We're studying electrostatics in class, and the teacher introduced us to Gauss' Law a few days ago as $$\int \vec{E} \cdot \mathrm{d}\vec{A} = \frac{Q}{\epsilon_0}$$

Now suppose that the entire universe had a charge density $\rho$ and there was no other charge. By symmetry, the field everywhere must be $0$. Then by taking any Gaussian surface, my teacher claims, we can see that $Q_\text{enclosed}$ must be zero so $\rho$ must be zero. This means that charge has a fixed true zero point, similar to acceleration but unlike position and velocity (under classical non-relativistic mechanics). This is surprising to me.

  1. Is this idea correct?
  2. Is this proof correct?