writing papers: definition in word or formula? If we write papers, is it or is it not desirable to write definitions in formulas AND words. 
So if I want to define the following set:
$$S:=\{ x \in \mathbb{N} : P(x) \}$$ where $P$ is some predicate (more concrete in the context of paper). 
Should I write:
A) Let $S:=\{ x \in \mathbb{N} : P(x) \}$ denote the set of all natural numbers for which P is satisfied. 
B) Define $S:=\{ x \in \mathbb{N} : P(x) \}$.
C) Let $S$ denote the set of all natural numbers for which P is satisfied. 
Which is the best Version?

I like to make things clear. So personally I would go for the following option.

D) Let $S$ denote the set of all natural numbers for which $P$ is satisfied.
  $$
S=\{x\in \mathbb{N}:P(x)\}
$$
  Blah blah blah...

As this is a definition, so it is nice to be precise, clear, and to make it stand out. However, this all depends on context. For example, is it an important definition, or a minor technical one? Is the property $P$ very complicated, or simply "$x$ is even"? (I exaggerate about the simplicity, but putting the definition on its own line will perhaps allow you to write out $P$ in all its gory detail if you wish.)
Also, it is helpful to read about what others have written. People keep telling me that the Benson Farb school write beautifully (and I tend to agree with these people) so I read his (and his colleagues) papers to see how the "good people" do it. I don't really understand the maths, but seeing how they present their work is very helpful.
Finally, if the editor doesn't like it then listen to them. But this would be an obscenely harsh reason for rejecting a paper...
