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Hyperreals as a vector space?

  1. Aug 31, 2010 #1
    Can the hyperreal numbers be described as a vector space over the reals with a basis (0,...,e2,e1,e0,e-1,e-2...), where e1 is a first order infinitesimal number, e0 = 1, and e1 a first order infinite number?
     
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  3. Aug 31, 2010 #2

    Hurkyl

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    Nope!

    Omitting zero, that is a basis for a real sub-vector space of the hyperreals. However, it certainly does not span the hyperreals -- your basis has countably many vectors, but the hyperreals are a real vector space of uncountable dimension.

    A specific hyperreal that is not in the span is [tex]\sqrt{e^1}[/tex].



    Also, I think you have some misconceptions about infinitessimals. In particular, for any nonzero infinitessimal e, there are other infinitessimals (such as [itex]\sqrt{e}[/itex]) that are infinitely larger in magnitude. I don't think "first order infinitessimal" makes any sense -- unless you have previously chosen a reference infinitessimal to compare things to.


    After pondering it for a bit, I think (but have not proven) any basis must actually contain two hyperreals whose ratio is finite, but not infinitessimal.
     
  4. Aug 31, 2010 #3
    Ah, okay, I didn't realise there wasn't a natural choice of "reference infinitesimal". Would it be possible to somehow arbitrarily define a particular infinitesimal to play this role, perhaps allowing the exponents to be real numbers?
     
  5. Aug 31, 2010 #4
    For an infinitesimal e, unlimited numbers H and K, and an appreciable number b, according to Goldblatt: Lectures on the hyperreals, p. 51, e/b, e/H, b/H are infinitesimal, b/e, H/e, H/b are unlimited, for e, b not equal to 0, while e/b, H/K, eH are undetermined. So if the ratio x/y is finite (limited), wouldn't that imply x is real, and y real and nonzero? But then x/y would be a multiple of 1 = e0, or whichever other nonzero real number we had in our basis.
     
  6. Aug 31, 2010 #5

    Hurkyl

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    You can choose a reference infinitessimal. Arguments I've seen often start off with something like "choose an infinitessimal e" or "choose an infinite hyperinteger H".




    Let e be a nonzero infinitessimal. The ratio
    [tex]\frac{e+1}{e}[/tex]​
    is finite and not infinitessimal, but not standard, and not a real multiple of 1.


    Oh, also of interest may be that, assuming we chose e positive, ee is an infinitessimal bigger than er for all positive standard numbers.
     
    Last edited: Aug 31, 2010
  7. Sep 1, 2010 #6
    Another snag: I just remembered, Goldblatt also calls H + K undetermined where H and K are both unlimited.
     
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