Well, the event horizon is the boundary where our conception and model of physics breaks down into uselessness. So in a way you can say black holes cannot exist within our model of physics (which is generally what "the math all just becomes infinities" actually means), and be kind of correct. But not really. That's a philosophical question though, not a scientific one.
I've read enough offhand laymans physics pop-culture shit to know that the one and ONLY thing any cosmologist can say conclusively about black holes is that the geodesic becomes so warped within the swartzschild radius that what reality exists within that radius is, literally, incomprehensible to us for now. Further, approaching that limit on the near side quickly devolves into nonsense. Hence the entire reason (beyond being crippled) that Stephen Hawking is a physics superstar. He unraveled one small knot.