# The set of all vector spaces [duplicate]

How can I prove that the set of all vector spaces doesn't exist? (In other words, if I gather all vector spaces, then it cannot be a set)

• Do you know the proof that the set of all sets doesn't exist? – Lee Mosher Jul 30 '20 at 14:07
• Yes I do know briefly. – Sphere Jul 30 '20 at 14:17

Take some fixed vector space $$V$$. Then for any set $$A$$ the set $$V\times \{A\}$$ is also a vector space in an obvious manner. Since the class of all sets is a proper class, and not a set, the collection of all vector space $$V\times\{A\}$$ is also a proper class. Thus the collection of all vector space which is a super class of the class above is also a proper class.
You can see that the set of all sets don't exist in this answer. On the other hand, every set $$S$$ defines a vector space, namely the free vector space on $$S$$. Then if there existed a set of all vector spaces, its elements would be in 1-1 correspondence with sets, i.e. it would define a set of all sets. Contradiction.