Are Non-Standard Analysis Elements Countable and Real?

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In summary, the conversation discusses the countability of *N, the nature of limited elements in *R, and a proof involving infinitessimal numbers and the transfer principle. The expert summarizes the main points and provides additional information on transferring predicates with free variables.
  • #1
jem05
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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.
 
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  • #2
jem05 said:
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
[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.
 
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  • #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.
 
  • #4
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.
 
  • #5
thank you so much, this was a great help!
 

What is Non-Standard Analysis?

Non-Standard Analysis is a mathematical theory that extends the standard real numbers to include infinitesimals and infinitely large numbers. It was developed by Abraham Robinson in the 1960s as an alternative to traditional calculus.

How is Non-Standard Analysis different from traditional calculus?

Non-Standard Analysis differs from traditional calculus in that it allows for the use of infinitesimals, which are numbers that are infinitely small but still greater than zero. This allows for a more intuitive and rigorous approach to calculus concepts such as limits, derivatives, and integrals.

What are the applications of Non-Standard Analysis?

Non-Standard Analysis has many applications in fields such as physics, economics, and engineering. It allows for a more accurate and efficient way of modeling and solving problems that involve infinitesimal and infinitely large quantities.

Is Non-Standard Analysis widely accepted in the mathematical community?

While Non-Standard Analysis has gained popularity and acceptance in recent years, it is still considered a non-mainstream approach to calculus. Some mathematicians have raised concerns about the consistency and rigor of the theory, but it continues to be studied and used by many researchers.

Are there any limitations to Non-Standard Analysis?

Like any mathematical theory, Non-Standard Analysis has its limitations. It is not suitable for all mathematical problems and may not always provide the most efficient solution. Additionally, the use of infinitesimals and infinitely large numbers can sometimes lead to counterintuitive results, which may be seen as a limitation by some mathematicians.

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