Is the electric field inside a conductor zero a result of Gauss's Law?

In our lecture course on Electromagnetics we were told that for a conductor charge resides on the surface and hence if we apply Gauss's Law to a surface inside the conductor the enclosed charge is zero. Hence, the electric field inside the conductor is zero (which makes sense for the charge on the conductor). However, the lecturer then proceeded to explain that this is an explanation for why Faraday's cages work, i.e. there is no electric field inside a Faraday cage, because the charge enclosed by a Gaussian surface is zero. This doesn't make an intuitive sense, as surely when you apply Gauss you have to take into account superposition. In other words, you only calculate electric field due to the charge enclosed, but there could be a charge outside of the field leading to a net electric field inside the conductor / Faraday's cage.

• The given argument works if you take "charge goes to the surface" as a postulate, but we got there in the first place by arguing that the internal field is zero for the reason Shawshank gives below. – dmckee May 11 '17 at 18:43