By Gabriel Riera
An creation to Badiou's philosophical inspiration and its implications for different humanistic disciplines and the social sciences.
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Additional info for Alain Badiou: Philosophy and Its Conditions
Admittedly, Badiou's position has not been constant, and its current version is tentative at best. Within the scope of mathematics itself there is awareness, indeed even accusations, that set theory has based itself on the innovations introduced by category theory. For the sake of homology with mathematics, it is important to notice how Badiou per forms a double move, both of which favor set theory. Regarding the identity the sis, there is a set-theoretic co-opting of category theory, while regarding the argument leading to the notion of " transitory ontology," there has been a legiti.
What is important here is that Peano set out to show how prior to these axioms nothing is demonstrable. This axiomatic commitment is, ofcourse, a key no tion for set theory, one that has significantly fulfilled its program of providing math· ematics with a foundational theory. So as we count, we recursively schematize both the repetitive drive of the number chain as well as the reformulation of its predeces· sor an incremental step ahead. These propositions cannot themselves be demon· strated as logical theorems, meaning they stand as axioms.
The transformation of such entities along a sequential chain produced definitions that relied on an idea best exemplified by Giuseppe Peano in his postulates or principles of mathematical induction. For the set of all positive integers, these postulates state that: 1 . 0 is a natural number, 2. every natural number has one successor, which is also a natural number, 3. no natural number has 0 as its successor, 4. different natural numbers have different successors, 5. if 0 has some property, and if when a number has that property then its successor also does, then all natural numbers have that property.
Alain Badiou: Philosophy and Its Conditions by Gabriel Riera