# Epsilon is as significant as infinity. Is the bracketing [in, oo] equivalent to (0, oo)?

Aug 23, 2016

No, but close. If $\epsilon$ is an infinitesimal, then $\left[\epsilon , \infty\right)$ includes all positive Real numbers.

#### Explanation:

If $a$ and $b$ are Real numbers with $a < b$, then $\left(a , b\right)$ denotes an open interval, consisting of all values from $a$ up to $b$, but not including $a$ and $b$ themselves.

$\left[a , b\right]$ denotes a closed interval, consisting of all values from $a$ up to $b$, including $a$ and $b$ themselves.

Other combinations are called half-open intervals, e.g.:

$\left[a , b\right) = \left\{x \in \mathbb{R} : a \le x < b\right\}$

What about $\pm \infty$ ?

First note that neither $+ \infty$ (a.k.a. $\infty$) nor $- \infty$ are Real numbers. Intuitively they lie beyond the ends of the Real number line, with $+ \infty$ to the right and $- \infty$ to the left.

We can take the set of Real numbers $\mathbb{R}$ and add these two objects as members to get the set $\mathbb{R} \cup \left\{\pm \infty\right\}$. They cannot be incorporated fully into the arithmetic of the Real numbers, since expressions like $\infty - \infty$ and $0 \cdot \infty$ are indeterminate. So they cannot fully be treated as numbers.

They can however be fully incorporated into the total order of the Real line, so that for any $x \in \mathbb{R}$ we have $- \infty < x < + \infty$.

So we can use the notation $\left(0 , \infty\right)$ to denote all of the positive Real numbers.

Note that the notation $\left(0 , \infty\right]$ is bogus in that it includes the 'value' $+ \infty$, unless you are specifically talking about $\mathbb{R} \cup \left\{\pm \infty\right\}$ rather than $\mathbb{R}$.

You can add infinitesimals and their reciprocals (which are infinite values) to the Real numbers $\mathbb{R}$ in a way that is consistent with arithmetic and has none of the strangeness of $\pm \infty$.
What you end up with is called a totally ordered non-Archimedean field. Let's call it $\overline{\mathbb{R}}$.
Suppose $\epsilon$ is an infinitesimal in $\overline{\mathbb{R}}$. Then in some ways the interval $\left[\epsilon , \infty\right)$ is similar to $\left(0 , \infty\right)$. It includes all positive Real numbers - i.e. Real numbers greater than $0$. However, it does not include all positive numbers in $\overline{\mathbb{R}}$. For example, it does not include $\frac{\epsilon}{2}$.
Note that the interval $\left[\epsilon , \frac{1}{\epsilon}\right]$ is well defined in $\overline{\mathbb{R}}$ and also includes all positive Real numbers. It does not include all transfinite positive numbers in $\overline{\mathbb{R}}$, e.g. $\frac{2}{\epsilon} \notin \left[\epsilon , \frac{1}{\epsilon}\right]$.