The Truth of Badiou

From Alain Badiou's essay, "Philosophy and Truth":

"A truth is, first of all, something new. What transmits, what repeats, we shall call knowledge. Distinguishing truth from knowledge is essential. It is a distinction that is already made in the work of Kant: the distinction between reason and understanding. It is a capital distinction for Heidegger: the distinction between truth - aletheia - and cognition or science - techne.
If a truth is something new, what is the essential philosophical problem concerning truth? It is the problem of its appearance and its 'becoming'. A truth must be submitted to thought, not as a judgment, but as a process in the real."

If truth is outside of established knowledge, then it is undecidable in terms of that knowledge. Hence, it requires an unfounded decision to follow that truth (from the perspective of knowledge):

"For a truth to affirm its newness, there must be a supplement. This supplement is committed to chance. It is unpredictable, incalculable. It is beyond what is. I call it an event. A truth thus appears, in its newness, because an evental supplement interrupts repetition."

"An event is linked to the notion of the undecidable. Take this statement: 'The event belongs to the situation.' If it is possible to decide, using the rules of established knowledge, whether this statement is true or false, then the so-called event is not an event. Its occurence would be calculable within the situation. Nothing would permit us to say: here begins a truth. On the basis of the undecidability of an event's belonging to a situation a wager has to be made. This is why a truth begins with an axiom of truth. It begins with a groundless decision - the decision to say that the event has taken place."

The ensuing work of bringing about a truth is what constitutes a subject in Badiou's terms. It requires fidelity to the event. But whereas a subject is always finite, the effects of a truth are always infinite, always outstripping the limits of a single subject or even a single era (limited only by the point of the Real, the 'unnameable'). However, this raises a question that has been mentioned in other places - how can one make any judgment concerning the truth? If a truth is necessarily situated outside of knowledge, then how can one use any established categories to judge the worthiness of an unfounded axiom? In particular, with regards to Badiou's project, how can we judge his seemingly unfounded axiom that 'mathematics is ontology'?

Perhaps a clue is offered later in the same essay:

"This decision [to mark an event] opens up the infinite procedure of verification of the true. This procedure is the examination, within the situation, of the consequences of the axiom that decided upon the event." (emphasis added)

If fidelity to the event entails examining the consequences of an axiom within the situation, then perhaps knowledge (changed by the introduction of a truth) can be used judge its value. The worthiness of Badiou's axiom is to be judged by reference to its effects on knowledge. Does this reduce a truth to knowledge, however?