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Help recalling theorem

  1. May 26, 2008 #1

    Gokul43201

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    This is something of an odd request, I guess.

    I have a very foggy recollection of a theorem in some field of applied mathematics - probably information theory. I think it has to do with a seemingly surprising result about the information needed to describe any real in some closed interval compared to the information needed to describe all the reals in that interval.

    I can't recall anything more definite about this, and what I've said above may itself be more wrong than right. Hopefully there's just enough correct stuff there to help ring a bell with someone. Does anyone know what I'm rambling about?
     
    Last edited: May 26, 2008
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  3. May 27, 2008 #2

    CRGreathouse

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    The information needed in both cases is uncountably infinite -- in particular, [tex]\beth_1:=2^{\aleph_0}.[/tex]
     
  4. May 28, 2008 #3
    Wouldnt you be able to describe a single real with a sequence that converges to it (which is countable)?
     
  5. May 28, 2008 #4

    CRGreathouse

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    There are [tex]\beth_1[/tex] sequences consisting of [tex]\aleph_0[/tex] rational numbers. So, you could describe a real number that way, but it wouldn't be countable.
     
  6. May 28, 2008 #5

    Gokul43201

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    Thanks CRG. I was about to ask how you get beth_1 for both cases... but I ought to give it some thought first. Also, this tells me that I probably haven't recalled the theorem correctly.
     
  7. May 28, 2008 #6

    CRGreathouse

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    A closed real interval is of the form [itex]\{x:a\le x\le b\}[/itex] for some real a and b. Thus describing the interval requires only giving two real numbers. Two real numbers can be combined 'for the price of one' in many ways, like interleaving digits:

    1.12345 (interleave) 2.24680 = 21.1224364850

    Work from the decimal point out, since real numbers can't be infinite.


    The second point is only that [tex]\aleph_0^{\aleph_0}=2^{\aleph_0}:=\beth_1.[/tex]

    I'm just trying to jog your memory.
     
  8. May 28, 2008 #7

    CRGreathouse

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    Of course an arbitrary subset (rather than interval) of the reals has cardinality [tex]\beth_2=2^{\beth_1}.[/tex]

    If the result was information theory, then we're probably looking at the wrong stuff. Information theory usually deals with finite numbers of bits, right? [tex]2^{2^{\aleph_0}}[/tex] doesn't strike me as particularly 'applied'.
     
  9. May 29, 2008 #8
    Perhaps they have infinities sitting around that get renormalized away?
     
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