You are not wrong. So I am not an expert in this area. My understanding is that we get different answers depending on what branch of math/physics you're using. Relativity theory says it is infinite, quantum theory says it isn't. The consensus of most experts in this field is that the center of a black hole isn't infinite, but that's based more on intuition than actual evidence. So the real answer is we don't know, but the educated guess is that it is not infinite.