Is the axiom of choice needed to define a collection of subsets of some set $V$?

Question

There's this sentence, reading: "Denote by $\mathbf{L}$ the collection of all linearly independent subsets of $V$." By $V$ I mean some arbitrary, finite dimensional, vector space.

I will flirt with naivety here: can we always pick whatever collection of subsets of $V$ we please? Can this statement be anything? i.e., can I -from a collection of subsets of $V$- collect all subsets such that they are linearly independent $and$ span $V$? What reassures me that I can collect this family. Is it the axiom of choice?

Question two: by the way; from the space of all subsets of subsets? Can I always collect any family I please?and so on...

Answer 1

I can define whatever I want. There are two options:

1. It's a set. In which case, it exists.
2. It's not a set, in which case it's a proper class, and I've defined something "too big".

The intuition for "too big" comes from the way we formulate the axiom (schema) of separation, or subsets, or bounded comprehension, as it is sometime known. We formulate it in the following way: If I can define a property, then given any set, the collection of elements of that set which satisfy that property is also a set.

Specifically, if you are only interested in collections of subsets of some fixed set, then you will always end up with a set, because you are bounding your "search for objects satisfying the property" within the power set of your given set.

So what about choice? Well, the axiom of choice hardly ever comes into play when you define objects. It comes into play when you want to argue that these objects are not empty. When we say, for example, that the axiom of choice is necessary to prove there is a Hamel basis for $\Bbb R$ over $\Bbb Q$, we mean that it is necessary for the proof that the set of all Hamel bases is not empty. Not that it is needed for defining that set.

Answer 2

Yes, you can take the collection of subsets of a vector space $V$ that are linearly dependent and span the space.

To get a precise definition of which sets you can form, you need to look at the axioms of set theory. The "separation" axiom scheme says that if you have a set (say the powerset of your vector space $V$) and a formula $\phi(x)$ in the formal language of set theory, you can form a set consisting of the elements of the original set that satisfy the formula. The formula can even mention other sets that you already have.

Now "$x$ is a set of linearly independent vectors that spans the vector space $V$" is expressible as a formula of set theory, and so in this case you can form the collection of subsets of $V$ that are independent and span $V$. You get this by applying separation to the powerset of $V$, as above. As long as you apply separation to something that's already a set, you end up with a set, rather than a proper class.

The difficulty in this method is that few people other than set theorists want to look at formulas in the formal language of set theory. Fortunately, we know from experience that essentially all "ordinary" mathematical constructions are expressible in that way. In particular, any property you can express about a vector space while only quantifying over vectors and sets of vectors, and using the operations of the vector space, will be fine. This includes concepts such as independence and span.

The things that can cause problems tend to have a more "logical" flavor, like "the set of natural numbers that cannot be defined in less than 50 words". In practice, mathematical questions about vector spaces are not problematic.