Can you mathematically prove that if there is a cavity in a conductor and no charge is placed in it, the field at all points in the cavity will be zero?
We can prove this. The proof is in two stages...
Since no charges are suspended in the cavity, no net charge will be induced on the cavity wall. This follows from Gauss's theorem, using a gaussian surface inside the conductor and totally surrounding the cavity. The electric field is zero everywhere inside the conductor (otherwise charge would be moving and we wouldn't have an electrostatic situation). So the flux through the gaussian surface is zero, and there can be no net charge inside the surface. Since there is no charge suspended in the cavity, the net charge on its walls must be zero.
The previous argument doesn't rule out equal and opposite charges on different parts of the cavity wall, and an electric field between them. What does rule these out is the fact that electric fields are conservative: the work done taking a test charge from one point to another is independent of route. [This can be deduced from the inverse square law.] Suppose point A on the cavity wall had a positive charge, and point B had a negative charge. Then work would be done by the field on a positive test charge going from A to B. [The charge would acquire KE.] But no work would be done (due to an electric field) if the test charge went from A to B through the conductor. This is inconsistent with the field being conservative. Therefore we cannot have positive and negative charges on different parts of the cavity wall, and there can be no electric field in the cavity.