Thread: Why/wat is "arbitrarily" used (for) and is there a mathemical way to "write it"

Hi everyone,

First of all, I am not really sure wether I have posted this topic in the right section of the forum. If not, moderators, please move this topic to the correct forum :-).

I'm not from a country where English is the native language, so especially with more specific subjects such as math I'm having trouble understanding everything. One of these things is the use of "arbitrarily", which appears to have the meaning "limitless/with no limit, but still finite". I am wondering: how can something be limitless but not infinite, as it's finite?

Besides that, I'm working on an essay which is about chaos theory, and I have come accross a definition which states a set M (by a dynamical system) whose elements are the same as N's or come arbitrarily close to it, which is defined as dense orbits. What does arbitrarily mean in this sense, and how could I get "arbitrarily" into an equation (what symbol would be used aswell)?


Regards,

Jesper

Originally Posted by Jesper

Hi everyone,

First of all, I am not really sure wether I have posted this topic in the right section of the forum. If not, moderators, please move this topic to the correct forum :-).

I'm not from a country where English is the native language, so especially with more specific subjects such as math I'm having trouble understanding everything. One of these things is the use of "arbitrarily", which appears to have the meaning "limitless/with no limit, but still finite". I am wondering: how can something be limitless but not infinite, as it's finite?

Besides that, I'm working on an essay which is about chaos theory, and I have come accross a definition which states a set M (by a dynamical system) whose elements are the same as N's or come arbitrarily close to it, which is defined as dense orbits. What does arbitrarily mean in this sense, and how could I get "arbitrarily" into an equation (what symbol would be used aswell)?


Regards,

Jesper

Let me see if I can give you an example of getting "arbitrary close." Suppose the distance between you and the wall is say 8 ft. Then divide the distance in half and move to that point. You are now 4 ft from the wall. Now divide that distance in half again and move to that point. You are now 2ft from the wall. Now continue in this fashion cintinually divide your distance in half. So you then move to points

1ft, 0.5 ft, 0.25 ft, 0.124 ft etc.

Question: Will you ever reach the wall - exactly? Well no - you'll never have the distance between you and the wall = 0. But you sure will be close. Will you be within say 0.001 ft of the wall? Or say 0.000001 ft from the wall? How about $\displaystyle \varepsilon $ where $\displaystyle \varepsilon$ is any very small number? If so, then you can get as close as you wish or "arbitrary close."

If you want to write this mathematically then if you let n be the nth step towards the wall the the distance is

$\displaystyle
\left( \frac{1}{2}\right)^n \cdot 8
$

and if you want this to be less than $\displaystyle \varepsilon $ then

$\displaystyle
\left( \frac{1}{2}\right)^n \cdot 8 < \varepsilon.
$

Thank you for clearing this up for me, Danny, I really appreciate that :-).

Even though you have cleared it up, I am still looking for a way to put that in a better mathematical form.

The case is, there is a set M and I want a way to mathematically write the set N which contains all the elements which are arbitrary close to the elements of M, thus the set N which is arbitrary close to the set M. Would the way to go with this be limits or? Thanks!

Originally Posted by Jesper

The case is, there is a set M and I want a way to mathematically [write the set N which contains all the elements which are arbitrary close to the elements of M, thus the set N which is arbitrary close to the set M.

There is a mathematical way of doing exactly that.
You need to know some topological concepts, in particular about metric spaces.
In a metric space if $\displaystyle M$ is a set then its closure of $\displaystyle \overline{M}$ is the set of all points a distance of 0 from $\displaystyle M$.
That concept is often described as the set points arbitrarily close to $\displaystyle M$.
So your set is $\displaystyle N=\overline{M}$.

Thanks very much, that's the second thing I needed!

A big thanks to both of you :-D!