Are Non-Standard Analysis Elements Countable and Real?

[jem05]

hello,
Im reading goldblatt's NSA book, and i just finished the first part.
i have what i think are some trivial questions:
1) I am just wondering if *N is countable.
2) are limited elements of *R real?
3) I am 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.

 

[Hurkyl]

1) I am just wondering if *N is countable.

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

2) are limited elements of *R real?

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

3) I am 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?

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

$$\forall x \in R : (|x| < \delta) \implies (|cos(x)-1|< 1/n)$$​

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

$$\lim_{x \rightarrow 0} \cos x = 1$$​

and invoke the nonstandard definition of limits in terms of the standard part function.

 

[jem05]

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.

 

[Hurkyl]

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:

$$f(n, \delta) = \begin{cases}<br /> 1 & \forall x \in R : (|x| < \delta) \implies (|cos(x)-1|< 1/n) \\<br /> 0 & \neg \forall x \in R : (|x| < \delta) \implies (|cos(x)-1|< 1/n)<br /> \end{cases}$$​

which can be transferred in the normal way.

 

[jem05]

thank you so much, this was a great help!