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Which is preferable to say, "the set of integers" or "the set of the integers", when I want to refer to the set {0, ±1, ±2, ...}?

Intuitively, I would say "the set of the integers", because "a set of integers" could refer to just any set consisting of integers, say {1, 2, 3}, and so "the set of integers" could refer to some set consisting of (not necessarily all) integers which was introduced earlier. It was recently pointed out to me that the form "the set of integers" is better, and now I am confused.

I tried googling both phrases, and both produce a lot of results.

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Depending on context, "integers" can mean all integers. If you say "the race of men", you're talking about all men, everywhere. If you say "the race of the men", you're talking about some men you referred to earlier (and whether they're black or white). If you say dogs are my favorite animals, you're talking about generic dogs. If you say the dogs are my favorite animals, you're talking about some dogs you referred to earlier. –  Peter Shor Mar 8 at 12:13
    
@PeterShor Likewise, the history of men would give me an idea of "the history of all of us". –  Damkerng T. Mar 8 at 13:56
    
I have described in this answer how to use the zero, indefinite (a/an) and definite (the) articles. In this case, since "integer" is a countable name and refers to a class in general, the use of the zero article is justified. I think @DamkerngT. does a good job explaining this case. –  Nico Mar 8 at 15:05

2 Answers 2

To me, I would use a phrase such as "a set of integers" or "this set of integers", for an arbitrary set of integers, for example, "a set of odd negative integers, where every integer in the set is greater than -99". Once such a set has already been introduced, it's fine to say "the (mentioned) set (of integers)" to refer to the previously stated set.

"The set of the integers" sounds awkward. And though "the set of integers" implies all integers, it can be ambiguous. And because it can be misread, it will be misread.

If I wanted to refer to the set {0, ±1, ±2, ...}, which is often denoted by Z, I would say "the set of all integers". This would avoid any confusions that may arise.

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+1 for "the set of all integers." That works much better than "the set of the integers." –  J.R. Mar 9 at 10:06

The set of integers we looked at earlier...

Refers specifically to all of the integers you mentioned before.

If you wish to address all of them you could also say:

The integers we looked at earlier.

Using "the" twice in this example, doesn't sound very good. It's unnecessary. "The set" specifies the set you mentioned, or "the integers" also covers this.

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