Can we prove that the thermodynamic state of a system is completely determined by any two out of the three factors $P,V$ and $T$ ? (Without using statistical mechanics. Only using Thermodynamics)
NB: I have not learnt the axiomatic formulation of Thermodynamics.
There exist systems with other degrees of freedom other than pressure temperature and volume, for example in magnetic systems you can also talk about the degree of magnetization and the applied field. So the completely general answer is no, because it is not always true.
Once you have limited yourself to systems that only have these degrees of freedom, we can point to the equation of state $$ p = -\left(\frac{\partial F}{\partial V}\right)_{T} $$ where $F(T,V)$ is the Helmholtz free energy. This links the three quatities, so at most 2 of them can be independent.
In terms of why at least 2 of them must be independent, this is again really a matter of definition. If we have a system in contact with a heat bath at a fixed temperature or in a thermally insulated container then we do have an extra equation linking the variables (the fixed temperature or the adiabatic equation respectively) and so only have 1 degree of freedom. We tend to think of these relations, however, as constraints on a more general $pV$ system, rather than fundamentally different systems.
