There are increasing attempts in the AI research community to address the "intersectionality" of racial, gender, and other biases in AI, instead of tackling each bias separately. What does it mean, then, to make AI intersectionally fair? This paper analyzes the dominant interpretation of "intersectional fairness" in recent studies and examines three fundamental problems with it. The paper goes on to distinguish a strong sense of AI fairness from a weak sense that is prevalent in the literature, and concludes by envisioning paths towards strong intersectional fairness in AI.