Infinitely small number added infinitely?

Discussion in 'The Cesspool' started by John J. Bannan, Jul 14, 2008.

  1. CptBork Valued Senior Member

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    Firstly, chaotic systems have nothing to do with it. Like I've said before, you should learn chaos theory to understand what it's about, but that's way, way, way off topic and requires plenty of university math background of its own before you can even get started. Secondly, you make the logical fallacy of treating infinity as a number. It's not.
     
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  3. John J. Bannan Registered Senior Member

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    How about a website, or maybe a quote from a text? Otherwise, I have to believe you, which I don't.
     
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  5. Myles Registered Senior Member

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    There are so many well written books for the intelligent layman on most of these subjects that I cannot understand how anyone who is interested can have such little knowledge.
     
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  7. John J. Bannan Registered Senior Member

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    I don't doubt that you believe you know the answer. I am testing that answer, however. And from what I've heard from you so far, I have every reason to question your answer. The mathematical answer, if there is one, should not be so hard that you can't put it in simple terms. It seems more likely to me that you've simply been taught certain things were true - without understanding why they supposedly are true.
     
  8. Reiku Banned Banned

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    Cpt -

    The chaotic systems, are completely relevant. Even Dr Hawing in his book, ''Black Holes and Baby Universes,'' states that such systems may limit our entire ability to determine a theory of everything. And a theory of eveything would mean we knew the sum-over all the histories of the universe, which not only seems impossible, but is remedial, next to our innability to know the absolute past, and the absolute future simultaneously.
     
  9. CptBork Valued Senior Member

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    Look up the density property of real numbers. The link I've provided discusses the relevant details, see corollary 1, subsection iii) on page 18.
     
  10. Guest254 Valued Senior Member

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    Reiku: on a side note, would you be willing to help me out with some chaos stuff? I tried to learn the subject a while back, but got stuck. If I dig out my notes, would you be willing to give a few pointers?

    (sorry for derailing thread)
     
  11. Myles Registered Senior Member

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    And we have even more reason to question the level of your understanding
     
  12. CptBork Valued Senior Member

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    Or the opposite could be true- I've questioned these things in such detail as to be able to see your logical fallacies, and to know what you need to do in order to learn their resolution.

    No they're not relevant. Chaos theory has nothing to do with mathematical infinity or Zeno's paradox.
     
  13. CptBork Valued Senior Member

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    Oh and Reiku, I explained to you a month ago why your claims about chaos theory are unrelated to what Dr. Hawking is saying. Chaos theory limits our ability to precisely describe what actually occurs in the universe, i.e. what will be the exact location of the moon relative to the Earth, accurate to 1 picometre, 1 million years from now? Chaos has nothing to do with our ability to come up with a grand unified theory of physics. I already explained this to you, you conceded the point, and I'm offended that you forget what I explained to you after I spent so much time doing so, even though you said you agreed with it at the time.
     
  14. CptBork Valued Senior Member

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    Now John, after you look at the source I provided, and how much work is involved, can you understand why we're not all that interested in explaining to you the axiomatic construction of real numbers or all the ins and outs of infinity? I can point out your logical fallacies and how they don't correspond to what's described in mathematics, but it's up to you to do the hard work of learning the actual math before you decide to challenge it.
     
  15. temur man of no words Registered Senior Member

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    Reply to Reiku's last post: I believe this was in a different context. Even if you have a Theory of Everything, because of a chaotic behavior you may not be able to predict everything with sufficient accuracy. This is so with today's theories.
     
  16. Nasor Valued Senior Member

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    Sure.
    http://everything2.com/node/1283443

    You are proposing that there is zero (α) and some other number (β) that has no other number between it and zero. If it is true that there is no number between α and β, then r does not exist, and it is therefore impossible that α<β. Hence they are the same number.
     
