Is $i\in n\Leftrightarrow i\in\{0,\ldots,n-1\}$ a common knowledge? Is $i\in n\Leftrightarrow i\in\{0,\ldots,n-1\}$ for every $n\in\mathbb{N}$ a common knowledge?
I am to publish a research article which uses this notation for convenience.
The question: Should I explicitly explain the reader what $i\in n$ for $n\in\mathbb{N}$ means?
 A: It is the standard construction of the natural numbers in set theory, known as von Neumann ordinals. If what you're doing is otherwise set-theoretic, you can use it (but it would still me merciful towards your readers to remind them of how it works, unless there are ordinals everywhere). Otherwise the conception of a natural number as the set of the numbers that come earlier is generally supposed to be an "implementation detail" that one should not have to think of when using the naturals.
Rule of thumb: If it makes sense in your context to consider the naturals defined by the Peano axioms rather than set theoretically, don't assume that your reader will be prepared to consider them sets.
Any particular reason not just to write $0\le i<n$?
A: Natural numbers are a set that satisfies Peano Axioms (they're the same up to isomororphism). Your construction is a common one by the great mathematician Von Neumann. However, do you see why it takes almost no effort to change the construction to make sure that the inclusion / membership between numbers / successors will not hold.
The reason it is particularly nice is that two sets are either the same or the subset of another set - this is a great way to define order.
