Unattainability of absolute zero

In a regular thermodynamic formulation, a state at absolute zero cannot be reached from positive temperature by finitely many cooling operations with finite control resources. The entropy-difference form of the Third law of thermodynamics makes regular entropy curves approach a common limiting entropy. An isentropic process starting above that entropy cannot intersect the common zero-temperature endpoint.

For an illustrative pair of settings with $S-S_0=a_{1}T$ and $S-S_0=a_{2}T$, $0<a_1<a_2$, an isothermal entropy-reduction step followed by reversible adiabatic cooling gives $T_{n+1}=(a_1/a_2)T_n$. Every finite number of stages leaves $T_n>0$. The model illustrates the entropy geometry; a general finite-time impossibility statement additionally needs assumptions about available interactions and cooling rates.