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Mind problem

  1. Jul 18, 2010 #1
    Take a long stick, lets say close to infinity, meaning infinity-0.000001, and ignore all the classical physics acting upon this stick, mount this stick on a rotation bar on earth and let it expand into the univers, now install a sensor at each point along this stick, and a person at the end of the stick somewhere in the univers, now rotate this stick 30 degrees on earth in the fastest time possible. the stick does not bend, break, burn or freeze, never.

    what will the result felt by the person on the end, and that of the sensor measuring the if the stick bends or not.
    And if the results of general relativity were ignored, will this experiment produce a person travelling faster than the speed of light.

    JM Maritz
     
  2. jcsd
  3. Jul 18, 2010 #2

    russ_watters

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    Staff: Mentor

    Well, yes, if you ignore the rules of physis, you can break the rules of physics in your mind. But is that really a useful thing to think about?
     
  4. Jul 18, 2010 #3
    sure
    no problemo

    okey-doke

    got it

    consider it done

    yup

    gotcha

    (a) undefined, (b) undefined, (c) undefined.
     
  5. Jul 18, 2010 #4
    There is your problem: relativity places limits on maximal "stiffness" of your stick and your 30° twist travels along the stick at finite speed.
     
  6. Jul 18, 2010 #5

    mathman

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    Science Advisor

    The fastest time possible will be limited by the velocity at the other end of the stick, being less than the speed of light.
     
  7. Jul 18, 2010 #6
    If we ignore all classical physics and all the results of General Relativity (and for good measure we might as well ignore all Special Relativity as well) then anything is possible and it is possible that this experiment will produce a person travelling at faster than the speed of light and it is also possible that the Moon is towed around the Earth by invisible pink unicorns too. Unfortunately the universe we are in is not like this. Special Relativity excludes the possibility of an infinitely rigid rod, so in this universe, your experiment can not produce person with superluminal motion (the rod bends).

    On a mathematical nit-picky note, infinity-0.000001 = infinity and infinity plus/minus (any real number) = infinity.
     
    Last edited: Jul 18, 2010
  8. Jul 19, 2010 #7
    Are you sure they aren't blue unicorns???


    On infinity:............I would completely agree with your definition on "common sense" or physical grounds but is it neccessarioly a mathematical definition??

    Where infinity(set of all integers) + (some) real numbers = not infinity but transinfinity

    WHere some real numbers =(transinfinite set of all real numbers) - (the infinite set of all positive integers).............????:confused:
     
  9. Jul 19, 2010 #8
    Look up cardinality. It may help you with the concept of infinite numbers and how they behave and different degrees of infinity. Its as strange and as interesting as Relativity but probably easier to grasp.

    Matheinste.
     
  10. Jul 19, 2010 #9
    Hi I am fairly familiar with cardinality and Cantors ideas. WHile I also found them interesting in the end I thought them somewhat absurd given that infinity is inconceivable by the human mind in the first place and so is completely indefinable.
    AS for countability. It created a picture in my mind of two mathematicians ,one counting integers and one counting reals, sitting there for as long as it takes; meaning relative eternities.

    As for the diagonal proof. I failed to see how it improved on the obvious truth that between any two positive integers there must be an infinity of positive reals.
    But that just MHO
     
  11. Jul 19, 2010 #10
    PS
    Having given myself many headaches and vertigo from thinking about infinity I always assumed Cantors trips to the funny farm were a consequence of getting too far into it.

    But this just MHO
     
  12. Jul 19, 2010 #11
    The obvious truth is less obvious when you look closely. Your statement above makes a lot of hidden assumptions. This also applies to learning relativity, the obvious is not always true.

    Matheinste.
     
  13. Jul 19, 2010 #12
    Hey - *your* human mind perhaps :tongue:

    Wha---? It is, like, *so* definable

    Tip: Forget your mind's pictures, focus on the definition.

    Mainly it improves on it by showing that, in a very precise sense, there can be different sizes of infinity, and that the size of the set of real numbers is strictly larger than the size of the set of integers, yea, even though both sets are infinite.