    Last edited: Jul 14, 2008
  17. John J. Bannan Registered Senior Member

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    Well, now we're talking. I notice that the corollary you cites says "between any two distinct real numbers lie infinitely many distinct real numbers." This is the Density Property of Real Numbers. In other words, "Given any two real numbers α, β ∈ R, α<β, there is a rational number r in Q such that α<r<β." However, it is not a corollary of this principle that if there is no r in a<r<B, then a=B. You have cited no authority for that proposition, nor any definition of equal that so states. An infinitely small number added to itself an infinite number of times would cancel out the infinities and create the smallest number possible. The smallest number possible would not have an r between it and Zero. The proof for the Density Property of Real Numbers does not take into account an infinitely small number added to itself an infinite number of times, and therefore, does not prove that number would equal zero. In fact, your proof assumes "a" < ny. But if "y" is the smallest number possible, than "a" could never be less than y, where "y" is the smallest number possible multipled by 1. "a" would at best equal "y", and therefore, the proof fails.
     
    Last edited: Jul 15, 2008
  18. CptBork Valued Senior Member

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    No, your understanding fails. Again. There is no infinitesimally small number other than zero itself. I will fill in the gaps for you on your objections, since you seem to miss some important details.

    One of the defining axioms of real numbers is the trichotomy property. That is, given real numbers \(a\), \(b\), we have either

    1) \(a=b\),
    2) \(a<b\),

    or

    3)\(b<a\).

    Suppose \(a\neq b\). Then case 1) is ruled out, and so we're only left with possibilities 2) or 3). In case 2), the density property shows that we can find a real number \(r\) such that \(a<r<b\). In case 3), we can find a real number \(r\) such that \(b<r<a\), again the density property guarantees this, no ifs ands or buts. So this proves that any time we have \(a\neq b\), we can always find a number \(r\) that comes strictly between them.

    Hence if we cannot find a number strictly between \(a\) and \(b\), that means that \(a=b\) must be true, otherwise we would contradict the combination of density and trichotomy properties which define the real numbers. This proves that there is no smallest number greater than zero. Nada, it doesn't exist, you can't define it. So all your arguments about infinitesimally small but non-zero numbers, as if such numbers existed now get tossed out the window. My condolences on your loss.
     
  19. Reiku Banned Banned

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    Bingo.

    But Guest, who is actually either a sneaky robo-alphacop, or indeed the very man himself, and cpt knew exactly what i was implying. If i was confusing, difficult to understand, or unclear, then speed my love to you. Otherwise, i say this out loud: Chaotic Systems -- and that can take the whole universe into retrorespect, that you cannot have a unified field theory, especially without some of the needed variables, like knowing exactly what happened in the past and the future. We exist here and now. And a sudden change, or flux in the past cone of the universe, will shift and alter it's future, and we would be oblivious to these changes, due to conservation ~ that is, if consciousness abides by such rules.

    Now Guest.../Alphanumeric,

    Something is wrong with you... seems, very wrong. I can tell. The subtle tones in which you are asking questions, indeed the correct ones, in which i have tried to answer correctly, but you just asked me for a direct help in an area me and alphanumeric have been known to have blown into extreme powers... (more from myself), but in this sense, you truely are a coward whoever you are, for doing what he wants. Either yet another one of his ''yes'' men have jumped in to help him, or perhaps just hoping that his works, which have been very loose, would go unnoticed. Trust me. They haven't.

    In the words of Guest, sorry for derailing the thread. :bugeye:
     
  20. Reiku Banned Banned

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    If i was a Borg, i would have said this post seems far too three-dimensional. Why do more active mathematicians fail to see the cost of a=b, so long as b plays the same roles but as conjugates?
     
  21. Pandaemoni Valued Senior Member

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    You can't say what the answer is to the question as posed, but...

    The basics of integral calculus involve adding approximating the area under a curve by breaking the area under that curve down into a series of smaller and smaller subsections. As you so so, the area of each subsection (obviously) gets smaller taken individually, there are more and more such subunits to add.

    In the limit, the areas of each subunit approaches (but does not reach) 0 and the total number of those subunits approaches (but obviously does not reach) infinity.

    The result of that process, which can be likened to adding up an infinite series of infinitely small boxes (technical distinctions aside), adds up to a number representing the area under the curve, and can range from negative infinity, up through 0, and on up to positive infinity, depending on the curve being measured.
     
  22. Reiku Banned Banned

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    Well said.
     
  23. funkstar ratsknuf Valued Senior Member

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    What magnificent nonsense.
     

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