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Non-Standard Analysis

  1. Jun 13, 2010 #1
    hello,
    Im reading goldblatt's NSA book, and i just finished the first part.
    i have what i think are some trivial questions:
    1) im just wondering if *N is countable.
    2) are limited elements of *R real?
    3) im trying to prove that if x is infinitessimal then cos(x) -1 is infinitessimal.
    i thought i can *-transfer the statement:
    (\forall n \in N) (\exists \delta \in R+) (\forall x \in R) (|x| < \delta) -->(|cos(x)-1|< 1/n)

    i thought if x infinitessimal, then |x| , 1/n for any natural integer, and that makes it < \delta
    so it will satisfy the transfer of the statement, making |cos(x)-1|< 1/n fr all n in *N.
    is that valid?

    thank you in advance.
     
  2. jcsd
  3. Jun 13, 2010 #2

    Hurkyl

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    No. It is the *-transfer of countable, though.

    Generally not. Every limited hyperreal number, however, can uniquely be written as the sum of a standard real number and an infinitessimal hyperreal number.

    I'm not 100% sure what you're arguing. One possibility is, in fact, a valid argument -- but one that only works for standard real infinitessimals: i.e. zero.

    What you want to do, I think, is to apply transfer only to
    [tex]\forall x \in R : (|x| < \delta) \implies (|cos(x)-1|< 1/n) [/tex]​
    Why just this part? Because you want to prove something for standard n and standard delta, but for nonstandard x.

    Actually, it would be a lot easier to transfer the statement
    [tex]\lim_{x \rightarrow 0} \cos x = 1[/tex]​
    and invoke the nonstandard definition of limits in terms of the standard part function.
     
    Last edited: Jun 13, 2010
  4. Jun 13, 2010 #3
    hello,
    for the *N countable part, i agree, but how does that prove to me that *N is not countable itself?

    for the cos(x) - 1 part, i wasn't 100% sure if what i did was correct because on what you said, wanting the epsilon delta part stay standard,
    what's confusing me is when do i have the right to just get them out of my sentence and transfer the rest.
    thanks a lot again, that was very helpful.
     
  5. Jun 13, 2010 #4

    Hurkyl

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    For *N, I was just stating the fact, I didn't know you were looking for a proof. A short proof sketch is that every standard real number is infintiessimally close to a hyperrational number.



    There are a couple of ways to deal with transferring a predicate with free variables.

    One way is to realize that, for each particular choice of value for the free variable, you get a sentence. i.e. if P(x) is a predicate in the real variable x, and *P(x) is its transfer, a predicate in the hyperreal variable x, then P(a) iff *P(a) for any real number. In other words, *P is an extension of P to the hyperreals that has the same truth value on all standard numbers.

    Another way is to model predicates is to write truth values as 0 and 1, and view the predicate as a function:
    [tex]f(n, \delta) = \begin{cases}
    1 & \forall x \in R : (|x| < \delta) \implies (|cos(x)-1|< 1/n) \\
    0 & \neg \forall x \in R : (|x| < \delta) \implies (|cos(x)-1|< 1/n)
    \end{cases}[/tex]​
    which can be transferred in the normal way.
     
  6. Jun 13, 2010 #5
    thank you so much, this was a great help!
     
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