Definite article with numbered nouns

I often encounter a problem with adding "the" or not. I had a proofreading teacher last year that gave us a concrete example as follows:

Cows eat grass = Cows, in general, eat grass

The Cows eat grass = Those specific cows eat grass

Yesterday I was writing an email and hesitated in adding "the" or not in this phrase:

"Although weeks 31 and 32 are not problematic anymore, we still need volunteers for weeks 34 and 35."

I finally decided to add "the" as I think it was a request for specific weeks...

Solution 1:

There are two common ways to show that a noun is numbered, i.e., that it occupies a particular number in a list of things of the same type. Those two ways are:

• using an ordinal number before the noun
• adding the number right after the noun, optionally preceded by the word number

Ordinal numbers (first, second, third, etc.) function more or less like adjectives, and as such, they can be used as part of both definite and indefinite noun phrases—the difference being whether the noun phrase as a whole should be considered definite or not in the context it appears.

Since ordinal numbers are most commonly used to limit a range of things down to one specific one (the one that occupies the nth place in the list), which makes the noun phrase definite, the definite article is much more common than the indefinite article.

The other way to express a certain number in a list or range functions differently. The number (with or without the word “number” itself) added after the noun does not act as an adjective as such, though they are quite similar to; what exactly it is generally categorised as syntactically, I don't know offhand.

Crucially, though, this postpositive number inherently marks the noun phrase as definite. In that sense, it functions like Saxon and pronominal genitives.

Saxon genitives act like determiners (like articles) and mark the noun phrase as definite—but since you can't have multiple determiners and you can't mark a noun phrase for definiteness more than once, you cannot combine a Saxon genitive with an article: *my the house doesn't work.

Postpositive numerals are not, as far as I know, generally considered determiners, but they do have the effect of marking a noun phrase as definite and blocking at least articles: *a week 33 doesn't work, nor does *the week 33.

Oddly enough, they don't block other determiners like Saxon and pronominal genitives: My week 33 is looking terribly busy is a bit awkward and not how most people would phrase it, but it's grammatical. A more likely scenario would be a student in class who, upon being told to flip to page 20 in the class book, raises his hand and says, “But my page 20 has been torn out!” (meaning “page 20 in my book”).

Solution 2:

When a designation follows a common noun, that noun phrase does not allow the definite article.  It doesn't matter whether those designations are numbers, letters, proper names, or anything else. 

• weeks 31 and 32
• columns A and B
• team members Alice and Bob
• operations Charlie and Delta

These designations are restrictive appositives. As restrictive appositives, they are definitive determiners and preclude the addition of the definite article. 

However, using the same destinations as adjectives (or, more correctly, as attributives) does allow the definite article:

• the 31st and 32nd weeks*
• the A and B columns
• the Alice and Bob team**
• the Charlie and Delta operations

The one exception to this pattern that comes to mind involves placenames:

• the cities Chicago and Boston

I'm tempted to think that the placenames don't stand as an appositive but rather as a genitive.  The phrases "the cities Chicago and Boston" and "the cities of Chicago and Boston" mean the same thing.  We can't say the same about pairings like "columns A and B" & "the columns of A and B", or "team members Alice and Bob" & "the team members of Alice and Bob". 

_______________ 

* The ordinal numbers are adjectives, but plain numbers don't act like attributives. Instead, they act like cardinals, which are adjectives in their own right.

** It may be grammatically correct to speak of "the Alice and Bob team members", but that phrasing is also unnaturally dehumanizing.