Can an object be infinitely small? I read somewhere that the earth has to be smaller than 1 cm to become a black hole, according to Schwarzschild. Since big bang came from a singularity, I am wondering, is there any minimum volume for anything? 
 A: Infinity is a mathematical term, very useful, but the history of physics has shown us that when we make mathematical extrapolations that lead to infinities of one sort or another, a different mathematical model will eliminate those infinities ( call me quantum mechanics).
In thermodynamics the black body radiation leads to the ultraviolet catastrophe, and quantum mechanics saves the day.
In classical electromagnetism, a point like electron would tend to an infinite potential at (0,0,0) as it goes with 1/r. Quantum electrodynamics saves the day.
That is because quantum mechanics has inherent probabilistic indeterminacies when sizes  become of order of h(the planck constant). 
Even though elementary particles are postulated as point particles, they are not classical particles, the wave/particle duality saves the day, so the minimum volume would be of dimensions compatible with h in the variables examined and the measurement methods used.
Once gravity is quantized, the set will be complete, taking care of minimum black hole volumes too, in a similar way.
A: Yes it can. An object can become infinitely small if it is compressed below its Schwarzschild radius. Gravitational singularities, which make a black hole, are objects with infinite density and 0 size.
