That the transcendental of any given world is itself a set of degrees, ergo a real set of multiples, means that the logic of appearance is itself an ontological postulate. That is, the logics of worlds derive from the science of being-qua-being, i.e. ontology. Mathematics is the ontological science because its technical statements are subtracted from any particular presentation. It would seem that phenomenology (made into a rigorous science of calculation by way of category theory’s relational arrows) depends on a certain intra-presentational relation (set theory being the presentation of presentation, i.e. the ontological discourse that implicitly axiomatizes the real of being-in-itself). So we should not presume to ascribe any content to the Greater Logic of being-there, and see it rather as a further elaboration on the presentation of presentation that maintains the necessary absence of any particular sense or content. Yet the irreducibility to ontology of the order-structure of appearing, which is to say the intelligibility innate to sensibility, entails some minimal existential content. Consequently, appearance’s non-ontological structure (i.e. some world’s transcendental) paradoxically is at once a set of multiples treatable in ontological situations and also a many-relational “machine” that orders some region of existence according to an intelligibility which makes sense only beyond the ontological strictures of set theory. What are we to make of this simultaneous fissure and connection, which Badiou himself tacitly adopts when assigning set theory to onto-logy and category theory to onto-logy? How do we contend with the latter’s irreducibility to the former while also maintaining that the latter’s essential operator, the category-theoretical transcendental, is nothing but an ensemble constructible from the former’s axiomatized relations, the one and only set-theoretical relation of belonging?
An adequate response ultimately demands an accounting of mathematical science that provides the resources for a synthesis of mathematical structures. While we will not endeavor to unpack the French rationalist tradition of mathematical philosophy to which Badiou owes his own original take on mathematics and its role as a condition for philosophy, we can provide a brief and schematic rejoinder to this conundrum. There must be a way to approach the relationship shared by ontology (set theory) and phenomenology (category theory) that does not sacrifice the rigorous simplicity of mathematics, namely the mathematical absence of content beyond its own internal rules and exigencies for creative development. Although Badiou may not make it overtly clear in Logics of Worlds, a quick reference to Being and Event assures us of the correctness of objective phenomenology’s innate emptiness and hence necessary recourse to actual experience. Both texts thematize appearing to a different extent and to different effect: the “count” in Being and Event, also called the effect of structure, becomes the “transcendental functor” in Logics of Worlds. The transcendental functor is an “operator of regional ontological consistency,” a global relation “between the structure of the transcendental and the structure that is retroactively assignable to the multiple insofar as the latter appears in such and such a world,” just as the count-as-one has always already structured the inconsistent multiplicity of pure multiple being into a consistent situation or classical world.  Thus the hyphen which separates two alternative emphases of onto-logy can be read as an inexorable connection of the global to the local, i.e. of the being-qua-being of some multiple to the regions of existence wherein that multiple contingently appears. Set theory’s grasp on being-qua-being is, as Badiou tells us, at best implied in its axiomatization of inconsistent multiplicity. Category theory’s multiplication of relations beyond that of belonging takes as many steps away from such an axiomatic implication of the real. It introduces into the neutrality of the pure multiple, without effacing it, a further ordering, indeed an actual ordering, since set theory’s formalization is subtracted from all but the barest requirements for consistency. The categorial’s rational connection to the ontological derives from a continuity of a real multiple’s set-theoretically deployed relation of belonging that extends from onto-logy into the onto-logy of an apparent multiple.
Now, consistency does not depend on which particular multiples are presented.  According to the Axiom of Separation, existence provides the multiples which language (here, mathematical formalisms) will separate out as consistent.  Prior to this axiom’s introduction, set theorists had to contend with the implication that to say a set or to parse a multiple implied its existence, which echoes the uncritical realism of certain network scientists whose simulations are framed as the science of the real world. Thus what seems to be a thoroughly networked orientation to the exposition of being and of appearance includes an organic centering via the axiomatic prohibition of presuming the objective reality of discursive formalization. This centering, designated in Being and Event by both the Axiom of Separation and the count and in Logics of Worlds by the too easily forgettable phrase “the one who measures,” pertains to the “subtractive” aspect of Badiou’s ontological treatise. Pure multiple being, whether of being as such or of a world’s transcendental operator, is that from which existence is subtracted. Mathematics, as the science of being, therefore proceeds infinitely from the inexhaustible, implicitly thought real and ever on the basis of what is already subtracted from the real of being. Even mathematical ontology’s minimal axiomatic of the generic “there is” presupposes an existent multiple, an extant effect of the count’s structure. If philosophy is to take “care of truths” in accordance with its conditions, among which is the mathematico-network condition that presents being as such and the logic of its appearance, philosophy’s “speculative metaontology” must introduce or fall back on at every turn the concept of organism.