What are the set theoretic properties of vector spaces?

Question

I am reading introductory Quantum Mechanics, where it says-

For a classical system, the space of states is a set (the set of possible states), and the logic of classical physics is Boolean. The space of states of a quantum system is not a mathematical set; it is a vector space.

Further, in the footnote, it says-

To be a little more precise, we will not focus on the set-theoretic properties of state spaces, even though they may, of course, be regarded as sets.

What does "set-theoretic properties of state spaces" mean?

Answer 1

Although I think @Michael's comments may be onto something, I have my own guess as to what Susskind means. A vector space is an ordered $4$-tuple $\mathcal{V}:=(V,\,\Bbb K,\,+,\,\cdot)$, with $+$ telling us how to add elements of $V$, and $\cdot$ telling us how to multiply them by elements of the field $\Bbb K$. I won't repeat all the vector space axioms you probably already know. The important point is that the vector space is not just $V$ itself. But we cheat the set theory a little:

• elements of $V$ will often, in an abuse of notation/terminology, be called elements of $\mathcal{V}$, which in turn may just be called $V$;
• we don't care about the details of how your favourite set theory defines an ordered $n$-tuple.