mixing. If the assumption that the particles in an ideal gas are indistinguishable is not appropriately taken into account, the calculated entropy is not an extensive variable as it should be. Olbers' paradox shows that an infinite universe with a uniform distribution of stars necessarily leads to a sky that is as bright as a star. The observed dark night sky can be alternatively resolvable by stating that one of the two assumptions is incorrect. This paradox was sometimes used to argue that a homogeneous and isotropic universe as required by the cosmological principle was necessarily finite in extent, but it turns out that there are ways to relax the assumptions in other ways that admit alternative resolutions. Mpemba paradox is that under certain conditions, hot water will freeze faster than cold water even though it must pass through the same temperature as the cold water during the freezing process. This is a seeming violation of Newton's law of cooling but in reality it is due to non-linear effects that influence the freezing process. The assumption that only the temperature of the water will affect freezing is not correct. Paradoxes relating to unphysical mathematical idealizations A common paradox occurs with mathematical idealizations such as point sources which describe physical phenomena well at distant or global scales but break down at the point itself. These paradoxes are sometimes seen as relating to Zeno's paradoxes which all deal with the physical manifestations of mathematical properties of continuity, infinitesimals, and infinities often associated with space and time. For example, the electric field associated with a point charge is infinite at the location of the point charge. A consequence of this apparent paradox is that the electric field of a point-charge can only be described in a limiting sense by a carefully constructed Dirac delta function. This mathematically inelegant but physically useful concept allows for the efficient calculation of the associated physical conditions while conveniently sidestepping the philosophical issue of what actually occurs at the infinitesimally-defined point: a question that physics is as yet unable to answer. Fortunately, a consistent theory of quantum electrodynamics removes the need for infinitesimal point charges altogether. A similar situation occurs in general relativity with the gravitational singularity associated with the Schwarzschild solution that describes the geometry of a black hole. The curvature of spacetime at the singularity is infinite which is another way of stating that the theory