2nd Law of Thermodynamics

Today what bothers me is the legitimacy of the 2nd law of thermodynamics.

I cannot fathom, that in a closed system, the entropy can increase but never be destroyed. If every particle has a nonvanishing size, than there is a maximum number of such particles that fit within that space and a (nonessentially finite) cardinal number describing that maximum.

If now the Entropy is proportional by the Boltzman constant to the logarithm of the maximum number of states in the system, the entropy cannot become higher than that cardinal number.

Also I read, that there is still no proof for the 2nd law of thermodynamics, but that it was deduced from empirical observations. This sounds somehow wrong as well; there were a couple of proves for the contrary for the law, but they didn't hold for closed systems, while the only evidence for the law being right is empirical. Now if I am right, the Heisenberg relation says, that you cannot observe a closed system without being part of that system, but if you are part of the system, it is impossible to observer the whole closed system. So I wonder how Mr. Boltzmann would have ever been able to gather evidence about a law valid for a closed system.