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Is it bad style to use $n\in\mathbb N$ as a shortcut/abbreviation for "n is natural number" in a context where one does not deal with sets and elements in particular?

For example: "For all $m, n\in\mathbb N$ we have $m+n=n+m.$" Should one write "For all natural numbers $m$, $n$ we have $m+n=n+m.$" instead?

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    $\begingroup$ Nothing wrong with the set-theoretic notation. $\endgroup$ – lulu Aug 15 '16 at 10:59
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It's perfectly common notation. How symbol-heavy you want your writing to be is a matter of style, but both the example sentences should be alright by most standards.

An exception might be a context where you are worried the notation "$n \in \mathbb{N}$" will not be (easily) understood by your audience. But this particular notation is rather common. Thus, in most moderately advanced contexts it should be alright.

A main reason to use the notation, as for much notation, is brevity and clarity. See Example of a very simple math statement in old literature which is (verbatim) a pain to understand, for where you can get without notation.

For the most part, the two statements simply express the same idea.

Let me turn your question around (as in the comment by Ennar), when you say "for all natural numbers $n,m$" what exactly do you mean by that? Or, what is the difference between "For all natural numbers $n,m$" and "For all elements $n,m$ of the set of natural numbers"?

Of course, there are some mathematicians that reject the notion of a set of all natural numbers and maybe those would see some problem with the notation.

In the other direction, one could argue that in common rigorous developments of mathematics each natural number was constructed as a set, and the mathematical structure one deals with just are sets.

But really, I feel both are tangential to the question at hand for most purposes, where "for all $n,m \in \mathbb{N}$" and "for all natural numbers $n,m$" are just two ways to express the same idea.

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  • $\begingroup$ And could you explain why one is using this notation? After all, the fact that "For all natural numbers m, n we have m+n=n+m.m+n=n+m." has nothing to do with sets and the ∈ relation. Why should one then use them in the formulation? $\endgroup$ – user361467 Aug 15 '16 at 11:11
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    $\begingroup$ @user361467, mathematically "$n$ is a natural number" precisely means "$n$ is element of set of natural numbers". $\endgroup$ – Ennar Aug 15 '16 at 11:26

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