According to Wikipedia,
The third law of thermodynamics states that the entropy of a closed system at thermodynamic equilibrium approaches a constant value when its temperature approaches absolute zero. This constant value cannot depend on any other parameters characterizing the system, such as pressure or applied magnetic field. At absolute zero (zero kelvins) the system must be in a state with the minimum possible energy.
It seems like the 3rd law is really just saying that, as the temperature of a closed system goes to absolute zero, the entropy and energy must both approach minimum values. That makes intuitive sense based on the definition of entropy in terms of probability and microstates vs macrostates, since the lower the temperature of a system is, the lower the average kinetic energy is, and hence there are fewer possible states the system could be in. But why does that mean it's impossible for any system to reach absolute zero? Obviously it can't happen to a closed system, since lowering the temperature at all requires putting in work to reduce the entropy locally and of course the lost energy has to go somewhere else. But I don't see why that would imply it's then fundamentally impossible for even an open system to reach absolute zero. I suspect it's because it would require arbitrarily large amounts of energy to keep decreasing the entropy towards that minimum value, and Wikipedia backs this up:
The laws of thermodynamics indicate that absolute zero cannot be reached using only thermodynamic means, because the temperature of the substance being cooled approaches the temperature of the cooling agent asymptotically.
And yet, I don't see why this is the case. What's the actual math showing that a system can only approach absolute zero at a (presumably vertical) asymptote?