Question

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

Answer 1

No, but close. If `epsilon` is an infinitesimal, then `[epsilon, oo)` includes all positive Real numbers.

Explanation:

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

`[a, b]` 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.:

`[a, b) = { x in RR : a <= x < b }`

What about `+-oo` ?

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

We can take the set of Real numbers `RR` and add these two objects as members to get the set `RR uu { +-oo }`. They cannot be incorporated fully into the arithmetic of the Real numbers, since expressions like `oo - oo` and `0 * oo` 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 RR` we have `-oo < x < +oo`.

So we can use the notation `(0, oo)` to denote all of the positive Real numbers.

Note that the notation `(0, oo]` is bogus in that it includes the 'value' `+oo`, unless you are specifically talking about `RR uu { +-oo }` rather than `RR`.

What about infinitesimals?

You can add infinitesimals and their reciprocals (which are infinite values) to the Real numbers `RR` in a way that is consistent with arithmetic and has none of the strangeness of `+-oo`.

There's a description of how at https://socratic.org/s/axhXNryX

What you end up with is called a totally ordered non-Archimedean field. Let's call it `bar(RR)`.

Suppose `epsilon` is an infinitesimal in `bar(RR)`. Then in some ways the interval `[epsilon, oo)` is similar to `(0, oo)`. It includes all positive Real numbers - i.e. Real numbers greater than `0`. However, it does not include all positive numbers in `bar(RR)`. For example, it does not include `epsilon/2`.

Note that the interval `[epsilon, 1/epsilon]` is well defined in `bar(RR)` and also includes all positive Real numbers. It does not include all transfinite positive numbers in `bar(RR)`, e.g. `2/epsilon !in [epsilon, 1/epsilon]`.