    In short, though matheinste said it better: it rocks!
     
  14. Jul 19, 2010 #13
    That tip also applies to anyone struggling with tensors, as I found out after many wasted hours trying to visualize them.

    Matheinste.
     
  15. Jul 19, 2010 #14
    Your example is interesting, what if the stick is only 600.000 kms long not infinite minus 1(No idea what that means).

    So when i move the stick, the other end would feel the force instantly, and that force being felt instantly is FTL by itself. But this is of course considering the stick don't get bend :)

    And unfortunatly it will be so that pretty much gets us to nowhere at all.
     
  16. Jul 19, 2010 #15
    I am sure you are right . Most statements by everyone contain a host of hidden assumptions.

    I would be curious to hear your view on what you see of mine in this particular case????
     
  17. Jul 19, 2010 #16
    .......... I was essentially joking. I would not seriously question a mathematicians
    pleasure in the cleverness of the proof or their validity within the abstract realm of mathematics.
     
  18. Jul 19, 2010 #17
    yeah, me too - apart from the last two paragraphs
     
  19. Jul 19, 2010 #18
    You are assuming that the cardinality of the positive reals is greater than the cardinality of the positive integers. That is not very well put mathematically because we are talking degrees of infinity and need another symbology for this maybe. Yossell is better informed than me on mathematics.

    Matheinste.
     
  20. Jul 19, 2010 #19

    Gib Z

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    Homework Helper

    By definition we can compare Cardinalities of sets comparatively by proving if there exist bijections from one set to another. We can prove there is no bijection from the positive integers to the positive reals, and so the latter set has a greater cardinality. There exist symbols for these cardinalities, which we call Cardinals, but they are not required for this.
     
  21. Jul 20, 2010 #20
    Maybe I could share the joke if I knew which last two paragraphs you are refering to??

    No comeback for the blue unicorn???
     
  22. Jul 20, 2010 #21
    post 12, the post you quoted - esp: explanation of the extra information Cantor's diagonal proof gives over your statement.

    Your original point was that infinity wasn't definable - which is false. `Blue Unicorns' is definable too. Good work.
     
  23. Jul 20, 2010 #22
    Yes it is to see and think about the results of GR and the speed of light limit on space time, and how space and time interlocks,
     
  24. Jul 21, 2010 #23
    On the diagonal proof. As a formal proof I am sure you are absolutely right. I am certainly not qualified to dispute it. I have no idea whether anyone ever attempted a proof on the basis I suggested. Probably not as Cantor was there.

    On the unicorn: Fine I am perfectly willing to accept I was wrong about the definability of infinity if you are willingly to agree that the definitions and proof of existence in both cases have an absolutely equivalent truth value ;-)

    Ill even ignore the logical fallacy of taking my original comment which was related to definabilty in the normal sense and not formal systems , out of context and implying a different interpretation. :-)
     
    Last edited: Jul 21, 2010
  25. Jul 21, 2010 #24
    Hi Austin0

    After reading one of your posts in another thread addressed to Dalespam, I'm a little cautious about continuing with our banter and not sure if you feel I've stepped on your sensitivities. Everyone who is interested in difficult things should be encouraged. So, and remember that this is meant well, let me make a few comments.

    You originally wrote:
    I tried to explain that this is not what Cantor purportedly showed; then you wrote:
    As usual, you have given me no idea whether you have understood. It's not a question of formalism or formal proof, it's a matter of content: the theorem does not *state* 'there are infinitely many positive reals between any two positive integers. My response said nothing about a FORMAL PROOF - I haven't defended its legitimacy.

    I know you are joking - but there should be no *if* about it. We're not bargaining here, this is not a negotiation. There can be good reasons to think a definition is no good, maybe because it uses concepts that are inadequate or badly understood, or because it captures an empty concept.

    It's again hard to understand what you mean here. A proof is a sequence of statements - I don't think of it having a truth-value. A definition is...a definition. In so far as the definition is done in terms of a statement 'x is a unicorn by definition iff x...', it is trivially true just by what the words mean. Proof of existence of what? The set of real numbers? Of points on a line? I shouldn't have to spend so much time guessing what you mean.

