What Postulate States Entropy must be Positive? I've been reading the Postulates of Classical Thermodynamics, and I haven't found anywhere to be said that the Absolute Entropy of a system has to be a positive number.
The third one states that the Temperature, $dU/dS$, must be Positive, since $S$ is monotone increasing with $U$, and that as $U(S)$ approaches flatness, $S$ approaches zero. But this doesn't mean that $S$ can't be negative. Consider for instance the function $U = kS^3$, which is monotone increasing, and where $S$ approaches zero as $dU/dS$ approaches zero from both sides, meaning $S$ could be negative.
Im pretty sure $S$ can't be negative, but I'd like to see the actual proof or Postulate that says so in CLASSICAL (not Statistical) Thermodynamics. Maybe it's included in the definition of entropy and I missed it?
 A: In classical physics, the absolute entropy is not well-defined - only entropy differences matter, so shifting your entropy by an overall additive constant doesn't matter.  So entropies can indeed be negative, just as energies can, and that doesn't say anything at all about your system, just about your choice of coordinates.
For example, the microcanonical ensemble for a system with $d$ degrees of freedom is given by $$\Omega(E) = \int \frac{d^dx\, d^dp}{h^d} \delta(H(\{p,x\}) - E) \delta E.$$
In quantum mechanics, $h$ is Planck's constant, but in classical mechanics it can be anything you want - it's just there to get the units right.  So if you decide to make $h$ large enough, then $\Omega(E)$ will become less than 1, and so the entropy $S(E) = k_B \ln \Omega$ will be negative.
A: A better definition of Entropy is given by the statistical mechanics definition of the Gibbs Entropy: 
$ S = -k_B \sum{p_i \ln{p_i}}$
Where $S$ is the entropy, $k_B$ is Boltzman's constant and $p_i$ is the probability of the $\mathrm{i^{\mathrm{th}}}$ microstate to be occupied. For any probabilities less than 1, the quantity is clearly always positive. It can be shown that this definition is consistent with the classical thermodynamic entropy 
