What is the proof of the fact that for an isolated system, entropy can never decrease?
There is no proof, it's just a "near-truth" that can be justified by statistical arguments for a system that has a very large number of molecules/atoms in it. The probability that the entropy of a macroscopic (visible to bare eye) system would be observed to decrease even for a very short period of time is tremendously small and can be completely ignored in practice.
In classical thermodynamics, the molecular constituents of matter are not even mentioned at all and the second law of thermodynamics is presented just as an experimental fact.
Ok, let's make a thought experiment.
Suppose I have a box witchcontains a bunch of particles and at t=0 they are all confined to be in the corner of the box. But since we classically can determine the initial state of all the particles we can calculate where they will be and how fast they will be moving at some later time. So in this case the entropy will not have increased because my uncertainty about the state of each particle have not changed.
But suppose now that when I measured the state of the particles I had some uncertainty in the position and momenta for some or for all of the particles. According to Liouvilles theorem the phase space volume will not change as the system evolves and therefore after some time t my uncertainty about the position and momenta of the particles will be the same, hence the entropy will not have increased either.
In the third case my initial knowledge about the particles is only that they are confined to be in some volume V which is small compared to the volume of the box. Then after some time t the particles will have spread out and the uncertainty in the position is much greater then before and so in this case the entropy have certainly increased.
What is the fundamental difference between the second and the third example? Why did the entropy increase in the third example but not in the second? Did the entropy increase beacuse my uncertainty about the initial state was uncountable?
Entropy is not defined for a microstate (combination of states of individual molecules) - it's defined as the number of microstates that correspond to a given observed macrostate (given by a combination of pressure and temperature fields in the system, etc.).
It's the most important fundamental result of the Boltzmann (or in its quantum extension Boltzmann-Uehling-Uhlenbeck) equation. What you need as input is the unitarity of the S-matrix, leading to the principle of detailed balance. You find a very nice discussion of transport theory (both classic and quantum) in
Landau Lifshitz, Course of Theoretical Physics, vol. X, Physical Kinetics
Separate names with a comma.