I would provide a mild corrective to that piece: something exists. For the time being, the evidence - mathematical, physical, and empirical - seems to suggest that anomalous bodies which we call "black holes" are responsible for the gravitational structure, shape, and motion of the universe.
However, the very mathematical, physical, and observational properties of these objects do lead us into paradox, which is why there has been so much effort lately to "solve" the black hole problem.
First, I believe that Mersini-Houghton's math is correct: the very conditions necessary to bring about a black hole also seem to preclude the possibility of a black hole.
Second, the quantum mechanic runs into the dilemma that black holes appear to destroy (perhaps "trap" is a better word to choose) information, which violates a basic law of quantum mechanics: the persistence of complete information regarding a physical system.
Third, black holes are literal instantiations of the observer's paradox (this is the empiricist's dilemma). We cannot observe black holes directly because the medium by which our vision works - light - cannot escape a black hole. Actual observation of a black hole would necessitate the death of the observer and would prevent him from sending any information back out of the black hole. Our only way to "observe" them is in the second degree; via satellite images of gravitational anomalies that can only be explained by massively dense objects that, for some reason, elude our vision.