Some musings on the nature of infinity...

Discussion in 'Math' started by cmartinez, Dec 15, 2015.

  1. cmartinez

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    My son is a fan of Michael Stevens. He's the geek's perfect geek who who can actually make geekiness look cool. And he can make any subject seem extremely interesting, even to those who are not too much into science.

    Anyway, in this video of him discussing supertasks he talks about the Ross-Littlewood Paradox (mentioned at 14:22), which I found fascinating, but I just can't wrap my head around:


    If anyone here actually understands it, I'd appreciate it if he/she took the time to explain it to me in layman's terms.

    BTW, keep watching if you have the patience, he gives his talk a very interesting twist at 17:45
     
  2. WBahn

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    I think the big assumption that is creating the paradox is the unstated claim that all infinities are the same while at the same time treating them like they aren't. Basically, parts of the paradox are worded to imply that any set that is infinite in size is the exact same size and any other set that is infinite such that subtracting one from the other leaves an empty set, yet in other parts of the paradox things are worded to imply that one infinite set can be, say, nine times the size of another infinite set.

    Aside: I don't think I'm gonna be spending a lot of time watching his channel -- right from the beginning I found the theatrics annoying (waving to a wildly enthusiastic audience that exists only in his mind) and the business with pulling off sweaters all the time I just found stupid. But them I admit that I'm a very pragmatic person not impressed by flash.
     
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  3. cmartinez

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    And I'm pretty much that way too, but then again, we're not his target audience. My son and his friends (all in their late teens) are fascinated by the way he touches every subject.
    If you listen closely, he doesn't dwell too much on one single idea, but rather he quickly moves to another point. Pretty much planting the seed of an interesting subject so that his audience later feels the need to learn more by doing some research by themselves, all the while trying not to saturate their attention span. And I have to admit it's worked for my kid.

    And about those annoying sweaters... you and I both failed to notice the point of his taking them off throughout the video... but my kid didn't... at first I thought it was about the patterns, then about the colors, and then I had to give up and ask him straight out what it meant.
    You see, he was taking the sweaters off at an accelerated zenodian pace, that is, he always took his next sweater off in half the time he took the previous one off.

    As I already said... it's kids that are his target audience, not us.
     
  4. WBahn

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    I figured the sweater-taking-off had to do with some lame attempt at demonstrating an infinite process. I didn't notice that he was taking them off at an accelerated pace.

    You're right, I'm not the target audience. And I'm also more than willing to tolerate and even applaud efforts to try to hook an otherwise disinterested audience. There are good ways to do it and bad ways to do it and I am far from an authority on either, but I think this guy's approach to that is okay. What I don't like -- and what I see increasingly -- are these fast-edit context switches that attempt to match the microscopic attention spans of today's kids -- precisely because I think that it is these kinds of things that foster and reinforce microscopically short attention spans in today's kids!
     
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  5. cmartinez

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    You're right... but then again we're dealing with a generation accustomed to the instant gratification provided by video games and social media, among other things. One interesting fact about my kid is that he loves parkour, that's the acrobatic sport very popular with youngsters nowadays. That sort of thing matches perfectly with his fast-moving mind, and it also does him good, as far as his physical health goes.

    I've been trying to instill in him the same love for reading that I have, but haven't succeeded yet. He's far more interested in the audio-visual experience. But his career choice is now forcing him to develop his concentration skills, and hence also his patience and attention to detail. He's maturing at an accelerated pace.
    He's just finished his first semester on Chemical Sciences and Nanotechnology Engineering at the Tecnológico de Monterrey University. He not only passed every subject with flying colors, but he was also granted a 60% scholarship.

    I'm so proud of him, he's a far better kid than I ever was.
     
  6. Kermit2

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    I was taught that infinity occurred everywhere one cares to look. In between the 1 inch and 2 inch mark on a ruler for example, how many points exist? Is that infinity equal to the infinity of a tangent function at 90 degrees?
    They are everywhere but they are not equal.
     
