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Math Help - Infinitesimals and Completeness Axiom

  1. #1
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    Infinitesimals and Completeness Axiom

    Under the hyperreal number system, infinitesimal numbers are nonzero, so then how would it be consistent with R being complete?

    If completeness axiom says that every bounded set has a least upper bound, then I know that the rationals don't have the completeness property since the open interval from 0 to pi doesn't have a least upper bound in Q. But under the hyperreal system, what about analogous interval from 0 to, say, some infinitesimal plus pi?
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  2. #2
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    I don't quite understand what you are asking here. Yes, R is complete but R*, the hyperreals, are not. There is a proof that the reals, as a subset of the hyperreals, is complete here:
    Math Forum: Ask Dr. Math: A Mathematical Essay
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  3. #3
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    Basically I'm saying that if you can show (0, root 2) interval has no least upper bound that is a rational (because root 2 is irrational between two rationals and makes the rational set incomplete), why can't you say the analogous with (o, root 2 + infinitesimal)? What would the least upper bound real number be for that interval?

    If R is complete, then anything between real numbers should also be real numbers. But if *R contains R, then where would the infinitesimal elements be located? If, say, root 2 + infinitesimal is between root 2 and [(root 2) + 1].
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  4. #4
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    Quote Originally Posted by anomaly View Post
    Basically I'm saying that if you can show (0, root 2) interval has no least upper bound that is a rational (because root 2 is irrational between two rationals and makes the rational set incomplete), why can't you say the analogous with (o, root 2 + infinitesimal)? What would the least upper bound real number be for that interval?

    If R is complete, then anything between real numbers should also be real numbers.
    No, this does not follow.

    But if *R contains R, then where would the infinitesimal elements be located? If, say, root 2 + infinitesimal is between root 2 and [(root 2) + 1].
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  5. #5
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    Then where does x + dx go if dx is infinitesimal? Is *R even ordered?
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