This issue is for logicians and operator algebraists (but also for anyone who is interested).

Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]):

Let $H$ be a separable Hilbert space and $B(H)$ the algebra of bounded operators.

**Definition**: A von Neumann algebra is a *-subalgebra $M \subset B(H)$ stable under bicommutant:

$M^{*} = M$ and $M'' = M$.

**Theorem**: The abelian von Neumann algebras are exactly the algebras $L^{\infty}(X)$ with $(X, \mu)$ a standard measure space. They are isomorphic to one of the following:

- $l^{\infty}(\{1,2,...,n \})$, $n \geq 1$
- $l^{\infty}(\mathbb{N})$
- $L^{\infty}([0,1])$
- $L^{\infty}([0,1]\cup \{1,2,...,n \})$
- $L^{\infty}([0,1]\cup \mathbb{N})$

**Noncommutative philosophy**: There are various *schools* of noncommutative philosophy, here is the *school* close to operator algebras. This issue is not about philosophy, so I will explain it quickly (for more details see *for example* the introduction of this book). First an intuitive idea : in the same way as there are *classical physics* and *quantum physics*, there are *classical mathematics* and *quantum mathematics*. What does it mean in practice ? It means the following : in the *classical mathematics* there are many different structures, *for example*, the measurable, topological or Riemannian spaces. The point is to encode each structure by using the framework of commutative operator algebras. For the previous examples, it's the commutative von Neumann algebras, C$^{*}$-algebras and spectral triples. Now if we take these operator algebraic structures and if we remove the *commutativity*, we obtain what we call *noncommutative analogues* : **noncommutative measurable, topological or Riemannian spaces**.

*This school* explores noncommutative analogues of more and more structured objects, it goes in one direction. My point is to question about the other direction (back to the *Source*) :

What's the noncommutative analogue of a set (called a **noncommutative set**) ?

What is a

noncommutative set?

The von Neumann algebras of the standard measure space $[0,1]$, $[0,1]\cup \{1,2,...,n \}$ and $[0,1]\cup \mathbb{N}$ are not isomorphic, but as sets, these spaces are isomorphic (i.e., same cardinal).

Is there a natural equivalence relation $\sim$ on the von Neumann algebras, forgetting the *measure space* but remembering the *set space*, on abelian von Neumann algebras?

**Remark**: If $M \sim N$, we could say that they are isomorphic as **noncommutative sets**.

The equivalence class could be called the **quantum cardinal** (a link with cyclic subfactor theory?).

Are there *noncommutative analogues* of the ZFC axioms ?

What I'm looking for seems different of what is called *quantum set* in the literature...