    My guess is: you think infinite sets are as much an invention as blue unicorns. This is a reasonable point - I don't necessarily dispute it - but it wasn't your initial objection. For what it's worth, my own view is that the continuity of physical fields, including the metric field, the infinite divisibility of lines in space, the thought that the space-time manifold really is isomorphic to R^4 is a genuine physical possibility that we have no reason to a priori rule out; the thought that somehow everything is fundamentally discrete is, in my view, a pious hope, a philosophical prejudice. I don't think there is anything incoherent about the infinite, and I think Cantor's work played a large role in this, and it pain me to see him and his work dismissed out of hand. As such, I think that we have a lot more reason to take infinite sets, including uncountable sets, seriously than blue unicorns. But I know that's not an existence proof.

    Ok - now, that post was a whole lot more serious and pedantic and po-faced than I like it to be. After this, I'm going back to rough and tumble kidding around, and I hope it won't hurt your feelings or put you off. I can't speak for everyone, but I do hope that this post gave you some idea of the difficulties I've faced answering your posts, but without putting you off in any way.
     
  26. Jul 22, 2010 #25
    then later post

    I never suggested that that was the case.
    I certainly knew that that was not the basis or part of his proof
    If you are given an allowed concept of transfinite sets then it appeared to me obvious that the reals would be greater as there is 1 infinity per integer. This was purely my idea. Perhaps I would have been better to say was better than???

    I am not suggesting that the plain logical obviousness constitues a proof.




    As above; you disregarded my telling you that I had some familiarity with the concept and the proof. SO you misinterpreted my words .
    I did not state the theorem was based on or ever stated the above.

    The proof refered to was the existence of transinfinites

    OK Getting down to brass unicorns I think a case could be made for all of the above.
    He did not really define transinfinity or infinity , he defined sets.

    ANd then defined a procedure to quantify those sets. A procedure which has no meaning without the a priori assumption of relative infinities and quantifiable infinities.

    In my opinion the concept quantifiable infinity is inherently oxymoronic and illogical.

    But this is not dealt with in the proof.



    Proof of existence of transinfinite sets. There is no problem defining infinite sets or sets which we conceive of as infinite eg. real numbers etc.

    I am sorry but as the subject of this discussion was transinfinite sets I assumed it would be clear.
    On truth. Isnt it true that in either mathematical or physical logical systems a theorem is true if developed through valid operations on allowable arguments and does not prove inconsistent with the systems other theorems or axioms? ANd false if this proves not the case?


    As I said no question on infinite sets. I also agree with your basic statements regarding continuity etc. WHen you get to the infinite divisability of lines in space I might not agree.
    Also on discreteness. I myself think it is possible that even space itself may ultimately be quantized or as you put it ;there is no physical or logical basis to rule it out a priori.
    I certainly would never question the existence of the infinite.

    On Cantor ,,,I must appologize for my lack of politesse. I have tremendous respect for math, mathemeticians in general and certainly for Cantor.

    That does not mean that I dont see a problem with the logic of the proof. In my mind Russells,, set of all self exclusive sets, showed a flaw in the logic of the formal system and Godels responce was a further demonstration of the limitations of such systems. I am sorry but I see this as another. While still recognizing it as a brilliant mental construction.
    Sorry if I steppeed on your sensitivities :-}


    I dont mind serious and pedantic at all,i learn froim it. I think you misread my post to DaleSpam. It is not about being sensitive,, it is about communication and logic.

    AS demonstrated in this post above. I am aware that my idiosyncratic visualizations and casual terminolgy is part ot it. But communication goes both ways.
    ALL actual points are ignored and the only responce is to some little technical error or terminology misuse.
    I am not particularly sensitive personally but it gets very frustrating when communication and rational discourse is not proceeding as it could.
    SO no need to worry about offending me and absolutely no worry about calling me on specific points, that is part of why I am here.

    I have enjoyed and benefitted from our exchanges and I appreciate it.
    And your sense of humor
    Thanks
     
    Last edited: Jul 22, 2010
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