Infinity

Discussion in 'Physics & Math' started by wegs, May 30, 2019.

  1. QuarkHead Remedial Math Student Valued Senior Member

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    B
    No. The set of integers is infinite but countable - they can be put into 1-1 correspondence with the Natural Numbers. That's a definition.

    Read AlphaNumeric's excellent post.

    The Real Numbers, by contrast, cannot - they are uncountable. Cantor's proof of this is delightful. Check it out
     
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  3. DaveC426913 Valued Senior Member

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    I was simply agreeing with you. Physically, a fraction of the smallest unit might be a problem, as you point out. But in the abstract, it's not a problem.
     
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  5. Write4U Valued Senior Member

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    Question:
    a) point (mathematics); a dimensionless coordinate (location)
    b) point particle (physics): a dimensionless particle with non-zero value.

    But then;
    c) In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle, because even an elementary particle, with no internal structure, occupies a nonzero volume.
    For example, the atomic orbit of an electron in the hydrogen atom occupies a volume of ~10−30 m3.

    There is nevertheless a distinction between elementary particles such as electrons or quarks, which have no known internal structure, versus composite particles such as protons, which do have internal structure: A proton is made of three quarks.

    Elementary particles are sometimes called "point particles", but this is in a different sense than discussed above.

    https://en.wikipedia.org/wiki/Point_particle

    I count two distinctly different patterns related to the physical definition of "point particle"?
     
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  7. wegs Matter and Pixie Dust Valued Senior Member

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    Okay, so when the example of the yardstick came up in this thread, I was thinking of that as being an analogy of the universe. (as finite) So, it didn't make sense to me how there could be an infinite number of galaxies, etc in a ''finite'' universe. (Points on a yardstick; I assumed that was where you guys were going with that, but think I was mistaken) I've been conflating math with physics, and that is why the yardstick example didn't make sense to me. *blush*
     
  8. DaveC426913 Valued Senior Member

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    Ah. Right.

    Galaxies have a non-zero extent. You cannot fit an unlimited number of them in a limited space.
     
  9. wegs Matter and Pixie Dust Valued Senior Member

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    Yes! I'm so glad that we have worked this out, because I was like...why am I not understanding this yardstick example?

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  10. QuarkHead Remedial Math Student Valued Senior Member

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    But surely, in English, at least, "smallest"means there is nothing smaller.

    How in any form of imaginable logic, can you have a fraction of "smallest" in science or mathematics?
     
  11. DaveC426913 Valued Senior Member

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    Because in mathematics, "smallest" doesn't exist. That's a limitation of the physical world.
     
  12. Write4U Valued Senior Member

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    But don't the mathematical limitations of the physical world dictate what is reality and human mathematics are symbolic attempts to explain or describe physical reality?
     
    Last edited: Jun 2, 2019
  13. Neddy Bate Valued Senior Member

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    So, let's say that we physically have a square which is one Planck length on each side. I assume the diagonal distance across the far corners is √2=1.414... Plank lengths. I suppose this is fine because it is greater than one Planck length.

    Now, let's say we physically have an equilateral triangle which is one Planck length on each side. I assume the altitude (height) from any side is √3/2=0.866... Plank lengths. This does not seem to be fine, because it is less than one Planck length.

    All of this is off-topic, but someone invoked Planck lengths as a limit to how many times we can subdivide a yard stick, which diverted the conversation a little bit.
     
  14. Write4U Valued Senior Member

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    Seems you are applying human theoretical maths to a RW problem. IMO, this is a matter of physical "permission".
    What if spacetime fabric allows only an altitude of one Planck length as the smallest permissible size of occupiable space? That would make the individual sides larger than Planck length and that IS permissible, no.....

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    May I ask if you have any comments on the Loll's CDT (causal dynamical triangulation)?
    I find that a most intriguing hypothesis of an evolving (unfolding) spacetime, which I assume would have to obey the Planck length constant.
     
    Last edited: Jun 2, 2019
  15. DaveC426913 Valued Senior Member

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    The physical world doesn't have mathematical limitations; it has physical limitations.

    Applied mathematics might, sure.
    Theoretical mathematics does not have to nave any correlation with the physical world.
    The Mandelbrot Set for example is infinitely regressive.
     
  16. Write4U Valued Senior Member

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    Is it real or descriptive?

    But are we talking about what can physically be real, or what can be theoretically constructed without regard to actual physical limitations?
     
  17. Neddy Bate Valued Senior Member

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    Yes, that would avoid the problem of the altitude (height) being 0.866... Planck lengths.

    But that would mean that two lines which are each 1.000 Plank length long cannot have an angle of 60 degrees between them. It is like saying they can only be 90 degrees (like my square example) or parallel. As soon as you try to make 60 degrees between them, they have to grow to 1.154... Planck length each, so that the forming equilateral triangle is not less than 1.000 Planck length in altitude. Maybe that is the case, I don't know.

    Sorry but I am not familiar with it. I might look into it when I have more time.
     
  18. Write4U Valued Senior Member

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    Why should we assume that 1.000 Planck length is required at all? It is a restriction, not a requirement, no?

    IMO, it simple. No dimension of an object can be less than 1 Planck length. If any part of an object's configuration becomes smaller than 1 Planck length, the entire object simply does not materialize in reality.

    I would refer to Bohm's Implicate to explain the mathematical implications in the formation of physical patterns which have mathematical restrictions. In a deterministic universe, the probability factor becomes reduced to zero.
     
  19. Michael 345 New year. PRESENT is 72 years oldl Valued Senior Member

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    Are we setting out onto the unicorn and god and imagination ocean?

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  20. Write4U Valued Senior Member

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    I'm trying not to. Seems to me mathematical physical permissions and restrictions are paramount in the formation of possible patterns. Mathematically, something cannot be physically smaller than what it physically can be.
     
    Last edited: Jun 2, 2019
  21. DaveC426913 Valued Senior Member

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    Theoretical mathematics has little to do with reality.
     
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  22. DaveC426913 Valued Senior Member

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    You guys are reading way too much into it. The Planck length is not a hard physical limit.

    It is simply the distance light travels in one Planck unit of time.

    The lower limit on physicality may be there, or somewhere near, but no where does it say it is some kind of hard limit. Nor will any scientist agree.

    Superstrings (if they exist) are on the order of the Planck length, but no one envisions them as square loops one Planck length on a side.
     
  23. Write4U Valued Senior Member

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    Good point. That brings up the question if quantum dwells at this level.

    True, but a string can only vibrate at a wave-length and frequency smaller than its physical length. Can a superstring smaller than a Planck length create a energetic wave at all, that is the question, seems to me.

    IMO, by definition a quantum packet cannot be smaller than a Planck length. It would lack energy.
    The only exception to this is the photon which has a true rest mass of zero, but derives it's mass from energy.
    ..........
    https://www.physlink.com/education/askexperts/ae180.cfm
     

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