  The third law of thermodynamics has a controversial past and a number of
formulations due to Planck, Einstein, and Nernst. It's most accepted version,
the unattainability principle, states that "any thermodynamic process cannot
reach the temperature of absolute zero by a finite number of steps and within a
finite time." Although formulated in 1912, there has been no general proof of
the principle, and the only evidence we have for it is that particular cooling
methods become less efficient as the temperature decreases. Here we provide the
first derivation of a general unattainability principle, which applies to
arbitrary cooling processes, even those exploiting the laws of quantum
mechanics or involving an infinite-dimensional reservoir. We quantify the
resources needed to cool a system to any particular temperature, and translate
these resources into a minimal time or number of steps by considering the
notion of a Thermal Machine which obeys similar restrictions to universal
computers. We generally find that the obtainable temperature can scale as an
inverse power of the cooling time. Our argument relies on the heat capacity of
the bath being positive, and we show that if this is not the case then perfect
cooling in finite time is in principle possible. Our results also clarify the
connection between two versions of the third law (the Unattainability Principle
and the Heat Theorem), and place ultimate bounds on the speed at which
information can be erased.