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  7. cmartinez

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    That's pretty much what I understood from the video too... not all infinities are alike
     
  8. WBahn

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    That was one of the implications. But then in several of the paradoxical problems the paradox itself was the direct result of subtracting one infinity from another and getting zero. That only makes sense if both of those infinities are not only alike, but exactly identical.

    As far as I know (and I'm sure that abstract math takes it to far greater detail) you have countably infinite and uncountably infinite. For instance, the number of natural numbers is countably infinite and so is the number of rational numbers. So, to that degree, they are both the same size because you could put a 1-to-1 correspondence between them. But the set of real numbers is uncountably infinite because you can't. But while the size of the sets of natural numbers and rational numbers is the same (namely countably infinite), that does not mean that there is any meaningful interpretation of things like the difference in the size of the sets or the ratio of the size of one set to the other. Those operations simply aren't well-defined for infinite operands.
     
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  9. djsfantasi

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    This topic is one that totally intrigued me in my undergraduate applied mathematics studies; the concept of Aleph numbers. They are numbers which represent the cardinality (measure of the number of a set) of infinite sets. They assign a number w-n, to each infinite set. Not an easy read, but Wikipedia describes it as such: https://en.wikipedia.org/wiki/Aleph_number
     
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  10. BR-549

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    I don't think infinity qualifies as a physical concept.

    Unless we can find a charge manufacturer, the energy is set.

    That cancels infinity.
     
  11. WBahn

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    It doesn't have to be a physical concept to be extremely useful. For instance, the implications of different types of infinity allow you to prove that there are problems that fundamentally can't be solved algorithmically -- the prove basically comes down to showing that the number of possible algorithms is countably infinite while the number of possible problems is uncountably infinite.
     
  12. ian field

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    My theory is: there was more than one big bang.

    Infinity may even have had an infinite number of big bangs - each one having beings on various planets that think their universe is the only one.
     
  13. Kermit2

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    This was touched on in a book I read about string theory. It claims a need for 10 dimensions and that some of the dimensions are "inward". They extend into the realm of smaller vs. The three we live in which stretch out into the realm of larger. Seems in the math that both are infinite, but one is contained in the set of the other. We cannot access it directly but all matter contains it. It moves with us and maybe the energy pushed into these other dimensions of smaller at the big bang is what other scientists are researching and calling dark energy. Lots of secrets left to discover about reality. :)
     
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  14. cmartinez

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    Would you mind sharing the name of the book with us?
     
  15. Kermit2

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    It's been about 10-12 years since I read it, but I will look for a link to it.
     
  16. Kermit2

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    Damn. It's only been 5 years since I read that.
    Anyway, Brian Greene - The Elegant Universe, is the name of the book.
     
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  17. joeyd999

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    If one insists that time does not exist in the absence of a universe, then, no, there can be only one. I'd love to elaborate on my reasoning for this, but I simply do not have the "time".
     
  18. djsfantasi

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    Thought experiment. If there are 10 dimensions, why not 20? The restriction that I see is our mathematics, not reality.

    But at least, it is now thought there exists more than 4 (Hawking wrote about that back in the 80s). As mentioned, some are small (and pointed inward).

    But small by whose perspective? What if along those dimensions, is a universe where our dimensions are small and can't be seen. Multiple universes, one time line (maybe) and hence multiple big bangs.
     
  19. joeyd999

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    Math is limited to a finite number of dimensions?
     
  20. djsfantasi

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    First it was 7, then 10, now some claim 26 dimensions of our universe...

    My point being (albiet explained poorly) is that how the mathematics are manipulated results in an artificial upper limit on the number of dimensions.
    Perhaps it is NOT the mathematics, but our understanding of it.

    In any case, start with the assumption there are dimensions that we do not directly perceive as far as we know. These dimension may be understood as small - but from whose perspective? Everything is relative! Another universe could be imagined as existing in our small dimensions, where in that universe, we appear the be very small dimensions.
     
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