Just re-read the sections "An introduction to Alain Badiou's philosophy", "Philosophy of Truth", "Philosophy of art", and "The definition of philosophy" as a converse to The Life of the Mind -- not that Arendt mentioned Badiou, because she didn't.
Upon second reading, I noticed some inherent fallacies Badiou makes that careens his arguments off the mark. As an example, when expressing the difference between Badiou's philosophy as opposed to poststructualists, there is an assumption that an event happens independent. Although Badiou appears to explain this away as part and parcel to being (that is, subjects in response not just subjects in existence) as a multiplicity rather than as a unity (the favoured response of Aristotle), he still does not actually explain when an event becomes out of the ether or what generated it. Even if he did, he doesn't explain how the being -- even in its multiplicities -- chooses. There's a tension in that Badiou doesn't actually really respond to the idea of choice as part of a component to willing.
And... I'll be honest, I don't particularly find mathematics as a answer to philosophical endeavours as convincing in of itself. Explaining away how and why man behaves through mathematics does nothing to explain the why at all, it just shows how it is formulated (well or not) since a formula states what is not why it is.
The other issue -- and this is more me than Badiou, is after reading Arendt, Badiou's structured assertions followed without any explanation or rooted in anything other than an argument (seemingly for arguments' sake), seems wanting.
However, there are still moments in which Badiou seems to touch on something quite extraordinary, especially in relation to thought and how emotions can provoke an event and thus, the event does, in this way, express itself as a choice. The willingness to do or not, instigated by a feeling, rendered through thought then expressed in judgment is exceptional work. If only he followed his ideas in this way more often rather than proving his worth by arguing to win rather than to discuss...