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Mathematicians often use the phrase "Denote the object X by the symbol Y". For example, I usually denote the set of numbers 1,2,3,4... by the boldface letter N. However a friend of mine uses a different phrase I have never seen before: "Let us write N for the set of numbers 1,2,3,4...". Is this a correct usage of the verb "write"?

NOTE: The "let us" is also awkward, but it is standard mathspeak and hence I consider it allowed on the basis of common usage. This is not the case for "write X for Y", which I have never read elsewhere.

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"Let us write N for the set of numbers 1,2,3,4..."

This looks like standard usage of the words "write" and "for".

It makes sense in an ordinary way which doesn't require knowledge of specialized mathematics terminology.

Is that acceptable in a mathematics paper? I'd say, probably yes. Here is another example, from https://www.zweigmedia.com/RealWorld/logic/logic7.html

Let us write N for the predicate "is a number" and use the standard notation ">" for "is greater than." Our statement is then:
∀x[(Nx(x>1)) → (x>0)].

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