy?

March 18th, 2018, 06:43 AM  #15 
Senior Member Joined: Jun 2014 From: USA Posts: 644 Thanks: 55  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 metacomment and very handwavy. 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 counterintuitive results. If you recognize that there are no known physical correlates to infinity and irrationals, you can also recognize that the counterintuitive 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 nonPlatonic, 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 nonPlatonic 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.) 
March 18th, 2018, 04:05 PM  #18  
Senior Member Joined: Jun 2014 From: USA Posts: 644 Thanks: 55  Quote:
Quote:
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. 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 (noninstantaneous) 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  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,696 Thanks: 2681 Math Focus: Mainly analysis and algebra 
Too much to parse in one go, but I'll comment a bit now and return later. Quote:
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 
Senior Member Joined: Jun 2014 From: USA Posts: 644 Thanks: 55  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.


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