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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?

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Depends on the level you expect your readers to be at. For example if you expect undergraduates to read your paper, then I would put the information in. If you expect phd's to read your paper, then you should be okay, perhaps just mention "using von Neumann's definition of ordinals..." – tomcuchta Dec 30 '11 at 17:47
I think that in the title and elsewhere you should write "notation" where you are writing "knowledge". – Dylan Moreland Dec 30 '11 at 17:48
Considering your profile your time would be well spent reading current mathematics and speaking to a tutor / a senior individual in the hope of guidance. I mean this in a constructive way, but I think you're slightly deluded with the output of material on your website - how is it useful. I admire your enthusiasm though am puzzled by your efforts. – Adam Dec 30 '11 at 18:18
up vote 6 down vote accepted

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 $0\le i<n$?

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I deal with index sets. I want numbers to be index sets (of finite cardinality). – porton Dec 30 '11 at 17:55
In combinatorics and computer science, I have seen the notation $[n]$ used for $\{ 1, 2, \ldots, n \}$ (sometimes for the set $\{ 0, \ldots, n-1\}$). – Srivatsan Dec 30 '11 at 18:18

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.

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