Minimal number of variables in a ZFC-undecidable sentence?

107 Views Asked by At

Let $\phi$ be a sentence of set theory. In Prenex form, $\phi$ can be written

$$ {\bf Q}_1 x_1 {\bf Q}_2 x_2 \ldots {\bf Q}_n x_n \ \ \psi(x_1,x_2, \ldots ,x_n) $$

where each ${\mathbf Q}_i$ is one of the quantifiers $\forall$ or $\exists$, and $\psi$ is a function of the Boolean values $x_i \in x_j$ and $x_i=x_j$.

If we want $\phi$ to be undecidable in ZFC, what is the smallest possible value for $n$ ?

Edit : We assume that ZFC is consistent, of course, otherwise the question is uninteresting.