Mathematically, the classical theory of general relativity predicts that when 'R' goes to zero, the curvature of space increases to infinity. This defines the singularity as it exists inside black holes and at the instant of the Big Bang. This is a mathematically well-defined condition within the general relativistic theory of gravity, and even Einstein did not dispute its mathematical reality .The issue is whether nature and the physical world can ever create such a condition. We do not know the answer to this question because we are not in any position to examine first hand what goes on inside black holes or at the Big Bang! Having said this, there is a caveat.
Physicists firmly believe that gravity will eventually be unified with the other three fundamental forces known to us. These forces are described by what are called quantum field theories and constitute the 'Standard Model'. Unifying gravity with these forces means that we have to find a description for gravity which is also quantum mechanical. The implication of such a quantum theory for gravity is that space-time, which is mathematically the same as the gravitational field, will also become a quantum mechanical object. This means that statements about what space-time is doing at a specific instant at zero separation, will be replaced by a quantum mechanical version of Heisenberg's Uncertainty Principle. The net effect is that all singularities will get smeared out over a small but finite volume of space and time defined by the so-called Planck units of length...10^-33 centimeters... and time...10^-43 seconds.
In quantum gravity, singularities are replaced in the physical world by very dense concentrations of mass-energy with densities of 10^94 grams/cc. This is a long way from the infinite-density state predicted by classical, mathematical singularities.