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Badiou's got some mathematical training; reading back and forth between the relevant sections of Goldblatt's Topoi and Badiou's account of Ω-sets in Logics of Worlds, for example, you can see that the one tracks the other closely. It's not just blind quotation, followed by hand-wavy inference-drawing either: you could actually learn about Ω-sets from Badiou's presentation of them alone, and not be too horribly surprised or confused when you came to read the technical presentation in Goldblatt (this was, in fact, the order in which I did it).

On the axiom of choice and "infinite liberty":

The AoC says that given a set {A, B, C...} none of whose members are the empty set, there exists a set {x ε A, y ε B, z ε C...} which takes one element from each of the first set's members. The point here is that the AoC "freely" chooses an element from each set rather than (for example) identifying a "least" element and choosing that: even when there's no rule that can tell you which element should be chosen, the AoC says that a set exists representing some choice.

The AoC only has any work to do in situations where no rule can be found (for example, no-one knows of a rule that will well-order the reals, but the AoC entails that a real can be chosen, then another from the remaining reals, then another etc. - so "axiomatically" a well-ordering of the reals exists, provided one accepts AoC) - hence it represents, in this sense, the possibility of a predicatively undetermined choice. That's the "infinite liberty" he's on about. It is nowhere asserted that AoC "proves" that such a liberty exists, but rather that introducing AoC into ZF makes such a liberty thinkable within the confines of its axiomatic system (this is in line with Badiou's general program of treating mathematics as "ontology", as a means for systematically demarcating what is thinkable of "being as such").

In terms of "interest to mathematicians": Badiou's early text The Concept of Model is a good philosophical introduction to model theory, and his Number and Numbers is an interesting and accessible guide to the philosophy of number, covering Frege, Peano, Cantor, Dedekind and Conway (surreal numbers).

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Badiou's got some mathematical training; reading back and forth between the relevant sections of Goldblatt's Topoi and Badiou's account of Ω-sets in Logics of Worlds, for example, you can see that the one tracks the other closely. It's not just blind quotation, followed by hand-wavy inference-drawing either: you could actually learn about Ω-sets from Badiou's presentation of them alone, and not be too horribly surprised or confused when you came to read the technical presentation in Goldblatt (this was, in fact, the order in which I did it).

On the axiom of choice and "infinite liberty":

The AoC says that given a set {A, B, C...} none of whose members are the empty set, there exists a set {x ε A, y ε B, z ε C...} which takes one element from each of the first set's members. The point here is that the AoC "freely" chooses an element from each set rather than (for example) identifying a "least" element and choosing that: even when there's no rule that can tell you which element should be chosen, the AoC says that a set exists representing some choice.

The AoC only has any work to do in situations where no rule can be found (for example, no-one knows of a rule that will well-order the reals, but the AoC entails that a real can be chosen, then another from the remaining reals, then another etc. - so "axiomatically" a well-ordering of the reals exists, provided one accepts AoC) - hence it represents, in this sense, the possibility of a predicatively undetermined choice. That's the "infinite liberty" he's on about.