If we assume general relativity is correct, then singularities are an unavoidable result of black holes. The math simply does not allow for such a large density

If we assume general relativity is correct, then singularities are an unavoidable result of black holes. The math simply does not allow for such a large density to not form a singularity. Having said that, we know GR is not correct, as it fails to account for any quantum phenomena; and we have thus far been unable to combine GR and QFT into a unified theory. Attempts to combine the theories fall under the title of quantum gravity. Several proposed theories of quantum gravity predict that black hole singularities do not exist.