  No bootstrap assumption is needed to derive the exponential growth of the
Hagedorn hadron mass spectrum: It is a consequence of the second law applied to
a relativistic gas, and the relativistic equivalence between inertial mass and
its heat content. The Hagedorn temperature occurs in the limit as the number of
particles and their internal energy diverge such that their ratio remains
constant. The divergences in the $N$ particle entropy, energy, and free energy
result when this condition is imposed upon a mixture of ideal gases, one
conserving particle number and the other not. The analogy with a droplet in the
presence of vapor explains why the pressure of the droplet continues to
increase as the temperature rises finally leading to its break up when the
Hagedorn temperature is reached. The adiabatic condition relating the particle
volume to the Hagedorn temperature is asymptotic. Since it is a limiting
temperature, and not a critical one, there can be no phase transition of
whatever kind, and the original density of states used to derive such a phase
transition is not thermodynamically admissible because its partition function
does not exist.
